Logic and First Principles, 12: The crooked yardstick vs plumb-line self-evident truths

Yup kairosfocus

KF, Ah, I think I know what you are talking about now. daveS

DS, the exchanges were real enough, tracing to certain views on Popper. I hope that is not a trigger. KF kairosfocus

KF, Perhaps I wasn't present during those exchanges. In any case, I don't doubt the importance of inductive reasoning. daveS

DS, I am recalling all too well long exchanges on the subject of the credibility of inductive reasoning and its resulting weak sense knowledge, which dominates the real world. In seeing the exchange that developed above, it seems to me important to see that inductive reasoning will also naturally arise and lead to a next stage that perhaps was unanticipated. That seems to be significant. Of course, the overall unreality and disconnect from self-sacrificial behaviour in honor/shame cultures make ET's point far more practically relevant. One take-away for me is that induction is something that needs a bit more. Just what, I am not yet sure, but I think it is connected to double-standards of warrant linked to induction. For one, consider the alphanumeric code observed in cell based life, a language phenomenon. KF kairosfocus

KF,

The silver lining, I guess, is that it seems to reveal that there is a problem with how we respond to induction and to weak sense knowledge

I don't think anything of the sort has been revealed, has it? We all use inductive reasoning all the time. We are forced to. We can barely survive without it. We can't do empirical science without it. daveS

H, DS posed a logic puzzle which is hardly RW, which I mistakenly thought could be briefly answered, allowing the thread to go back to focus. The silver lining, I guess, is that it seems to reveal that there is a problem with how we respond to induction and to weak sense knowledge . . . which is what science gives precisely because of its inductive character. Which, is where this series began. The significance of plumb-line, self-evident truths remains. KF kairosfocus

daves:

It’s a logic puzzle.

And as I said- the first argument would stand. It would be highly illogical to try to determine the color of your eyes given that such knowledge leads to your committing suicide. ET

I haven’t paid any attention to this puzzle: I dropped out when it seemed kf didn’t want to talk about applying logic to the real world, but that is exactly where the discussion has gone anyway; At 145 Dave wrote,

There was some back and forth above about using logic to reason about abstract things vs. using it to reason about things in the “real world”. I think this puzzle shows that we really only reason about abstractions, ultimately. That is, we have to form an abstract model of the situation on the island. Hopefully a sufficiently accurate model; that can be difficult to do because the “real world” is not a blocks world whose state can be described with mathematical precision.

I absolutely agree with this. Once we make a model and use it to deduce conclusions, we then re-examine the world to see if those conclusions are borne out. If they are not, we revise our model. As as been noted, our model will include premises and other facts that we think are true, via induction, as well as definitions we think are sufficiently clear. But our model is always “fuzzy” in some ways, so our conclusions about the real world are always provisional, with some degree of uncertainty. Kf writes,

PS: deductive reasoning in valid ways can also arrive at false conclusions. The issue is the truth of relevant premises, and that is normally established in ways that are shot through with inductive experience.

Yes, as I said above. hazel

Anyway, FTR, I'm not saying that inductive reasoning is of a lower class than deductive reasoning. END daveS

DS, in a bit of a hurry but this is pivoting on what is knowledge. Knowledge, first is not equal to certainty or theoretical scientific knowledge and general factual knowledge on experience and report are not knowledge. In this context, deduction is normally not independent of induction and induction is intricately tied to deduction too. So, it is reasonable on this case to hold that the islanders empirically know a law that eyes they are aware of on island are blue or brown. On the givens we know that such an inductive conclusion is not just cogent but accurate. It is on this if not A then B pattern that the islanders will operate. I repeat, inductive knowledge is not a second class, second rate form of knowledge. KF PS: deductive reasoning in valid ways can also arrive at false conclusions. The issue is the truth of relevant premises, and that is normally established in ways that are shot through with inductive experience. Indeed, the concept of a background of experience and understanding is going to imply just such. PPS: Further eye colours would not affect the fate of the blues, but they would make a huge difference for the others. For they would then have uncertainty on their own eye colour. kairosfocus

PS to #146:

Again, we know there are only two eye colors present. The islanders don’t ever know that in this particular instance of 100 blue-eyed and 900 brown-eyed islanders.

We could have been given various instances where there are 1, 2, 3, 4, and so on up to 1000 different eye colors, and it would not change the outcome---the blue-eyed people would all commit suicide at some point, and the remainder would never know their own eye colors. daveS

KF,

DS, this is getting a bit tangential, but the givens as you link do state two colours are present on the island.

Again, we know there are only two eye colors present. The islanders don't ever know that in this particular instance of 100 blue-eyed and 900 brown-eyed islanders. We could be given the names, addresses, photographs, and eye colors of every islander, but it wouldn't change the outcome of the puzzle.

It seems there is an objection to inductive logic as though it were a second class level of suspect reasoning instead of a major means to knowledge.

I don't regard it as a second-class level of reasoning, just different. Such reasoning has a pretty important weakness: even correct inductive reasoning can lead to false conclusions. But sometimes it's all you have to work with. You can't always use deductive reasoning (which is one of its weaknesses). daveS

KF,

The relevance is, that this puzzle shows how important it is to get start-points and steps of reasoning right, and it shows — by the online disagreements and puzzlement — just how hard it is to get our thinking straight.

There was some back and forth above about using logic to reason about abstract things vs. using it to reason about things in the "real world". I think this puzzle shows that we really only reason about abstractions, ultimately. That is, we have to form an abstract model of the situation on the island. Hopefully a sufficiently accurate model; that can be difficult to do because the "real world" is not a blocks world whose state can be described with mathematical precision. In this puzzle, some of the crucial information can be given only implicitly (or else one gives away the solution), which adds to the difficulty. daveS

DS, this is getting a bit tangential, but the givens as you link do state two colours are present on the island. Next, we must credit basic sense of order, so a pattern observed without exception will be taken as established fact. It seems there is an objection to inductive logic as though it were a second class level of suspect reasoning instead of a major means to knowledge. That is itself an important issue if it is on the table, one relevant to ID and to Science in general. KF kairosfocus

ET, It's a logic puzzle. We're supposed to deduce what we can from the given premises, even if they aren't consistent with typical human nature. No one believes that all Cretans are liars, yet it's productive to examine what follows from that statement (a famous paradox, in this case). daveS

Look, guys, no one is going to work that hard just so they can commit suicide. "Oh no, if I know my eye color I must commit suicide. Let me follow the clues so that I can determine my eye color." Really? The first argument would stand or they would kill the traveler- argument #3. ET

ET, To echo KF's post, imagine there are 10 total islanders, only one with blue eyes, and that they are standing in a circle where they can all see each others' eyes. The single blue-eyed islander can see everyone else's eyes, none of which are blue, so he can tell his own eyes are blue after the foreigner's statement. None of the other 9 islanders can deduce his or her own eye color with the available information. daveS

KF,

There is a variant with a green eyed griot, which could drastically shift the dynamics — each person now being aware of a third option so the brown eyed cannot determine that by elimination if not blue then brown.

I haven't seen any versions of this puzzle where the islanders actually know there are only two eye colors. Certainly on this island, where there are in fact only two, each inhabitant knows there are at most three eye colors present.

it troubles me that it makes people repeatedly reflect on religious adherence in a very negative way, as though such were typically irrational, extremist and dangerous; fostering an invidious psychological association.

I agree. I also find the tired trope of the blue-eyed foreigner visiting a "tribe" of mostly brown-eyed islanders with weird religious practices somewhat cringeworthy. There is no need to include suicide in what is supposed to be an amusing puzzle. An update is in order. daveS

ET, I add, in the n = 1 case, each islander is aware of the stranger's eye-colour, blue [similar to sky and sea]. All but one will be directly, observationally aware of one islander with similar eyes. That one would see no other islander with such eyes and on the presumption that the visitor spoke truthfully, would infer that s/he is the one with blue eyes. On this recognition being publicly shows (the story is loaded), every other islander would realise that his or her eyes are the other evidently available colour, brown. (There is a variant with a green eyed griot, which could drastically shift the dynamics -- each person now being aware of a third option so the brown eyed cannot determine that by elimination if not blue then brown.) The relevance is, that this puzzle shows how important it is to get start-points and steps of reasoning right, and it shows -- by the online disagreements and puzzlement -- just how hard it is to get our thinking straight. Hence, the value of plumb-line, self-evident truths that are naturally straight and so correct our crooked yardsticks. We are finite, fallible, morally struggling, too often ill-willed and polarised. It is no surprise that we speak of an is-ought GAP that needs to be bridged. KF PS: The puzzle brings out the challenges of logic in an empirical world, and it troubles me that it makes people repeatedly reflect on religious adherence in a very negative way, as though such were typically irrational, extremist and dangerous; fostering an invidious psychological association. I therefore note that in some honor/shame culture societies, suicidal defence of the community or of one's honour is an act of defending or redeeming honour. Some elements of this remain in our civilisation, it is no accident that the highest military medals are often awarded posthumously. kairosfocus

F/N: Do I need to note that deductive chains often start from empirically, inductively inferred facts? KF kairosfocus

ET, now you see part of why my argument pivots on each blue seeing 99 others. KF kairosfocus

DS, While this is tangential to the main thread, it reveals on a side issue the patterns that often play out. KF kairosfocus

When n=1, the single blue-eyed person realizes that the traveler is referring to him or her, …

What? The single blue-eyed person doesn't know he/ she is blue-eyed so he/ she wouldn't know who the traveler is referring to. The brown-eyed people would but they cannot discuss it. And, if the single blue-eyed person commits suicide and the other 999 observe it, then they will know they have brown eyes and have to follow the one. ET

PS:

BTW, we are given as a fact that there are just the two colours, so the inductive inference is correct on our “god’s eye view.”

Yes, but it is not given that the islanders know there are just two colors. Neal in the stackexchange thread seems to assume that is the case. The brown-eyed people can never know for sure they have brown eyes, whereas the blue-eyed people can. daveS

DS, I added somebody's comment at Math Stack exchange. Yes, I see the Mathematical induction case. I also see something much more direct, each person sees 999 others. Of these, if a person is blue-eyed, s/he will see 99 others with blue eyes. Thus, on the inference day 99 is critical and if that day passes without event it means there is another invisible blue. For each blue that can only be oneself (the only invisible eyes) so on day 99 the blues all know they are blues. Then, on seeing the 100 blues remove themselves, the browns can each see 899 other browns and no blues. Each brown now knows s/he is a brown, on the premise that those are for all they know, the two possible states. The removal rule then applies to the browns. The problem has a gap in the formulation as given. Notice, it pivots on world models. This is also an illustration of the logic of signalling behaviour that carries implicit information. KF PS: This side exchange is illustrative of how ever so many exchanges get locked up. kairosfocus

KF, No, there is no inductive reasoning involved in this puzzle whatsoever. The solution is supported by a deductive proof showing that if the tribe had n blue-eyed people for some positive integer n, then n days after the traveller’s address, all n blue-eyed people commit suicide. We are given that there were 100 blue-eyed people and 900 brown-eyed people, hence we deduce that 100 days after the address, all 100 blue-eyed people commit suicide. daveS

