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

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Let’s propose a silly example, that a certain Emperor (maybe, just before he went out in his new invisible clothes) decides that a certain crooked stick is now the standard of length, straightness, uprightness and accuracy, a crooked yardstick. Suddenly, what is genuinely such things will be deemed the opposite. And then, suppose that somehow he and his publicists persuade the general public to accept the new standard. Will they not then find that those backward fuddy duddies that hold up their old yardsticks are ignoramuses and obstacles to progress and harmony?

Are we then locked into a war of competing imposed definitions and redefinitions? (That would for sure be a manipulator’s paradise.)

That’s where a plumb-line might help:

A plumb-line

Here, we see something that is naturally straight and upright, which will then clearly correct the crooked yardstick. It will even vindicate the fuddy duddies, even though the progressives won’t like it.

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.

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.

Likewise, for any distinct A — say, a bright red ball on a table — we see that the world can be dichotomised W = {A|~A} thus showing the triple first principles of right reason, Identity, Non Contradiction and Excluded Middle:

Here, A is itself in light of its particular distinct characteristics. No x in W can be both A and ~A. Any x in W will be A or else ~A. These laws are inescapably certain, indeed, any argument to object to them must rely on distinct identity and its corollaries to make an intelligible point. A classic case in point is a remark by St Paul:

1 Cor 14: Yet even lifeless things, whether flute or harp, when producing a sound, if they do not produce distinct [musical] tones, how will anyone [listening] know what is piped or played? And if the [war] bugle produces an indistinct sound, who will prepare himself for battle? So it is with you, if you speak words [in an unknown tongue] that are not intelligible and clear, how will anyone understand what you are saying? You will be talking into the air [wasting your breath]! [AMP]

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.

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.

Likewise, identity and the logic of being allow us to see how inductive reasoning and causality can be grounded.

So, too, as arguably there are self-evident, plumb-line moral truths, 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. Indeed, Cicero is back:

—Marcus [in de Legibus, introductory remarks,. C1 BC]: . . . the subject of our present discussion . . . comprehends the universal principles of equity and law. In such a discussion therefore on the great moral law of nature, the practice of the civil law can occupy but an insignificant and subordinate station. For according to our idea, we shall have to explain the true nature of moral justice, which is congenial and correspondent [36]with the true nature of man. We shall have to examine those principles of legislation by which all political states should be governed. And last of all, shall we have to speak of those laws and customs which are framed for the use and convenience of particular peoples, which regulate the civic and municipal affairs of the citizens, and which are known by the title of civil laws.

Quintus [his real-life brother]. —You take a noble view of the subject, my brother, and go to the fountain–head of moral truth, in order to throw light on the whole science of jurisprudence: while those who confine their legal studies to the civil law too often grow less familiar with the arts of justice than with those of litigation.

Marcus. —Your observation, my Quintus, is not quite correct. It is not so much the science of law that produces litigation, as the ignorance of it, (potius ignoratio juris litigiosa est quam scientia) . . . . With respect to the true principle of justice, many learned men have maintained that it springs from Law. I hardly know if their opinion be not correct, at least, according to their own definition; for “Law (say they) is the highest reason, implanted in nature, which prescribes those things which ought to be done, and forbids the contrary.” This, they think, is apparent from the converse of the proposition; because this same reason, when it [37]is confirmed and established in men’s minds, is the law of all their actions. They therefore conceive that the voice of conscience is a law, that moral prudence is a law, whose operation is to urge us to good actions, and restrain us from evil ones. They think, too, that the Greek name for law (NOMOS), which is derived from NEMO, to distribute, implies the very nature of the thing, that is, to give every man his due. [–> this implies a definition of justice as the due balance of rights, freedoms and responsibilities] For my part, I imagine that the moral essence of law is better expressed by its Latin name, (lex), which conveys the idea of selection or discrimination. According to the Greeks, therefore, the name of law implies an equitable distribution of goods: according to the Romans, an equitable discrimination between good and evil. The true definition of law should, however, include both these characteristics. And this being granted as an almost self–evident proposition, the origin of justice is to be sought in the divine law of eternal and immutable morality. This indeed is the true energy of nature, the very soul and essence of wisdom, the test of virtue and vice.

So, then, are we willing to acknowledge the problem of crooked yardsticks and the value of plumb-line, self-evident truths in our thinking, arguing, deciding and doing? END

Comments
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
March 5, 2019
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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. KFkairosfocus
March 5, 2019
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Argument 3- The islanders kill the foreigner :cool:ET
March 5, 2019
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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
March 5, 2019
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Yes, I agree with your conclusions, including day n + 1.daveS
March 5, 2019
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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
March 5, 2019
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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
March 5, 2019
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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. KFkairosfocus
March 5, 2019
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BB, true enough for some aspects. As noted, more advanced technologies are Math-heavy or even Math-based. For example aeronautics and computing. KFkairosfocus
March 5, 2019
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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
March 4, 2019
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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
March 4, 2019
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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
March 4, 2019
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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
March 4, 2019
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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. KFkairosfocus
March 4, 2019
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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
March 4, 2019
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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? KFkairosfocus
March 4, 2019
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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
March 4, 2019
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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. KFkairosfocus
March 4, 2019
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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
March 4, 2019
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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
March 4, 2019
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Hazel “Does that seem like a reasonable statement to you?” Yes Vividvividbleau
March 4, 2019
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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
March 4, 2019
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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
March 4, 2019
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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. Vividvividbleau
March 4, 2019
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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
March 4, 2019
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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?. Vividvividbleau
March 4, 2019
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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
March 4, 2019
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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
March 4, 2019
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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
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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. Vividvividbleau
March 4, 2019
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