Currently, one of the objections on the table to a demonstration on how certain structural and quantitative entities are implicit in there being a distinct possible world is the rejection, dismissal or doubting of certainty of conclusions. This again reflects one of the many problems with thought in our day.
Let’s add a quip, for those who doubt that warranted (as opposed to ill-advised) certainty is possible: are you CERTAIN that we cannot be justifiably certain?
Accordingly, I took the opportunity to comment in the fallacies discussion thread:
[KF, FDT 304:] One of the themes that keeps surfacing is “certainty,” which sets up the issues: warrant, knowledge, reliability, credibility, and responsibility.
Given that we ever so often use knowledge in a sense that is less than absolute, irrefutable certainty, as in science, I have put on the table that knowledge speaks of warranted, credibly true (so, reliable) belief. Obviously, this is normally provisional, but it leaves open cases where the degree of warrant and credibility are such that these claims are utterly certainly true and beyond doubt, save by the irresponsible. That is where self-evident truths and inescapable first principles of right reason live. Such includes mathematical truths of the order||| + || –> |||||.
Thus, we see that warrant comes in degrees and must attain to some degree of reliability that lends the credibility that leads to responsible belief. This may be less than certain, e.g. by and large scientific theories and models are not certain, they are open to correction on many grounds. Yes, Science is at a relatively low rung on the warrant ladder. Observations of science are another matter, they carry with them the credibility of witness, which can be morally certain.
So, certainty is now on the table, and just like warrant (from which it derives) it comes in degrees depending on cases, context and subject matter.
Moral certainty is a case where the grounds of warrant are sufficiently strong that one would be derelict of duty if one were to willfully treat something of that degree of credibility as though it were false, when something of great moment or value is on the table. For example, in a criminal case under Common Law jurisdictions, one must prove beyond reasonable doubt — this is a criterion of responsibility in the context of duty to justice. In commercial or civil matters, preponderance of evidence is a lower standard.
Beyond that everyday sense of certainty, lie the cases where in effect there is reason to believe that the judgement that x is the case has passed beyond room for reasonable, responsible doubt and is utterly unlikely to be reversed; something is true and is so grounded that there is no real room for doubt, but is not a necessary truth — one that must be so in this and all other possible worlds. Then, there is self-evidence, where x is so, is seen to be so by one with enough experience to understand the claim properly, and is such that the denial is immediately, patently absurd. That error exists, is a case in point, the attempt to deny instantly exemplifies that error exists. Likewise, one cannot be deluded that s/he is conscious, as to doubt is an act of consciousness.
Regrettably, we are so situated that it is impossible to build a whole worldview up from matters that are at least self-evidently so.
However, this degree of warranted certainty (and what lies beyond) serves to provide yardsticks and plumblines to test our worldview cores. For example, that error exists is undeniably true and warranted to self-evident certainty. This confirms that truth beyond opinion exists.Likewise, that some truths are intelligible, accessible by reason. As we observe and experience that error exists means that observation and experience can access truth. Similarly, we have warrant to undeniable certainty, so certain knowledge exists. If certain knowledge exists, knowledge (embracing weaker senses) exists also.
Further to this, beliefs, opinions, ideologies and worldviews that assume, argue, opine and assert that truth, or knowledge, or warrant or certain knowledge do not exist or that claims to such only serve “intolerance” and oppressor-classes — their name is Legion, are swept away wholesale as error. And yes, for cause I have the fell work of cultural marxism squarely in my sights, along with radical relativism and radical subjectivism.
Moreover, having warranted this point to certainty, I freely hold there is demonstrative warrant and that for cause opinion and rhetorical objection to the contrary avail nothing. Though in a politically correct era, many will take the vapours and will be frightened that I have announced a policy of right wing, Christofascist totalitarianism dressed up in Torquemada’s robes. That is how far ever so many in our civilisation have been misled.
That agit prop induced and/or mal-education induced reaction is unwarranted, the issue is to act responsibly and rationally in light of duty to truth, right reason, prudence, justice, etc.
However, there are higher yet degrees of warrant and certainty of knowledge.
Some truths are necessary, certain, intelligible and knowable to utter, incorrigible certainty and even absolute: the truth, the whole truth on a material matter, nothing but the truth on the material matter. Where, a necessary truth will be so in this or any other possible world. And what is more, many such truths are intelligible and warranted to similarly necessary certainty. Many core principles of reason and mathematics are of this order.
For relevant example, for a distinct world to be possible of existence, it must have in it at least one feature [A] such that it is different from all other possible worlds. We may then freely dichotomise W: W = {A|~A}. This already indicates that rationally intelligible structure and quantity are present in the fabric for such a world, we may readily identify here duality, unity (and complex unity in the case ~A), also nullity. The von Neumann construction then gives muscle to Peano’s succession from unity, and we have the natural counting numbers. From this, we may further recognise Z, Q, R, C and more.