DS, it involves both induction and deduction. Already, on the expected day, an observation is made (as each blue sees 99 blues) and a conclusion is inferred (that each other blue sees 99 also), leading to the conclusion that there is for each blue one more blue invisible to him or her self, thus that final blue is oneself. Also, agreement or disagreement/majority opinion is not a sufficient criterion of warrant. BTW, we are given as a fact that there are just the two colours, so the inductive inference is correct on our "god's eye view." What happens is the framing focuses on the blues, but the browns can see and infer just as well. KF PS: At Math Stack exchange there is a comment: "The day after the blue-eyed people commit suicide, everyone else gathers in the square and commits suicide, leaving the stunned foreigner alone on the island. – Neal Nov 15 '12 at 23:06" https://math.stackexchange.com/questions/238288/is-there-no-solution-to-the-blue-eyed-islander-puzzle PPS: It seems there are multiple versions of the puzzle. kairosfocus

The evidence in hand also suggests that solutions to the puzzle should involve deduction rather than induction. This puzzle is all over the web. Can you find anyone who shares your position? daveS

DS, the evidence in hand is that eye colour comes in two varieties. Absent one, then the other. KF kairosfocus

KF, Sure, but I wouldn't commit suicide based on such an inductive inference. I think that's not what the creator(s) of the puzzle intend either. Suicide is indicated only if an islander comes to know his or her eye color through deduction. daveS

DS, induction, where it is stipulated that the islanders have blue and brown eyes. Each would see 999 cases that confirm a two-colour law, with 99 or 100 blues and the rest brown. It would be most unlikely for them to imagine a third colour, which would undermine the prior inference process also. KF kairosfocus

Argument 3- The islanders kill the foreigner :cool: ET

PS: On the other hand, would the brown-eyed people ever know that none of them has green eyes, for instance? I'm reconsidering the events of day n + 1. daveS

Yes, I agree with your conclusions, including day n + 1. daveS

DS, an odd puzzle. Of course each blue eyed person is aware there are n - 1 blue eyed persons by inspection. It seems to me that if by day n - 1 plus 1 the others have not done the self destructive ritual, there is at least one more and so they are aware on day n that they must be also in the set of the blue eyed as they are also aware of all the brown eyed. Day n + 1 will be a day of horror. The same day the rest will realise their eyes are the remaining colour, to all but certainty (as these are the only two colours they see), and likely the rest will also suicide the next day. KF PS: The subtext about religion also begs to be pointed out. kairosfocus

Regarding the application of logic to questions about the "real world", the blue-eyed islander puzzle might provide an interesting diversion. It probably wouldn't illuminate any of the issues being discussed here, but it is one of the best logic puzzles in circulation, IMHO. daveS

MG 116 & H 114: MG points to the essential point. On whether Math or Logic contain the other, I suggest, overlap. Similarly, study of thinking overlaps between logic and psychology. KF kairosfocus

BB, true enough for some aspects. As noted, more advanced technologies are Math-heavy or even Math-based. For example aeronautics and computing. KF kairosfocus

KF&Hazel@61&62, I don’t disagree that some concept of mathematics is needed for technology, but I disagree that it has to be formalized, or even understood. After all, we can throw and catch a baseball, which requires calculus and advance math to simulate (I am not even aware that we have done it effectively yet), but a moderately talented ten year old can master that skill. All without understanding advanced math. Brother Brian

kf, vivid asked me a question about logic in the real world at 99, and Stephen at 102 and 110. That’s how the discussion about logic in the real world got started. hazel

KF@113 What a delightful non-standard compliment in your opening! Thank you. I have a day job that prevents me from posting regularly. It was easier being a lurker, but more frustrating being unable to respond to various pitiful non-sequiturs posted by A-Mats. math guy

H@114 You have stated on numerous occasions that abstract entities are real, but only as they exist within minds. That is a classic example of nominalism. Like it or not, you have earned the label. Furthermore, you (correctly, IMHO) state that logic is a compartment of mathematics, albeit the latter has axioms, definitions, and lots of other stuff. My question "do fields exist?" has a great deal to do with "applying logic to propositions about the real world". Your evasion of the question speaks volumes. math guy

H, pardon but in discussing necessarily true claims and necessary beings, no one has been discussing contingent facts or statistical correlations vs cause-effect identification. And BTW, your arguments above still suggest nominalism or conceptualism, with influences from the kantian ugly gulch concept. KF kairosfocus

To Math Guy at 109. First, the whole nominalist thing was several threads ago. I mentioned seeing the word and some ideas about it one time, and then kf said I was “championing it”. I made it clear then that if it meant what he said it meant, then it didn’t apply to me. I’m not interested in being labeled with a particular philosophy. I think that issue is dead. Also, those same discussions were about the relationship between math and the physical world. At 66 kf said to you those were “old grounds”, and I replied at 70, “This thread is about the significance of the fact that “false propositions exist” can be shown to be logically true. As kf said at 66, all the stuff about how math can describe the real world is “old ground” from previous threads.” So I’m not interested in reviving that topic either. I want to stay focused on the fact that there is no significant consequences that follow from proving that “false propositions exist”, or even “error exists”, as everyone knows that. The subject of applying logic to propositions about the real world, as opposed to within the world of logic, has come up in the last few hours, and I’ve said a few things about that at 112 and 107. hazel

MG, it is ever so refreshing to see you pop up again. Your remark is interesting:

My (somewhat rhetorical) question “Do fields exist?” was directed at Hazel. Your previously stated experience with electromagnetism already implied an affirmative opinion. So let me address a more concise version to Hazel the nominalist. Hazel, do fields exist as part of the physical universe (as opposed to solely within minds)?

This brings to the fore the problem of nominalism or even conceptualism vs realism. But also, it brings to mind one of my bits of educational radicalism. I have never liked simply teaching people the hand rules for electromagnetic interactions: draw your right hand that cranks the generator to do the generator rule, and the left hand for the motor rule. I bring up the Lorentz force expression, with its vector-field equations, especially on the magnetic side, whereby F = q * (v x B), B magnetic induction (taking into account that materials can align with and amplify an external magnetic field) and v the vector velocity of a charge. So, now we need to explore the effect of a rotating B-field, such as happens with electrical machines. I get an old fashioned cathode ray oscilloscope, the kind that deflects an electron beam electrostatically (and BTW, applies the classic in vacuo version of ballistics, substituting an E field for a gravity field to get parabolic arc deflection thus a linear deflection on screen). An expendable one as this is potentially messing with magnetic shielding, but it is important enough to be worth it. I get a bar magnet. Set the scope to XY mode. (And yes, physical graph plotting.) Put the beam to the point where the fluorescent dot is on the origin. Now bring up the bar magnet, sideways (so the B-field is across the beam), and lo, it deflects the beam sideways. Now, rotate the bar magnet by hand. The beam is pulled along and also rotates. Suddenly, squirrel cage induction motors are "easy" to understand, as are old fashioned TV's and Computer monitors: they use magnetic deflection (which would push around the arc of a circle BTW). Also, we can understand particle accelerators, bubble chambers and cloud chambers etc. E and B fields are very real, as are gravity fields. Indeed, we live in their presence as our weight testifies, as compass needles testify and as lightning strokes testify -- spark gap discharges on the giant scale. All of which deeply embed structure and quantity in an actual physical world. And of course, I again call attention to the mobius strip challenge. In your experience, how do nominalists respond to such a case? KF kairosfocus

to Stephen at 110: I'll try to be clearer. In the world of logic, to say “X is a P and x is not a P” is a logical contradiction, and is thus a false statement. This is a logical conclusion. If, in the real world, Johnny says “That is a cow and that is not a cow”, then Johnny is wrong; his statement is necessarily false because the law of non-contradiction applies to that situation in the real world. The difference is that within the world of logic, all valid statements about the elements of that world are either logically true or logically false. Nothing but logic is needed to determine their truth value. However, as my example about smoking causing cancer was meant to show, most statements in the real world can not be resolved with pure logic. The example about Johnny and the cow is wrong because of pure logical reasoning, but the truth value of the proposition about smoking can not be resolved by pure logic. Logic will undoubtedly be used in assessing the evidence as to whether smoking causes cancer, but pure logic by itself can’t resolve the issue. That’s the difference, I think. hazel

H, Let me follow up. First, the triple first principles of reason (which were cited as an example in the OP also) are inescapable just to communicate. That is of course a case of self-evidence. It is not the same as to assert or imply that therefore there are no other cases, it being taken for granted by people of reasonable experience and understanding that coherent communication is possible and actual. However, it is indeed a fact of life -- as I also pointed out -- that ever so many are taught to doubt and dismiss such. In enough cases we have seen over the years they have appealed to quantum concepts that this is a UD weak argument corrective. One that turns on how the Physicists rely on distinct symbols etc just to do their analysis. Next, you will recall, as I have again pointed out at 98, that in the OP, my illustrative case using Royce's example of agreed truth, is:

For simple example the Josiah Royce proposition, E = error exists, is undeniably true. To see that, try to deny it, ~E. That in effect claims it is error to propose E. So E must be true.

Notice, the simple context of inspection with understanding of what denial of a proposition entails. Where, too, we must recall that truth is not merely a logic variable value, it is a connexion between logical ponderings and the real world: an assertion Q is true if it accurately describes an actual state of affairs, such as where I am, it is raining as I type, and a slight breeze is blowing. As Aristotle put it, truth says of what is that it is; and of what is not that it is not. Being true is not merely a matter of conformity with or derivation from a set of axioms presumed as core truth but of accurate description of actual states of affairs of reality. The kantian ugly gulch fails. Going further, logic applies to being and to distinct identity. As I have repeatedly pointed out, if a candidate being B is such that its core characteristics x and y are in mutual contradiction, it is impossible of being, e.g. a square circle. (I recall, that long ago now, you did not like the categorisation of modes of being/non-being I presented: contingent beings that would exist in at least one possible world but not in another one (due to presence or absence of enabling causes), necessary beings (as part of the framework for any world), impossible entities (such as square circles).) In short, a purely logical criterion can and does specify that certain proposed entities B are impossible of being, because core characteristics x and y cannot both be true in any possible world. This reflects the point that truthfulness of a proposition is connected to accurate reference to reality. So strong is this, that the mere possibility of a world can be used to identify that a suggested B could not exist once x and y are in mutual contradiction. For the square circle, squarishness and circularity. In this context, I took time to draw out that for a distinct possible world W, there must be some characteristic A that marks it as separate from close neighbours W' and W", leading to partition of characteristics, W = {A|~A}, thus the observation on inspection that this exhibits unity, duality, nullity, opening up embedded structure and quantity through the von Neumann construction etc. Notice, this shows that such core mathematical entities are necessary beings, structurally embedded in any possible world. That is not at all the same as to imply that they are self evident. This is because an elaborate reasoning process rather than a trivial or simple explanation is required to draw such out, succession to get N, additive inverse to get Z, noting size + direction to get abstract vectors, ratios to get Q, infinite convergent sums of rationals to get reals, R, abstract rotations to get C, etc. At this stage, I infer that part of your objections turn on accepting necessity of being but disputing the difference between a simple explanation and a deduction from in effect axiomatic first principles. Strictly, once the first principles of reason are seen to be inescapably true and self evident, that suffices to establish my point that once a SET is, truth is. That is, the set that collects truths is not empty. Thus, similarly, we have warranted and even certainly warranted truth, thus both knowledge and knowledge to incorrigible certainty. That would already be enough to establish the bankruptcy of hyperskepticism, subjectivism and relativism, whose adherents are legion. So, too, would the mere necessity that error exists and that false assertions exist. However, it is apparent that you seem to tend to dichotomise logical truth (presumably per deductions on axiomatic systems) and truth as accurate reference to reality. The two overlap, for sound axiomatic systems and for cases of necessary entities discovered to be such through exploring abstract logic worlds or abstract possible worlds. That for example is so for the core mathematical entities I identified. (This embedding, in turn is part of my answer to the Wigner challenge.) However, there is a sufficient distinction that I need to stress it: ontological truth turns on accurate description of reality, not mere consequence on some set of axioms accepted as part of a logic game. Nor does ontological truth pivot on whether or not one believes, accepts or understands. Or, can warrant or recognise. That is the sense on which I have repeatedly pointed to the act of constructing ordinary and mobius strip paper loops and doing snip around the loop exercises, for a Mobius strip, there is a dramatic difference between cutting around in the middle and 1/3 way across. Here, one sees embedded structure and quantity manifested regardless of what one thinks, believes, imagines or understands. I would take it that for instance, a concrete demonstration like this is a case of empirical self-evidence. Once one has the experience, one cannot deny the reality, certainty or the truthfulness of accurate descriptions. On pain of absurdity. So, yes, we here see self-evidence that goes beyond inescapably true first principles of right reason. On other cases, let us again take up, error exists -- E for short. Notice, it is now part of the background of common good sense and experience, that distinct identity, ability to think and communicate truth, etc obtain. In that context, we recognise that an error is a missing the intended mark of truth (which yes points to implicitly known and accepted duty to truth). Here, we consider an empirically well known truth that in fact is pivotal to why we argue, debate, discuss: error exists. However, a subtlety lurks, it is a necessarily true claim. Why? As, on inspection, it asserts a proposition, E. Such comes with its coupled antithesis ~E, and ~E has a meaning: it is an error to claim error exists. Oops, self-falsifying. That is, E cannot be effectively denied, it is undeniably true. It is necessarily true in the full ontological sense that E holds truth in any possible world and so too in any actual world. (Given multiverse possibilities, we cannot assume this is the only actualised world.) So, we have a necessary truth which would be enough to again overturn hyperskepticism, subjectivism and relativism. But that is not enough, there are bigger fish to fry out there. Such as? Well, E is not only necessarily true as following on axioms in a logic game, it is not just ontologically necessary, it is self-evident. That is, for one able to understand, it is seen as true and necessarily so. Moreover, on the attempted denial, its necessity is patent on effectively immediate absurdity. Here, by undeniability, as the denial directly implies the truth of E. Going further, consider F, the proposition that false propositions exist. Again, that is empirically so, if I were to NOW say it is raining here, that would be false. The rain has stopped for the moment. It is also necessarily so as the entangled antithesis ~F implies that it would be false to assert F, thus directly implying F. Undeniable and self evident again. Where of course one could point out that within the axiom system, it denies LNC and so would be dismissed. But we are not just looking at in-world logic games but reference to reality. I have already underscored that we have in mind the sorts of hyperskeptical denials and dismissals that we have seen over the years, which are also manifest across the wider real world. I trust this is enough to bring out the concerns and adequately warrant the claims. KF kairosfocus

Hazel

Someone can’t say “that is a cow” and “that isn’t a cow”. We can be certain that statement is false.

Do you mean that it would be a false statement about the real world and not just a false statement "within the logical system." If so, why did you use the qualifying phrase "within the logical system?" StephenB

KF@66 My (somewhat rhetorical) question "Do fields exist?" was directed at Hazel. Your previously stated experience with electromagnetism already implied an affirmative opinion. So let me address a more concise version to Hazel the nominalist. Hazel, do fields exist as part of the physical universe (as opposed to solely within minds)? math guy

Hazel “Does that seem like a reasonable statement to you?” Yes Vivid vividbleau

re 105 to Vivid. Yes, if one says "smoking causes cancer" and "smoking doesn't cause cancer", and is using equivalent meanings and criteria for those words, then we would say that is false. THE LNC would be violated. I agree. You write,

Of course to claim that smoking causes cancer does not violate the LNC so logic does not even apply

It seems clear to me that if we were try to research the question of whether smoking causes cancer, we would use logical principles all the time in gathering and assessing the evidence. Logic would apply, but the question would not be a matter of pure logic. Does that seem like a reasonable statement to you? hazel

Stephen, you write

If that is true, then why do you also say that logical truths flow necessarily, with certainty, *within logical systems,* as if that same certainty did not apply *outside logical systems* – that is, in the real world.

See my answer to Vivid at 104. I think it answers your question, also. You ask,

Also, mathematical logic is not synonymous with philosophical logic. Why are you even talking about mathematics since it has nothing to do with the subject matter?

By “philosophical logic” I assume (but I may be wrong) you mean standard symbolic logic, starting with the three laws of logic, and moving on to statements about conditionals, conjunctions and disjunctions, etc. Math uses the tools of logic along with additional axioms. Often people refer to logico/mathematical systems because logic is foundational to math. For instance, in the OP kf wrote, as he often does,

Similarly, if we look at the world partition W = {A|~A} we see that A is itself, a unit distinctly different from the complex unity ~A, thus we find unity and duality. Where too the partition is empty and there is nothing in W but outside A and ~A, thus, nullity. This sets up the natural numbers, integers, rationals, reals, continuum, and even by using vector rotation, complex numbers.

That is, the development of mathematics starts with positing a unique, distinct unit and then using the laws of logic, along with other suitable definitions, builds the entire edifice of numbers and algebraic tools that go with them. That’s why I brought in math: because it is a related field where propositions are certainly true, and we can be certain of them because we can prove them logically. Both symbolic logic and math create a body of certain truths which flow logically from their starting point, but logic is the more basic, and then math builds using logic. hazel

Hazel “But if someone said “smoking causes cancer”, for instance, we could not rely purely on logic to determine whether that was true or false” Of course to claim that smoking causes cancer does not violate the LNC so logic does not even apply. Now if someone makes the counter claim ( all positive and counter claim being exactly equal )we can ,using the LNC ,be certain that someone is wrong. Vivid vividbleau

Someone can't say "that is a cow" and "that isn't a cow". We can be certain that statement is false. But if someone said "smoking causes cancer", for instance, we could not rely purely on logic to determine whether that was true or false. hazel

Hazel “But here, at least I have been discussing logical truths, which are certainly true, as opposed to using logic to work to establish other kinds of less certain truths about the real world” Using logic can we not be certain about certain things in the real world? For instance if a claim is made about the real world that violates the LNC can we not be certain that the claim is false?. Vivid vividbleau

Hazel

Yes, we use logic to understand the real world, so I am not beginning to say that logic doesn’t apply to the real world.

If that is true, then why do you also say that logical truths flow necessarily, with certainty, *within logical systems,* as if that same certainty did not apply *outside logical systems* - that is, in the real world. Also, mathematical logic is not synonymous with philosophical logic. Why are you even talking about mathematics since it has nothing to do with the subject matter? StephenB

re 98: it seems like we've been in complete agreement about that since post 3! :-) Error exists. No one would ever deny that, as far as I can tell. But how it is "pivotal" is another question. Did you read my post to Vivid at 96? hazel

Vivid, your question expands the discussion considerably. Yes, we use logic to understand the real world, so I am not beginning to say that logic doesn't apply to the real world. However, logical truths flow necessarily, with certainty, within logical systems, which has been, in my opinion, the topic of this thread. I agree with kf, who has said this multiple times - see 35 above, for instance, that warrant for other types of statements "comes in degrees, as does certainty." But here, at least I have been discussing logical truths, which are certainly true, as opposed to using logic to work to establish other kinds of less certain truths about the real world. To be clear, by “within the world of logic” I mean propositions that are made about logical (and mathematical, if we allow that in as part of logic) entities and established through logical principles using previously established facts. For instance, in logic, DeMorgans law “not (A and B) = not A or not B” is a truth within the world of logic. e^(i*pi) = -1 is a truth in the complex number system, and Morley’s theorem, mentioned earlier, that the three trisectors of the angles of any triangle intersect in an equilateral triangle is a truth in Euclidean geometry. hazel

Hazel Thank you for your thoughtful response. Permit me a question, you say “I think all the consequences of the laws of logic are true in the same way that ““false propositions exist” is true: as a consequence within the world of logic” Perhaps you already addressed this somewhere else so forgive me if I missed it. What do you mean by “within the world of logic”? For instance your not saying that logic doe not apply to the real world.? I have no reason to think otherwise but let me know. Vivid vividbleau

Folks, sometimes it helps to remind ourselves. Notice, again, the OP:

For simple example the Josiah Royce proposition, E = error exists, is undeniably true. To see that, try to deny it, ~E. That in effect claims it is error to propose E. So E must be true.

See how understanding is pivotal? KF kairosfocus

re 94: Vivid writes,

FYI there have been critics that have visited this site that don’t accept the laws of logic as self evident and real truths.

I know nothing about that. hazel

Hi vivid. Let me take it line by line: 1.

We have in hand a demonstration by example of a self evident truth.

Well, as Dave and I have been pointing out, the demonstration of “false propositions exist” is a logical deduction from the even more self-evident truth the of the law of non-contradiction. But I certainly take the three laws of logic as self-evident truths, by virtue of which they are the starting axioms of our system of logic. 2.

If one truth exists, truth exists.

I think all the consequences of the laws of logic are true in the same way that ““false propositions exist” is true: as a consequence within the world of logic 3.

If one self-evident truth exists, self evident truth exists.

Yes, but once you’ve established the three laws, I’m not sure how many more self-evident truths there are. There are a number of beginning axioms in math and geometry that are considered self-evident, but this can get on problematic, as we have found that in some cases you can assume a different axioms and get a different logical system which has its own truths. 4.

If one necessary truth exists, necessary truth exists.

I’m not sure how “necessary truth” differs from “self-evident” truth. 5.

If one warranted, credible truth exists and is accepted by one person, knowledge exists.

I assume that a “warranted, credible truth” is one established within the logical systems of logic and math, so once such are known by someone, knowledge exists. 6.

If the truth is warranted to certainty, certain knowledge exists.

Yes: there are many many facts in logic and math that we have certain knowledge of. 7.

If the truth is necessary, knowledge of necessary truth exists. If the truth is warranted to self evidence, known self evident truth exists.

Again, I’m not sure what all this means, especially “warranted to self evidence”. I consider consequences which flow logically from beginning principles, as kf quoted Copi about at 83, as necessary. Since they derive their necessity (maybe that is what he means by “warranted”) from the beginning self-evident truths, perhaps that is what he means by “warranted to self evidence”, but I’m not sure. 8.

Schemes of thought that deny such are therefore in that respect falsified.

I know of no “scheme of thought” which denies the laws of logic or the logical consequences which flow from them. 9.

Of course, as one SET in question is that error exists and another closely linked one is that false propositions exist, then this also implies that we must be prudent in our investigations and claims.

I’ve already said I agree with that. hazel

Hazel “If “such” refers to the proposition that “false propositions exist”, then I don’t think anyone has denied that from the beginning of the thread.” The “such” is quite clearly the implications he spells out in the first paragraph of 72. Do you deny the implications? If so what is it specifically do you object to? Vivid vividbleau

Hazel “But the laws of logic have already been accepted as real truths.” FYI there have been critics that have visited this site that don’t accept the laws of logic as self evident and real truths Vivid vividbleau

re 92: If "such" refers to the proposition that "false propositions exist", then I don't think anyone has denied that from the beginning of the thread. hazel

Hazel In 72 KF summarizes his thought process with this conclusion and I think this is what he is asserting. It seems that everyone is in agreement here. “ Schemes of thought that deny such are therefore in that respect falsified. “ Vivid vividbleau

This is all a little silly. Consider the first law of logic, the law of non-contradiction, F = “for any proposition P, P and ~P cannot both be true.” F is true, as it is the first axiom of logic. Therefore ~F is false. Therefore, false propositions exist. For the life of me, I don’t see how this leads to anything other than “so what”? kf says it establishes that there are real truths. But the laws of logic have already been accepted as real truths. We don’t even need to go through all this about the proposition “false propositions exist” (although we can) because all we need to do is show an example of a false proposition, which I just did. hazel

KF,

DS, kindly note the distinction between an explanation and a deduction.

Eh? It's a straightforward proof. You assume ~E and show that this implies E. daveS

And yet Copi says, "To determine whether an inference is correct, the logician examines those propositions that are the initial and end points of that process and the relationships between them." And yes, it is a stepwise logical deduction process. Back at 5, you wrote the following stepwise logical argument, although it wasn’t very clear.

DS, that is a useful reworking. F = false propositions exist. The denial ~F implies it is false to claim that a false proposition exists. This is or implies an assertion that is true or false, i.e. a proposition. So the latter implicitly affirms what it tries to deny, and is necessarily false — that’s stronger than one or the other is false

Here’s the same argument, more straightforwardly, in proof by contradiction form: Let F = “false propositions exist” Assume ~F is true: “false propositions do not exist”, which is to say “all propositions are true.” Therefore F is true. Thus contradicting our assumption that ~F is true. Therefore F is true. (Wrote this before I saw Dave's post. I have no idea what purpose kf has saying this is an explanation, as if that somehow adds to the conclusion. And his recent P.S. doesn't make sense, including the idea that "direct understanding", whatever that is, adds something to, much less is a substitute for, logical deduction.) hazel

DS, kindly note the distinction between an explanation and a deduction. By what it is, a proposition Q carries with it the entangled counter-assertion ~Q of coupled, opposed T/F value. So, to assert there are no false propositions directly fails on insightful explanation. Passing on to E, the assertion carries with it its counter ~E by simply being a proposition, where again, on insightful inspection, ~E must fail. I am not saying that one cannot construct a chain of claims to show a SET would follow on some premises or other but instead that on understanding it -- which may require explanation to build understanding -- one sees that it is so, must be so, and must be so on patent absurdity if denied. In both cases discussed, the false proposition undermines the meaning of what a proposition is. KF PS: In the usual case, the argument to a proposition may be ultimately less plausible than directly understanding, e.g. it will often involve the inescapable use of the proposition or may be abstruse, etc. kairosfocus

For simple example the Josiah Royce proposition, E = error exists, is undeniably true. To see that, try to deny it, ~E. That in effect claims it is error to propose E. So E must be true.

Isn't this a logical deduction? "Assume ~E. Then the proposition 'error exists' is erroneous. Hence ~E is false, therefore E is true". daveS

H, Copi shows that I am using the term proposition in the generally understood way: an assertion say Q that is assignably T or F, which will automatically bear with it its own antithesis ~Q (of coupled opposite value ~T or ~F) -- do I need to say, "entangled" to make the point clear? In that light, the proposition that no propositions are false is by insightful inspection necessarily false. This pivots on what a proposition is, rather than a chain of argument. I repeat, an explanation is not a stepwise logical deduction process. KF kairosfocus

To be clear, these are propositions within the realm of logical systems your sources are talking about. hazel

I don't disagree with Copi, and I have no idea why you think I might. I have certainly not said anything different than what Copi is describing. hazel

F/N: I cite Copi from my copy p 5:

"Inference is a process by which one proposition is arrived at and affirmed on the basis of one or more other propositions accepted as the starting point of the process. To determine whether an inference is correct, the logician examines those propositions that are the initial and end points of that process and the relationships between them. Propositions are either true or false, and in this they differ from questions, commands and exclamations. Only propositions can be either asserted or denied . . ." [Intro to Logic, Macmillan, 1990.]

In short, a proposition is what is asserted or implied by a thought, utterance, belief or doubt, etc, that is assignably T or F. Where obviously for any Q there is its denial ~Q by antithesis. The error of the proposition, There are no false propositions is then patent on the meaning of what a proposition is. Effectively, there is no possible world W in which some proposition Q is, that Q is false, where Q stands for any proposition whatsoever -- as in, no propositions are assignably False under any circumstances in any world. Explanation is not in this sense an inferential process but a drawing out of meaning and understanding. If you disagree with Copi et al (in effect, the field of scholarship), on what a proposition is, kindly explain. KF PS: Went digging, 14th Edn 2014 (mine is 8th) with now two co authors not just one:

Propositions are the building blocks of our reasoning. A proposition asserts that something is the case or it asserts that something is not. We may affirm a proposition, or deny it—but every proposition either asserts what really is the case, or it asserts something that is not. Therefore every proposition is either true or false. [p. 2]

kairosfocus

Yes, kf, no one is arguing that there is any weakness in the logical demonstration that "false propositions exist" is true. hazel

H, so far as I understand a proposition is by definition an assertion that is assignably T or F. It is part of the meaning of asserting Q that there is an alternative ~Q that is denied; I believe this is called antithesis. One will be T and the other F, part of the background. This brings up the part about being of suitable experience and understanding to see what is "in" a claim. In the relevant case, there is a special claim such that the denial is recognisably the one that is false, as it in fact undercuts part of the meaning of what a proposition is. KF kairosfocus

KF (#76), Certainly we know today that trisecting some angles with compass and straightedge is impossible. daveS

kf, you write, " Q is asserted, ~Q denies it, a proposition being assignably T or F by meaning, one or the other must be false. Here, False propositions Exist is denied, and we see that it is the denial which is false." That is a logical deduction. I don't know why you call it an explanation, any more than any logical or mathematical proof is an explanation. And yes, some SET exist: the laws of logic, for example, so the set of SET is not empty. But, as I write above,

Fine, but that doesn’t establish what those others are. Each one is a separate issue. So I agree that “this also implies that we must be prudent in our investigations and claims.”

hazel

KF (#76): Yes, agreed. daveS

H, no. There is not a deduction at work but an explanation. Q is asserted, ~Q denies it, a proposition being assignably T or F by meaning, one or the other must be false. Here, False propositions Exist is denied, and we see that it is the denial which is false. KF PS: I laid out several corollaries of one SET being recognised for what it is. One such is that if a SET exists, then self evident truth -- a category -- exists, i.e. it is meaningful and non-empty. kairosfocus

DS, so far as I gather, there were things they tried but could not do with said instruments (famously, squaring the circle). It was Algebra that provided a criterion for the rule. Of course, there is a trivial case of trisection, the 60 degree thirds of a 180 degree angle. There may be others e.g. as derived from it. But there is no general construction such as for bisection of an angle. I suppose an infinite chain of chosen bisections will converge on trisection, but the final value would require transfinitely many steps. KF PS: To construct 30 degree parts of a right angle, construct the perp bisector of the line and then do 60 degree angles, one on each leg of the right angle. That should work. But the trisection of 60 degrees is apparently not possible on such tools. Relax the rules and other things become possible. kairosfocus

PPPS: Let me clip the opening remarks from No. 10 in this series:

One of the issues we must face is whether there is enduring moral truth that can be warranted to such a degree that it rightly governs our thoughts, words (especially in argument) and deeds. Where, given that we have an inner voice (conscience) that testifies to duty under moral law, as well as an inescapable sense of duty to truth, right reason, prudence, justice, uprightness etc., if that intuition is false, then our whole inner life becomes tainted by grand delusion. A lot is at stake, in short. A quick first answer is, that we may recognise that grand delusion is self-referential, incoherent, self-falsifying — a case of reduction to absurdity. That is, we see the inescapability of being governed by moral truth as part of the same first principles that we cannot prove but must accept on reasonable responsible — note the self-reference! — faith, for all proofs, all arguing must start from such.

kairosfocus

Actually, kf, "false propositions exist" is a logical deduction. Calling it self-evident in this case only means that it's a simple deduction. The laws of logic are self-evident. No one denies this. Your argument is that if one self-evident logical truth exists, then other can. Fine, but that doesn't establish what those others are. Each one is a separate issue. So I agree that “this also implies that we must be prudent in our investigations and claims.” hazel

KF, Well, trisection definitely was studied by the ancient Greeks, and methods for trisection were known. They of course didn't know it was not possible using only compass and straightedge. daveS

H, I did not leap without warrant, I drew out the implications of an established point. We have in hand a demonstration by example of a self evident truth. If one truth exists, truth exists.If one self-evident truth exists, self evident truth exists. If one necessary truth exists, necessary truth exists. If one warranted, credible truth exists and is accepted by one person, knowledge exists. If the truth is warranted to certainty, certain knowledge exists. If the truth is necessary, knowledge of necessary truth exists. If the truth is warranted to self evidence, known self evident truth exists. Schemes of thought that deny such are therefore in that respect falsified. Of course, as one SET in question is that error exists and another closely linked one is that false propositions exist, then this also implies that we must be prudent in our investigations and claims. KF PS: I note, that logic of being is one class of logical investigation and that one result is that if a proposed entity Q is such that two core characteristics x and y are in contradiction then Q cannot exist in any possible world, i.e. it is impossible of being, e.g. a square circle. Logical considerations can and do have existential import. PPS: Also, once we find moral SETs, we can draw out a framework for moral knowledge. This was already discussed at no 10 in this series. It has been outlined above, starting with that reason and discussion inescapably pivot on known duties to truth, right reason, prudence, fairness etc. In addition as moral abuses exist, what they violate exists. E.g. it is self evidently wicked to kidnap, bind, sexually torture and murder a child for one's pleasure. From which, much can be drawn out. There has been no leap from certain simple truths to moral truths, further relevant considerations have been repeatedly brought to bear. Cf: https://uncommondesc.wpengine.com/ethics/logic-and-first-principles-10-knowable-moral-truth-and-moral-government-vs-nihilistic-manipulation/ kairosfocus

DS, trisection was not something doable with compasses, unscaled straight-edge and dividers -- one of the rules of classical Geometry, so this would not have been explored in classical times. KF kairosfocus

re 65 to MG. This thread is about the significance of the fact that "false propositions exist" can be shown to be logically true. As kf said at 66, all the stuff about how math can describe the real world is "old ground" from previous threads. kf writes, "The main purpose of this OP and thread has been to move the game forward." But that hasn’t happened. Kf jumps from "false propositions exist" to claiming we can “ find truth (real, objective truth not strongly held opinions), warranted truth, self-evident and certain truth reasonable, credible, acceptable. Where, arguably, such does involve a core of moral truths and instructive test cases.” But I can’t see how he has justified that leap. Logical truths exist. I think we all agree on that. Then what? What is one other logical truth that follows from “"false propositions exist"? Just asserting that other truths logically exist doesn’t establish what those truths are, or why they are true. hazel

Yep, Dave. I found out about it 20 years or so ago while teaching high school geometry. I spent a lot of time trying to prove it by using equations based on 180° in a triangle, and found out later that can't be done. I don't now why that approach doesn't work. Wikipedia did not exist at the time! hazel

Math Guy @ 64- “purely immaterial beings existing in a non-physical world” would be directly with all of the immaterial information there is, including all mathematics. They would all be super duper versions of Srinivasa Ramanujan ET

hazel,

One example that always fascinated me: the three trisectors of the angles of any triangle intersect in an equilateral triangle! Isn’t that cool?

Yes, that is very interesting. Is this the theorem? It's surprising that it was discovered only a little over 100 years ago. daveS

MG, it seems we are visiting old grounds. I take it from the non-reaction to the bulk of substance I summarised in 52 above, that it is clear that there is good reason to accept that there will be abstract structures and quantities that are necessarily present in any possible world as part of its framework. From the point we reach Z, we have direction and size, thus vectors. The step of the rationals plus infinitely continued converging power series of rationals gives us abstract continuum, R. The first rotation of R, by i*R gives us planar space. Modifying, the ijk unit vectors give us 3-d rotations and a 3-d vector space that is a continuum. Along the way, we may readily extend to the transfinites. Let me add: just recognise w as order type of the unending succession of naturals: {0,1,2 3 . . . } --> w A field can be understood as a quantity that is assigned values across a domain of space, i.e. at each point in a relevant region, it assigns a value. Sometimes, a scalar (temperature, local air pressure, local air density, potential such as a voltage etc, popping across to physics) or a vector (e.g. gravity, electrical, magnetic fields, flow fields for fluids etc). So, we can assign fields to conceptual spaces and readily observe them in the physical world. While I am at it, space itself has measurable properties connected to electromagnetism etc. which in that case set the propagating speed of electromagnetic waves. Fields are inevitable in abstract logic model worlds, as part of the core facts of structure and quantity that are antecedent to axiomatisations. They are seen in our world, working integrally with many spatially extended physical phenomena, which can be seen as ways in which objects influence space around them. A massive object notoriously distorts the space-time fabric, giving rise to gravitational phenomena. This is in fact the root of General Relativity, and it is why we accept that on the grand scale, our world dies not follow the abstract structures of a Euclidean space. But under relevant local circumstances, we can use familiar models that fit Euclidean expectations. So yes, fields are real and space is real. Similarly, the mobius strip shows that objects in space can and do exhibit quantitative and structural properties that do not depend on our axiomatisations and calculations to be real and even empirically demonstrable. We already saw that there are transfinitely many core basic mathematical facts that are tied to the bare possibility of a world, as they are part of the framework for such to be of distinct identity. So, too, I have no doubt that Godel was right: there are ever so many facts of Math unreachable by sufficiently complex axiomatic systems, that are coherent, and that there is no constructive process to establish a demonstrably coherent axiomatic system. So, bot, we must work by faith having learned of our limitations and there is in principle an escalating, endless hierarchy of mathematical systems that cannot exhaust the span of mathematical facts. Coming back to focus, it is important that we have confidence in the significant intelligibility of reality, that we see and accept that there are self-evident first truths, that there are truths, that some can be warranted so as to be knowledge, that some are so warranted beyond rational doubt, including that humbling truth: error exists. From this, we can see that in all of our explorations we find ourselves inescapably under government of duties to truth, right reason, prudence, fairness etc. Indeed, such is inextricably entangled with our life of responsible, rational, significant freedom. This points onward to the need to bridge the IS-OUGHT gap, which, post-Hume we know can only be done at world root. I have long since pointed out that to do so, there is but one serious candidate: the inherently good, utterly wise creator-God, a necessary and maximally great being. One who is arguably the prime, ultimate mind and who is worthy of our loyalty and of the reasonable, responsible service of doing the good that accords with our manifest nature. Of course, objectors are free to put up an alternative under comparative difficulties across factual adequacy, coherence and balanced explanatory power: ________ . The main purpose of this OP and thread has been to move the game forward, to the point where we find truth (real, objective truth not strongly held opinions), warranted truth, self-evident and certain truth reasonable, credible, acceptable. Where, arguably, such does involve a core of moral truths and instructive test cases. This, as radical hyperskepticism, subjectivism and relativism run riot all across our civilisation, wreaking havoc as they go. It is time for a sea-change, KF kairosfocus

Hazel (and Rob Sheldon, for that matter) We use infinite and continuous vector or tensor fields to great effect in physics. For instance, the "zero point field" describes the behavior of virtual particles, modeled via Feynman diagrams. The gravitational field seems to be almost as mysterious. Do fields exist? math guy

ET@63 I must disagree. Unless the “purely immaterial beings existing in a non-physical world” are capable of grasping all aspects of infinite sets (such as the integers) so as not to be bound by finite processes (like our current mathematical proofs), then Gödel's theorems imply an infinite hierarchy of formal systems, each containing true results that are unprovable in lower systems. The complexity of provably verifiable truths (i.e. theorems) form an infinite chain. Just like ordinal numbers, there is no "final most complicated theorem". math guy

daves, I would think that if we were if we were "purely immaterial beings existing in a non-physical world" then we would know everything. There would be nothing to pursue. ET

BB, technology long preceded significantly sophisticated mathematics, but rudimentary mathematics is part of our day to day existence. KF kairosfocus

Not very far, although "formalized" is a broad term, and so is "rudimentary technology", as technology usually has gone hand-in-hand with other advances such as trade, exploration, war, etc. That's what I think. hazel

Hazel and Dave, I have a question that I would be interested in your response to. Could a technology develop without some form of formalized mathematics? Very rudimentary technology would be possible, but how far could it be extended without mathematics? Brother Brian

Agreed, Dave: math is a delightful and fascinating field irrespective of any application to the physical world. One example that always fascinated me: the three trisectors of the angles of any triangle intersect in an equilateral triangle! Isn't that cool? hazel

ET, We wouldn't necessarily require it. Perhaps we would pursue mathematics as a recreational activity. daveS

DS, I suppose we often find vaguely mechanical contrivances easier to understand. KF kairosfocus

SB, the current resort to infanticide under false colour of law speaks volumes. Notice, I now start with moral government of our reasoning and discussion. Even the most cynical know that we are guided by duties to truth, right reason, prudence, fairness etc. They just hope to exploit it. KF kairosfocus

Hi daves- What makes you think that mathematics would require developing "if we were purely immaterial beings existing in a non-physical world"? ET

PS to my #50: A Universal Turing Machine in Minecraft This also leads me to wonder how mathematics would have developed if we were purely immaterial beings existing in a non-physical world. What would they think of Turing machines and the lambda calculi? daveS

KF @43, Yes, the objective moral law is self-evident *in itself* and *to those* who have not been [a] psychologically harmed, [b] brainwashed, [c] or morally weakened as a consequence of forming immoral habits. Also, fallen human nature temps us to shirk the responsibility of submitting our ego to the authority of binding moral truths. Our natural desire is to prefer our own desires over the truth. Subjectivism, Hyperskepticism, Relativism, Idealism, Nominalism, Communism etc, are all just different ways of denying the moral authority of the divine lawgiver so that we can be a law unto ourselves. StephenB

H & DS, Bottomline is, that we can see that there are possible worlds holding distinct identity. As I outlined earlier, this leads to the effects of that distinct identity where any possible world W is distinct from close neighbours due to at least one characteristic A (which may be reduced descriptively to a proposition). The world partition then shows that nullity, unity and duality are necessary corollaries of W being a distinct possible world. Apply von Neumann's construction and N follows thus Z, Q, R, and using rotating vectors C. In short, embedded structure and quantity are part of the fabric for any possible world. Which is to say such core mathematical entities are necessary beings, they did not begin to exist, they cannot cease, they are not dependent for their abstract reality on our reflection or conception. Their properties are independent of our spinning out abstract structural and quantitative logic-model worlds. Indeed, the utility of that in key part is that exploring such a world brings out such framework entities which will then be in any world, including ours. Other more specific features will extend to our world to the extent that there are in common characteristics. This sort of perception of real abstract entities inextricably part of any distinct world is sometimes called Mathematical Platonism. I add: thus in regards to certain core mathematical entities, properties, structures etc, we discover them, we do not invent them. The set C is an example where we thought we invented then found that a vector-rotation view naturally yields the key results. (This then invites in the complex exponentials approach with all its richness, where also the ghost of Fourier beckons.) I suspect the label is triggering, it is the substance that counts. I am pointing out that mathematical abstracta are necessarily embedded in the framework for any world, forming part of the logic of being. I am not imagining some sort of cloudy world out there with perfect cases of nullity, unity, duality, etc. So, while there is but one null set -- separate recognitions of it can obviously find no distinctions so it is all one and the same: { } --> 0 -- I don't think there is some ideal world repository out there where that abstractum is somehow concretised so that a sufficiently visionary person may somehow catch a glimpse like an Astronomer peering through a telescope. I do think that such realities are eternally contemplated by God as a facet of omniscience [God is the greatest possible Mathematician and Logician, Reason himself], but that is separate. I add, nope, I don't think there is some ethereal scroll or codex with perfect, complete Mathematics written out in perfect symbolism so that a visionary can inhale incense then go to the library, don magically translating spectacles -- an extension of virtual reality goggles I suppose -- and read it all. I have also been struck by the Mobius strip. Cut three strips of paper, make one common loop and two with a half twist. Cut the ordinary loop and one M-loop going around, in the centre. The first will separate into two loops, the second as you know doubtless will become a single, double length loop with several twists. Now, cut the third going around, 1/3 way across. It will separate into a narrower mobius loop interlocked with a longer loop. Those properties are independent of our expectations, ideas, understanding, axiom systems or whatever. They directly, empirically show how certain structural and quantitative properties are embedded in the fabric of our world. There are many other cases. Most recently I looked at how hearing exploits a mechanical implementation of a transform to the frequency domain. Where the Fourier Transform is about as abstract a thing as we will ever meet. KF PS: SEP, supplement to article on Mathematical Platonism:

Supplement to Platonism in the Philosophy of Mathematics Some Definitions of ‘Platonism’ Dummett 1978b, p. 202: Platonism, as a philosophy of mathematics, is founded on a simile: the comparison between the apprehension of mathematical truth to the perception of physical objects, and thus of mathematical reality to the physical universe. Dummett 1991a, p. 301: Platonism is the doctrine that mathematical theories relate to systems of abstract objects, existing independently of us, and that the statements of those theories are determinately true or false independently of our knowledge. Field 1989, p. 1: A mathematical realist, or platonist, (as I will use these terms) is a person who (a) believes in the existence of mathematical entities (numbers, functions, sets and so forth), and (b) believes them to be mind-independent and language-independent. Gödel 1995, p. 323 [Platonism is] the view that mathematics describes a non-sensual reality, which exists independently both of the acts and [of] the dispositions of the human mind and is only perceived, and probably perceived very incompletely, by the human mind. Maddy 1990, p. 21: [Realism or platonism is the view that] mathematics is the scientific study of objectively existing mathematical entities just as physics is the study of physical entities. The statements of mathematics are true or false depending on the properties of those entities, independent of our ability, or lack thereof, to determine which. Parsons 1983, p. 273: As is customary in discussing the foundations of mathematics, platonism means here not just accepting abstract entities or universals but epistemological or metaphysical realism with respect to them. Thus a platonistic interpretation of a theory of mathematical objects takes the truth or falsity of statements of the theory, in particular statements of existence, to be objectively determined independently of the possibilities of our knowing this truth or falsity. Shapiro 1997, p. 37: [Realism in ontology or platonism is the view that] mathematical objects exist independently of mathematicians, and their minds, languages, and so on. Resnik 1980, p. 162: Let us call an ontological Platonist someone who recognizes the existence of numbers, sets, and the like as being on a par with ordinary objects and who does not attempt to reduce them to physical or subjective mental entities.

The main SEP article summarises:

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. Just as electrons and planets exist independently of us, so do numbers and sets. And just as statements about electrons and planets are made true or false by the objects with which they are concerned and these objects’ perfectly objective properties, so are statements about numbers and sets. Mathematical truths are therefore discovered, not invented.

PPS: Let me add, that information and energy look to me like similar embedded abstracta that have structural and quantitative properties. Angular momentum is similarly a vector that produces astonishing effects yet is decidedly not concrete. PPPS: Let me further add, I am viewing Mathematics as having two aspects, a study and a substance. That is as (the study of) the logic of structure and quantity. I think both are distinct and important. The mobius strip for example is independent of our study but exhibits embedded structural and quantitative properties which historically helped to launch topology. kairosfocus

H, this thread is part of a general exercise on logic and first principles, which I have become convinced is necessary because the root disagreements and contentions are worldviews driven, not science. In this context science comes in fairly late. In that context, hyperskepticism, subjectivism, relativism and the like are major cultural forces. An important minority could be termed neural illusionism or the like where consciousness is viewed as an illusion. And yes, part of that has been the attempt to dismiss first principles of reason. Way back when I looked at boolean algebra I recall being taught that there was nothing special about the triple cluster LOI, LEM, LNC, such was just part of the seventeen laws, and of course at bottom truth tables could be trotted out. It is only on considerable reflection that I came to realise that there is more to the story. I don't know if you realise that my sliding scale treatment of what knowledge is -- with degrees of warrant, certainty and credibility/reliability -- is a deliberate weak form i/l/o how we speak of scientific, historical and common sense knowledge claims. Similarly, the weak, inquiry form form of the principle of sufficient reason and causality etc. KF kairosfocus

hazel, Those views are very close to mine, I believe. I have been trying (with limited success) to read about automated reasoning and automated theorem proving lately, and it has reinforced some of these views. Some of the sources I am looking at explain a concept verbally, then implement that concept in a programming language, which gives one a very concrete understanding of what is going on. As an example, one could build a computing machine which actually does compute the billionth decimal digit of pi (of course some already exist). In principle, this machine could be built out of legos or even 2 x 4s. Provided the machine is built correctly, the output of this machine, when it finishes the computation, is an objective "fact", independent of our minds. That is, if the physical universe operates as I understand it to. And I believe any other mathematical "fact" should be amenable to the same sort of mechanical verification. [Edit: By "fact" I mean "theorem" here; I don't expect mechanical verification of axioms or choice of logic]. daveS

Hi Dave. I don't know whether you (or I) want to reopen a can of worms, but here goes. After some thought, I had concluded last fall that my position is that mathematical truths exist within the logical symbol systems which we have developed to express them, but not in some eternal, pre-existing Platonic realm. Within those systems, the force of logic creates truths that exist (the one billionth digit of pi, base 2, for instance), that we could potentially discover, at least in theory. They are objective in the sense that anyone following the laws of logic and utilizing the proper math would reach the same conclusions. They are either unequivocally either true or false, even if we don't which at this point. More broadly, I also think that in general abstractions exist in our minds, and in the verbal and written systems we use to express them. I'm not interested in re-arguing this with kf, but I thought I'd share my basic view on these issues. hazel

Kf writes, “We can see this in for example how we are agreeing that truth is real and can be objective. … Similarly, the force of the undeniable truth, error exists, has led to recognition that hyperskepticism, radical subjectivism and radical relativism are not tenable … These are important points of progress.” If you are referring to me (and probably Dave), all we have agreed upon is that in the world of logic, it is proven that “false propositions exist” is true, that the laws of logic and associated mathematical structures exist, and we can use mathematics to understand the world, at least to some extent. I don’t think we have agreed that these logical truths are “real and objective” in the same way you do (although what that way is is not clear), or have the same implications as you think they have. Also I don’t think whatever you mean by “hyperskepticism, radical subjectivism and radical relativism” has even been addressed, other than in your rhetoric, unless you mean not accepting the laws of logic, which I don't think is true of anyone. hazel

hazel,

We discussed a whole bunch of this last fall, Dave, and I’m pretty sure I am one of those that kf doesn’t think falls in the minority that he thinks you are in. However, I think you would need to know more about what kf means about the existence and nature of abstract, objective truths before you agreed that you agreed with him about these matters.

Thanks, that's interesting. I should also admit that if someone pressed me on what I mean when I say the integers or real numbers actually exist independently of our minds, I don't know that I could make a convincing case. daveS

H, okay, I think it is 36 that I somehow must have connected to you. KF kairosfocus

kf, you write concerning Gab, "you also discussed and I noted to you as a result". No, I never discussed Gab until you mistakenly identified me as the person who had brought it up. See posts 32, 34, and 36. hazel

H, you also discussed and I noted to you as a result -- I do not like what Gab seems to be doing here though I understand frustration at say Wikipedia's moderators. Empowering unmoderated trollery and apparently tracking people all over the web opens up serious cans of worms, I add. I note, too, that while agreement is important, acknowledgement is not the same as establishment of truth. It is the establishment of key truths that can serve as plumb-lines to check socially agreed or imposed yardsticks that is the pivot of the OP. Hence, the story of the silly Emperor. KF kairosfocus

SB, thanks for your ever insightful points. You are quite correct to point out that error addresses those of logic and those regarding entities that are real, whether concrete or abstract. I have also noted that given logic of being and the metaphysical import of the principle of identity, if a proposed entity such as a square circle has contradictory core characteristics it will be impossible of being. This thread is serving as a forum for clarification and for building some key points of agreement. We can see this in for example how we are agreeing that truth is real and can be objective. That which says of what is that it is, and of what is not that it is not is true. Similarly, the force of the undeniable truth, error exists, has led to recognition that hyperskepticism, radical subjectivism and radical relativism are not tenable (though, doubtless, they are still very common or are even dominant views on the ground . . . having been drummed into us by the media and education systems). These are important points of progress. I note, too, that no one has actually extrapolated from or directly inferred from, that truth exists to therefore moral truths exist. What I did was to note that in discussing and arguing (even about the import of error exists), we mutually, implicitly and even intuitively recognise that our reasoning is morally governed by duties to truth, right reason, prudence, fairness etc. This is inextricably entangled with inferring, judging, rationality in general and is of course echoed in our typical understanding that knowledge must have truthfulness or at least credible truthfulness as a criterion -- we cannot properly know what is manifestly false, though we may err and believe it. So, we know first that truth is actual and then that it is in some cases not only intelligible but warranted so objective. And in the process of reasoning, we recognise another class of truths: truths regarding duties, in this case inescapable truths on the duties of rationality. Inescapable, on pain of instant, vicious absurdity that would destroy reason and discussion toward truth and sound reasoning. In short, we have here cases of inescapable and so self-evident truths of moral character, moral truths. Absent these, rationality collapses into chaos. Where, too, they are not subject to proof -- the attempt to prove or disprove or simply discredit already implicitly turns on the duties in question. They are part of where our proofs come from. Beyond, we have other self-evident moral truths that more directly shape or test our yardsticks of conduct. I have commonly used concrete manifestly undeniable evils as instructive test cases. For example, it is manifestly wicked to kidnap, bind, sexually assault and murder a child for one's pleasure. From this, we may build a considerable body of moral knowledge that is relevant to society. That is the point you drove at in writing:

We would have to say something like this: Moral violations exist, therefore moral truth exists. This would qualify as a self evident truth. Even people who deny objective moral truth complain – routinely and illogically – that they have been the victim of a moral violation.

Moral violations exist only if moral duties are true and binding. So, justice [a narrow form of the duty of fairness] and our recognition of its violation show that moral truths that are sufficiently objective to frame moral knowledge also exist. Justice of course is closely tied to rights (which require being in the right), and leads on to how moral truth and moral government reflect the law of our nature and guide the framing of civil law. Or at least, should guide such framing. KF PS: I note that deer is used in two senses, as a reference to particular species and as the wider family, Cervidae (in turn, distinct from Antelopes). We thus must recognise the principle of definition on what is common and what is different, genus-differentia -- now very important in computing given the object paradigm. Also, there are differences of local or regional names. Due to mistakes in the 1600's, what Americans call a Moose is what historically Europeans called an elk. The elk of North America has been known to inter-breed with the Red Deer of Europe (this happened when both were introduced to New Zealand). The picture is complicated. In the USA, deer refers to the white tail or the black tail or the mule deer. The black tail seems to be a sub-species of the mule deer. Your statement is true in the North American sense, once we reject the secondary use of "deer" to denote the family of related species. Ambiguity is ever the foe of clarity. kairosfocus

Hazel

The phrase “error exists” doesn’t distinguish between logical error and error about the world, and your statement does.

The statement "error exists" applies to both the realm of internal logic and the realm of extra-mental reality. Each realm is in perfect correspondence with the other.

If we just look at “error exists” as a statement about the world, then of course it does: Johnny says, “Look at that deer”, and Billy correctly says,”No, that’s an elk.” Johnny was in error. The difference here is truth within the world of logic and truth about the world.

Johnny errs both ways, both logically and metaphysically. Logically, an elk cannot be a deer because if violates the law of noncontradiction. Metaphysically, an elk cannot be a deer because it violates the law of identity. The logic of mind always corresponds to the logic of the universe. It is impossible for something to be logically true and metaphysically false. On the other hand, we cannot say that since logical and metaphysical error exists, it follows that moral truth exists. We would have to say something like this: Moral violations exist, therefore moral truth exists. This would qualify as a self evident truth. Even people who deny objective moral truth complain - routinely and illogically - that they have been the victim of a moral violation. StephenB

Kf, you write, “H, thanks re the commenter add-in, I don’t think I will go there. That was Tribune that mentioned that, not me. I agree with you that I wouldn’t want anything to do with it. Also, I am sure everyone in this discussion accepts the laws of logic as true, and necessary components of arguments both within math and logic and about the world. I don’t think that is in question. hazel

H, thanks re the commenter add-in, I don't think I will go there. I note also we are not looking at a derivation but an explanation as to why false propositions must exist. There is no extrapolation to moral truths, but there is a recognition of another bit of inescapability on pain of absurdity. In arguing, we depend on implicit acceptance of duty to truth, right reason, prudence, fairness etc, denial of which duties is instantly, viciously absurd. This is instantly known moral truth, regarding our actually being bound by such duties. What is then interesting is how such can be adequately sustained in a post-Hume world. Which requires bridging the IS-OUGHT gap at world root level. KF kairosfocus

BB, that is precisely the point, self-evident truths are where many proofs must start from, and in other cases they serve as plumb-line tests. For key example( as Epictetus pointed out) if one asks to prove the first principles of logic, the proof would have to use them, and we can add that the attempt to deny or dismiss such will also necessarily involve using them. They are inescapably true. Proofs depend on such, and so are necessarily weaker. KF kairosfocus

re 31 to Dave and 37 to Brian: I certainly don't see how the logical deduction that "false propositions exist" or the logical development of the number system extrapolate to there being objective moral truths. These don't seem connected to me, and I don't see how the metaphor of plumb lines or crooked yardsticks adds anything to any argument that they are. hazel

Self evident truths and objective moral truths may exist but there are no ways of proving them. Brother Brian

Dissenter isn't the same. It's more akin to Disqus but where you can't be censored. It is a Gab product though.True that Gab can be quite nasty. Anyway, keep the fight. (Just saw another Dissenter comment on the Wiki ID article. This might be fun) tribune7

H, I laid out the significance again, just above. In an age of radical skepticism, relativism and subjectivism, showing that there are truths, intelligible truths, warranted truths, knowledge and even certainty and self-evident truth is an important plumb-line check against some fairly crooked yardsticks: radical skepticism, subjectivism and relativism are undercut. A limited but important result, one that actually breaks the momentum of some major trends of how people have been led to think in this day and age. BTW, you may recall that I earlier showed that warrant comes in degrees, as does certainty. Specifically, scientific explanatory constructs do not -- rpt., NOT -- rise to even moral certainty. KF kairosfocus

Trib, longtime no see. Never heard of that stuff before. I gather Gab can get pretty hairy and nasty. I gather it is like Twitter, in which I have no interest. BTW, are they tracking you everywhere you go, showing you comments on the page? That would be a deal-killer for me. Thanks, though. KF kairosfocus

Hi Dave, I, like you I think, don’t see how proving the proposition that “false propositions exist” leads to certainty about any propositions about the world, much less about morality, or to the existence of “objective” truths other than those which reside in mathematical and/or logical system. In particular, you ask,

And are we still dealing with the same level of certainty we have about the truth of the proposition “error exists”?

I think the answer is no, we’re not, as I’ve mentioned. Propositions about the real world are a different kind of thing, established in a different way, and provisional in comparison to mathematical/logical propositions. Also, you said

FTR, I accept the LOI, LNC, LEM. Regarding the bit about {A | ~A}, I don’t follow how this “sets up” things such as the real number system. However I do accept that all those mathematical structures exist, and we can use mathematics to understand the world, at least to some extent. I’m not a radical skeptic, relativist, or subjectivist, fortunately.

Kf then writes of you,

DS, in the particular regards of interest, you seem to be one of the apparent minority who accept that there are objective truths, that such can be intelligible, that they can in certain cases be warranted sufficiently to be knowledge, that such truths embrace abstracta [specifically entities of structure and quantity] and more.

We discussed a whole bunch of this last fall, Dave, and I’m pretty sure I am one of those that kf doesn’t think falls in the minority that he thinks you are in. However, I think you would need to know more about what kf means about the existence and nature of abstract, objective truths before you agreed that you agreed with him about these matters. And FTR I also “accept the LOI, LNC, LEM. …[and that] all those mathematical structures exist, and we can use mathematics to understand the world, at least to some extent. hazel

Don't want to hijack the thread, KF, but you might want to do a post on Gab Dissenter which allows commenting on any post on the web. I've already put a comment on the article for Intelligent Design on Wikipedia. You have to have Dissenter to be able to see it. It's described here: https://www.pscp.tv/w/1yNxaOpVWPQGj You can download it here: https://dissenter.com/download tribune7

KF, Yes, I'm not so sure there is an objective moral code which exists independently of our minds. daveS

DS, in the particular regards of interest, you seem to be one of the apparent minority who accept that there are objective truths, that such can be intelligible, that they can in certain cases be warranted sufficiently to be knowledge, that such truths embrace abstracta [specifically entities of structure and quantity] and more. I have not seen enough to take it that you acknowledge moral government of our rationality and that there are thus inescapable moral truths, which sets up a context for objectivity of certain moral truths, i.e. objective albeit limited moral knowledge. The plumb-lines above are about sorting out some seriously crooked yardsticks in our civilisation. KF kairosfocus

KF, Thanks for typing that out. At this point I'm not persuaded I need to make any changes to my worldview, however. daveS

DS, Possible world W has to have in it some distinct feature A that marks it separate from nearby neighbours, say W' and W". Let us call it A and partition W = {A|~A}. A is a case of unity that must be present for a distinct world W, leaving the cluster of other features as complex unity ~A, which is a different unit. Thus, duality is necessarily present. As W has nothing between or beyond those two, we see nullity also -- the partition is empty. We see here 0, 1, 2, thus by succession N, thence Z, Q, R, and even C (considered as a set of vectors of rotation from the polar axis). We are not leaping from Math to induction, or objective morality, we are laying out further cases and classes. As was earlier explored, induction and analogy, as well as causality turn out to flow from distinct identity and properties in common, even when there is distinction. LOI is actually extremely powerful, once recognised as saying A is itself i/l/o its core characteristics that give it its distinct, specific, particular identity. Two neighbouring peas in the same pod are distinct, each being itself and not another. By contrast the Evening and Morning stars turned out to be a common planet. See linked. For moral knowledge, we exploit the inextricable entanglement of reasoning with its moral government through duties to truth, right reason, prudence, fairness etc. That is, the IS-OUGHT gap HAS to be bridged, which is only possible at world-root. Our existence as a rational responsible community is implicated. We are looking at inescapable truths of moral character. In that context, for human community, the self evident case of evil exploitation and destruction of a child for pleasure sets up a framework for drawing out the law of our nature, with broad implications. See linked. So, no, there is no leap beyond reason or prudence involved. And that step was addressed already, this OP is in effect setting a further stage, of how we go about reasoning in the face of crooked yardsticks backed up by power and influence. Thus the silly example that riffs on the fable of the Holy Roman Emperor's invisible new clothes. KF kairosfocus

KF, Fair enough. FTR, I accept the LOI, LNC, LEM. Regarding the bit about {A | ~A}, I don't follow how this "sets up" things such as the real number system. However I do accept that all those mathematical structures exist, and we can use mathematics to understand the world, at least to some extent. I'm not a radical skeptic, relativist, or subjectivist, fortunately. How you make the leap from logic and mathematics to morality and so forth escapes me. I suggest we leave that for the 2nd or 3rd step. I believe that I am a reasonable person and understand the material world as well as the average layman. Is there anything you mention above which should cause me to change my views? And are we still dealing with the same level of certainty we have about the truth of the proposition "error exists"? daveS

DS, it has already been explicitly laid out, several times from the OP on. Let me clip the OP:

So, now, let us lay on the table a key concept: there are self-evident first truths (including inescapably true claims) that can and do serve as plumb-line tests for various truth claims. And thus, such truths can allow us to sift through various worldview or ideological alternatives and schools of thought. Which then allows us to think, decide and act with greater soundness. For simple example the Josiah Royce proposition, E = error exists, is undeniably true. To see that, try to deny it, ~E. That in effect claims it is error to propose E. So E must be true. [it's not a matter for opinions and agree/disagree, this is an established undeniably true point of knowledge] Is this a trivial result? No, as E is an example of self-evident truth, of truth, of objectively warranted truth (thus knowledge), indeed of truth warranted to undeniable certainty (thus certain knowledge). Such immediately sweeps away radical skepticism, relativism and subjectivism, as well as a raft of linked common notions. . . . . [summary on the core, triple first principles of reason, LOI, LNC, LEM] This is again, hardly a trivial result. Ever so much of the modern skepticism towards reasoned thought pivots on dismissiveness towards precisely these three laws of thought. Where, BTW, Quantum Physicists rely on just these laws in order to do their work. [see the weak argument correctives] Similarly, if we look at the world partition W = {A|~A} we see that A is itself, a unit distinctly different from the complex unity ~A, thus we find unity and duality. Where too the partition is empty and there is nothing in W but outside A and ~A, thus, nullity. This sets up the natural numbers, integers, rationals, reals, continuum, and even by using vector rotation, complex numbers. That is a non-trivial consequence. [--> framing a lot of the logic of structure and quantity as embedded in the fabric of any possible world, discussed previously] Likewise, identity and the logic of being allow us to see how inductive reasoning and causality can be grounded. [links in OP, the legitimacy of sound analogy is also implied] So, too, as arguably there are self-evident, plumb-line moral truths,[link in OP] moral forms of radical skepticism, relativism and subjectivism, as well as a raft of linked common notions are also swept away. This re-opens the issue of intelligible laws of our morally governed nature, framing thought, speech, behaviour and law . . .

KF kairosfocus

KF, I think everyone here accepts that 'error exists'. What's the next step in the sequence? daveS

H, until people are willing to acknowledge objective truth and knowledge, principles of reason etc, it is futile to try to assume that acceptance and then build up consensus on what propositions are true, what ones are necessarily true, and how we may then frame reliable, credible bodies of knowledge. In short, we need plumb-lines and sound yardsticks, then we can address substance. KF kairosfocus

DS, a modest effort but with rather relevant consequences given common perceptions regarding truth, knowledge, principles of reason, morality etc. In short, a case of a plumb-line. KF kairosfocus

Thanks, Dave. Actually we did discuss the question of the nature of the existence of abstractions last fall: sorry you missed it.:-) But obviously the important question is which propositions are true, which, because some propositions can be false, always involves some work, as I commented on in in my third-to-last paragraph at 19. hazel

hazel, I also believe that establishing that false propositions exist is a rather modest achievement. We all have known that from a very young age, I would wager. Well, perhaps a philosopher will jump in soon and question whether propositions actually exist, but I don't anticipate that. daveS

H, the issue was clearly identified: establishing that there are knowable truths about the world, knowable to in this case certainty. In a world where there are such things as popular notions of hyperskepticism and more sophisticated radical relativism or subjectivism, that is already a significant corrective. It shows that such things, as a bloc, are falsified. If your worldview implies that truth beyond perception or opinion is meaningless or impossible, this is an answer. If your view is that truth is radically relative, this too is decisive. Thus, such serves to re-open minds to objective truth, warrant, knowledge. Of course that one of the pivotal certainties is that error exists, that should serve as a warning, leading to due prudence. This also highlights that naturally certain truths serve as means to correct crooked yardsticks, which are out there -- as was just exemplified. KF kairosfocus

Hi Dave. At 11, you write, "Ok, we’ve established that false propositions exist." My question is, now what, and to some extent, so what? At 6, I wrote, “The phrase “error exists” doesn’t distinguish between logical error and error about the world, and your statement does,” and you agreed at 8. I think this is a critical distinction, because we ascertain whether logical propositions are false and whether propositions about the world are false, in different ways. Kf says the distinction is not important if all we want to do is claim that “the point that error exists is undeniably true”, but I don’t see that much more follows. Of course, as my moose/elk example, shows we can make errors in propositions about the world, and of course propositions in logic can be false: kf gave a classic example involving if P then Q. But the mere fact that such errors exists doesn’t tell us anything about what is actually true, either logically or about the world: it just warns us that we can be wrong. kf does say the fact “that errors exist, implies that certainly knowable and known truths exist, and can be warranted beyond responsible doubt.” This is true, but it offers no guidance by itself as to what statements are true, either logically or about the world: those have to be found on their own merits. Furthermore, those merits are established in different ways and have a different kind of truth value, depending on whether they are in the domain of pure logic or in the domain of knowledge about the world. And last, logical propositions are either true or false–there is no question of “reasonable doubt”, but in respect to propositions about the world, what constitutes “reasonable doubt” adds a whole new element to judging whether a proposition is true or false. So, in summary, I don’t see any huge significance in establishing the proposition that false propositions exist. I'd be interested in your thoughts on this, if you're interesting in returning and replying, (which I fully understand you might not want to do.) hazel

The Benevolent and Protective Order of Deer? ET

Everyone has his or her pet views or reality, but you'll find out when you die. (Or not.) --Homer Simpson mike1962

Oops: How about Johnny says to his wife says, "Look, there's a moose, dear" and his wife says "No, that's an elk, dear." Will that work as an example? :-) I was thinking about my experiences in the Colorado Rockies where both mule deer and elk are common, and sometimes newcomers confuse the two. hazel

Johnny says, “Look at that deer”, and Billy correctly says,”No, that’s an elk.” Johnny was in error.

Except that both the American and European elk are both deer. :-) Bob O'H

ScuzzaMan, What is the next step in the sequence? daveS

"Which counts." Which is why some people refuse to take the next step in the sequence, and prefer to batter themselves senseless against the rock of the foundation, like a fly against a window, and with all the cognitive content thereof. They know. Most assuredly, they know. It isn't their being wrong that convicts them - it is that they know they're wrong, and persist anyway. ScuzzaMan

DS, enjoy lunch. However, it's a little more than that. It is true that errors exist, implies that certainly knowable and known truths exist, and can be warranted beyond responsible doubt. That cuts a wide swath across a world in which radical relativism and subjectivism are often put forth as though they were well-founded. Manifestly, they aren't. Which counts. KF kairosfocus

Ok, we've established that false propositions exist. I'm going to break for lunch. daveS

DS, in our discussions, self-evident is used to describe a truth T that once stated to one of appropriate experience and insight will be seen as true, as necessarily true and as true on pain of patent absurdity on attempted denial. We can create a taxonomy of cases (especially on what sort of absurdity results) but the point is sufficiently clear and crisp to be useful. As a classic example, that one is conscious is manifestly so to oneself and to try to deny it instantly lands in the absurdity: and so who is aware of attempted denial? Likewise, attempts to deny the triple cluster of first principles of right reason immediately must use these in the attempt; they are inescapably true. And so forth. Such truths never amount to enough to form the basis of a worldview but serve admirably as plumb-lines to test the yardsticks we use. Which is crucial in avoiding the morass of manipulation, confusion, chaos, polarisation and outright deceit that seems more and more manifest in our dark day. KF PS: The parallel line postulate is often given as a counter instance. I note that such often involves context switching from a plane characterised by members of R x R. In that context, take y = mx + c, and set m constant for a family of lines. Vary c, c1 c2, c3 etc, these lines will have strictly equal separation for any given x value. This brings out the force of the point. I resort to this to draw out that the intuition that the axiom holds has in it an implicit space being spanned. kairosfocus

H & DS, that error exists simply asserts that the set that collects errors is non-empty. By contrast the set of dreams of rocks is decidedly empty. Error, takes the force, a claimed truth that misses the mark of accurately describing reality. An error in or of logic counts as an error and an error regarding attempted description of the world counts as an error. For example, P => Q, Q so P is an error as P => Q is not an equivalence, so to claim P follows from Q is an error of reasoning. I note, too that if a claimed state of affairs X leads to a contradiction as X => y AND X => ~y, then we see that a logical result establishes an ontological falsification through a logic of being result. For example, if X were a square circle it would have circularity AND squarishness, which stand in mutual contradiction of core characteristics so the logic of being entails the ontological result that a square circle is impossible of being. Likewise, the Moon is made of Green Cheese is an error if meant as a truth claim, but that is an error of fact. I recall a Sci Fi novel that pivoted on the Moon being a disguised massive starship . . . logically possible but factually obviously false. No distinction is necessary for the purposes of the point that error exists is undeniably true. KF kairosfocus

hazel, Yes, I definitely agree on the second point, and perhaps also on the first. The term "self-evident" seems to be used in a number of different conflicting ways, so I use it reluctantly. daveS

H, pardon, but self evident does not mean axiomatic. It does mean that a claim C is such that once stated to a person of suitable experience and understanding, s/he will immediately see that it is so, must be so and must be so on pain of immediate patent absurdity once denial is attempted. That obtains for this case. Similarly, 2 + 3 = 5 is self evident -- || + ||| --> ||||| -- but 971^2 = 942,841 is necessarily so but is not self evident. Besides, in schemes of thought axioms need not be self evident. KF kairosfocus

Dave, I like that you framed the statement in terms of logical propositions. I have some comments, and am interested in your response, if you wish. 1. That the proposition "False propositions exist” is true follows quickly and logically, as you and kf have shown. However, I think this misuses the word "self-evident", because this is a deduction (albeit a simple one), not an axiom. A=A is self-evident, and an axiom of logic. I think the word "self-evident" is used in so many ways that it is better to find other ways to describe things. 2. The phrase "error exists" doesn't distinguish between logical error and error about the world, and your statement does. If we just look at "error exists" as a statement about the world, then of course it does: Johnny says, "Look at that deer", and Billy correctly says,"No, that's an elk." Johnny was in error. The difference here is truth within the world of logic and truth about the world. hazel

DS, that is a useful reworking. F = false propositions exist. The denial ~F implies it is false to claim that a false proposition exists. This is or implies an assertion that is true or false, i.e. a proposition. So the latter implicitly affirms what it tries to deny, and is necessarily false -- that's stronger than one or the other is false. The same consequences follow, and doubtless there are many other cases that show the same thing. KF kairosfocus

VM, unfortunately, the silliness addressed is out there in our world. Actually, it has long been so, as say the prophet Isaiah challenged 2700+ years ago:

Isa 5:20 Woe (judgment is coming) to those who call evil good, and good evil; Who substitute darkness for light and light for darkness; Who substitute bitter for sweet and sweet for bitter! 21 Woe (judgment is coming) to those who are wise in their own eyes And clever and shrewd in their own sight! 22 Woe (judgment is coming) to those who are heroes at drinking wine And men of strength in mixing intoxicating drinks, 23 Who justify the wicked and acquit the guilty for a bribe, And take away the rights of those who are in the right! [AMP]

A few centuries later, Plato spoke in The Republic on the parable of the cave, whereby we can see a systematically distorted perception of reality that is resistant to correction. He also wrote the parable of the Ship of State, which I now excerpt:

It is not too hard to figure out that our civilisation is in deep trouble and is most likely headed for shipwreck. (And of course, that sort of concern is dismissed as “apocalyptic,” or neurotic pessimism that refuses to pause and smell the roses.) Plato’s Socrates spoke to this sort of situation, long since, in the ship of state parable in The Republic, Bk VI:

>>[Soc.] I perceive, I said, that you are vastly amused at having plunged me into such a hopeless discussion; but now hear the parable, and then you will be still more amused at the meagreness of my imagination: for the manner in which the best men are treated in their own States is so grievous that no single thing on earth is comparable to it; and therefore, if I am to plead their cause, I must have recourse to fiction, and put together a figure made up of many things, like the fabulous unions of goats and stags which are found in pictures. Imagine then a fleet or a ship in which there is a captain [–> often interpreted, ship’s owner] who is taller and stronger than any of the crew, but he is a little deaf and has a similar infirmity in sight, and his knowledge of navigation is not much better. [= The people own the community and in the mass are overwhelmingly strong, but are ill equipped on the whole to guide, guard and lead it] The sailors are quarrelling with one another about the steering – every one is of opinion that he has a right to steer [= selfish ambition to rule and dominate], though he has never learned the art of navigation and cannot tell who taught him or when he learned, and will further assert that it cannot be taught, and they are ready to cut in pieces any one who says the contrary. They throng about the captain, begging and praying him to commit the helm to them [–> kubernetes, steersman, from which both cybernetics and government come in English]; and if at any time they do not prevail, but others are preferred to them, they kill the others or throw them overboard [ = ruthless contest for domination of the community], and having first chained up the noble captain’s senses with drink or some narcotic drug [ = manipulation and befuddlement, cf. the parable of the cave], they mutiny and take possession of the ship and make free with the stores; thus, eating and drinking, they proceed on their voyage in such a manner as might be expected of them [–> Cf here Luke’s subtle case study in Ac 27]. Him who is their partisan and cleverly aids them in their plot for getting the ship out of the captain’s hands into their own whether by force or persuasion [–> Nihilistic will to power on the premise of might and manipulation making ‘right’ ‘truth’ ‘justice’ ‘rights’ etc], they compliment with the name of sailor, pilot, able seaman, and abuse the other sort of man, whom they call a good-for-nothing; but that the true pilot must pay attention to the year and seasons and sky and stars and winds, and whatever else belongs to his art, if he intends to be really qualified for the command of a ship, and that he must and will be the steerer, whether other people like or not-the possibility of this union of authority with the steerer’s art has never seriously entered into their thoughts or been made part of their calling. Now in vessels which are in a state of mutiny and by sailors who are mutineers, how will the true pilot be regarded? Will he not be called by them a prater, a star-gazer, a good-for-nothing? [Ad.] Of course, said Adeimantus. [Soc.] Then you will hardly need, I said, to hear the interpretation of the figure, which describes the true philosopher in his relation to the State[ --> here we see Plato's philosoppher-king emerging]; for you understand already. [Ad.] Certainly. [Soc.] Then suppose you now take this parable to the gentleman who is surprised at finding that philosophers have no honour in their cities; explain it to him and try to convince him that their having honour would be far more extraordinary. [Ad.] I will. [Soc.] Say to him, that, in deeming the best votaries of philosophy to be useless to the rest of the world, he is right; but also tell him to attribute their uselessness to the fault of those who will not use them, and not to themselves. The pilot should not humbly beg the sailors to be commanded by him –that is not the order of nature; neither are ‘the wise to go to the doors of the rich’ –the ingenious author of this saying told a lie –but the truth is, that, when a man is ill, whether he be rich or poor, to the physician he must go, and he who wants to be governed, to him who is able to govern. The ruler who is good for anything ought not to beg his subjects to be ruled by him [ --> down this road lies the modern solution: a sound, well informed people will seek sound leaders, who will not need to manipulate or bribe or worse, and such a ruler will in turn be checked by the soundness of the people, cf. US DoI, 1776]; although the present governors of mankind are of a different stamp; they may be justly compared to the mutinous sailors, and the true helmsmen to those who are called by them good-for-nothings and star-gazers. [Ad.] Precisely so, he said. [Soc] For these reasons, and among men like these, philosophy, the noblest pursuit of all, is not likely to be much esteemed by those of the opposite faction; not that the greatest and most lasting injury is done to her by her opponents, but by her own professing followers, the same of whom you suppose the accuser to say, that the greater number of them are arrant rogues, and the best are useless; in which opinion I agreed [--> even among the students of the sound state (here, political philosophy and likely history etc.), many are of unsound motivation and intent, so mere education is not enough, character transformation is critical]. [Ad.] Yes. [Soc.] And the reason why the good are useless has now been explained? [Ad.] True. [Soc.] Then shall we proceed to show that the corruption of the majority is also unavoidable, and that this is not to be laid to the charge of philosophy any more than the other? [Ad.] By all means. [Soc.] And let us ask and answer in turn, first going back to the description of the gentle and noble nature.[ -- > note the character issue] Truth, as you will remember, was his leader, whom he followed always and in all things [ --> The spirit of truth as a marker]; failing in this, he was an impostor, and had no part or lot in true philosophy [--> the spirit of truth is a marker, for good or ill] . . . >>

(There is more than an echo of this in Acts 27, a real world case study. [Luke, a physician, was an educated Greek with a taste for subtle references.] This blog post, on soundness in policy, will also help)

The problem is real, indeed the fable of the Emperor's new clothes is quite similar. In more recent times, one can argue that Fascism, Nazism and Communism were large scale political delusions, and more. Yes, it is silly to fall into systematic irrationality but that has been of concern for thousands of years and is unfortunately again highly relevant in our day. KF kairosfocus

I find it easier to understand if we rephrase the argument. Consider the proposition "False propositions exist". Then clearly either "False propositions exist" or its negation "False propositions do not exist" is a false proposition. daveS

I was gonna write something, but this is so SILLY that's not worth the effort. vmahuna

Logic and First Principles, 12: The crooked yardstick vs plumb-line self-evident truths kairosfocus