Widening scope, and using reality in the widest sense, in reality (to include the case where there may be plural worlds as domains in reality) there will be some A, thus too ~A and a similar dichotomy obtains, R = {A|~A}. Instantly, A is itself i/l/o its core characteristics, perhaps a bright red ball on a table. This is the Law of Identity, LOI. Similarly, by the contrast and dichotomy, no x in R will be in A and in ~A, law of non contradiction, LNC. Thirdly (notice how counting numbers are implicit) any x in R will be in A or else in ~A, not in both or neither. Law of the excluded middle, LEM.
These three are inescapably true. We cannot prove them by appealing to something deeper, as to try to prove cannot but assume and implicitly use them. Likewise a claimed disproof or possible world in which they do not hold will on inspection be found to be implicitly using them. Such are the start-points for reasoning.
In short, we can see that the claim that we cannot be certain about anything is itself a claim to certainty, so it is advisable to instead explore the degrees of warrant we may obtain, for various types of cases. Once we do so, it becomes clear that there are degrees of warrant thus of certainty. Where, moral certainty is the first such degree, with self-evidence and necessary, undeniable or inescapable truths progressing upwards on the ladder. Such then allow us to have yardsticks and plumb-lines to test our reasoning and knowledge claims.
The common notion that associates certainty with oppression, intolerance etc and reacts by applying a hermeneutic of suspicion fails to properly address warrant and knowledge. END
PS: It is likewise worth pausing to point out the relevant demonstration on how considerable, rationally intelligible substance of structure and quantity are implicit in there being a distinct possible world on the table. Here, 257 in the same thread:
1: Consider reality, and within it some distinct entity, say a bright red ball on a table, B. Thus the rest of reality is the complement to B, ~B. Reality, R = {B|~B}
2: Immediately, B is itself (distinctly identifiable i/l/o its core, distinguishing characteristics), this is the fundamental law of thought, Law of Identity, which sets up the dichotomy and its corollaries.
3: Clearly, no x in R can be B AND ~B. Law of non-contradiction, a corollary.
4: Likewise, any x in R must be in B or in ~B, not between them or separate from them: B X-OR ~B, law of the excluded middle. The second corollary.
5: Now, ponder a possible world, W, a sufficiently complete description of a possible [coherent!] state of affairs in reality, i.e. in this or any other world that could be or is.
6: So far, we have set up a framework for discussion, including pointing out the key first principles of right reason that we must use so soon as we type out a message using distinct characters, etc. These are not provable, they are inevitable, inescapable and thus have a right to be presumed first truths of right reason.
7: Now, W, holds distinct identity, it is a particular possible world, different from all others. That is, if claimed entities W1 and W2 are not discernibly different in any respect, they are just different labels for the same thing W.
8: Notice, all along we are trafficking in statements that imply or assert that certain things are so or are not so, i.e. propositions and that relationship of accurate description of reality that we term truth.
9: All of these are not merely concrete particulars or mere labels, they are abstracta which are inevitable in reasoning. Indeed, the relationship of intentionality implicit in attaching a name is an abstractum, too.
10: Now, W is one of infinitely many possible states of affairs, and shares many attributes in common with others. So, we mark the in-common [genus] and the distinct [differentia].
11: So, we freely identify some unique aspect of W, A. W, then is: W = {A|~A}.
12: But already, we see rationally discernible abstract entities, principles and facts or relationship, quantity and structure; i.e. the SUBSTANCE of Mathematics. Namely,
13: first, that which is in W but external to A and ~A is empty, as is the partition: nullity.
14: Likewise, A is a distinct unit, as is ~A [which last is obviously a complex unity]. This gives us unity and duality.
15: So, simply on W being a distinct possible world, we must have in it nullity, unity and duality. These are abstract structural and quantitative properties embedded in the framework for W.
16: This is, strictly, already enough for the claim that there is an abstract substance of mathematical character that is necessarily embedded in any possible world, which is itself an abstract entity, being a collection of propositions. In at least one case such are actualised, i.e. it is possible to have an accurate summary of our world.
17: However, much more is necessarily present, once we see the force of the von Neumann succession of ordinals (which substantiates Peano’s succession), actually presenting the natural counting numbers starting from the set that collects nothing, which is itself an undeniable abstract entity:
{} –> 0
{0} –> 1
{0,1) –> 2
{0,1,2} –> 3
. . .
{0,1,2,3 . . . } –> w [first transfinite ordinal]
etc, without limit
18: We here have N. Define for some n in N, that -n is such that n + (-n) = 0, and we equally necessarily have Z. Again, rooted in the distinct identity of a world, we are studying, exploring, discovering, warranting (as opposed to proving), not creating through our culturally influenced symbolism and discussion.
19: Similarly, identify the ratio n:m, and we attain the rationals, Q.
20: Use power series expansions to capture whole part + endless sum of reducing fractions converging on any given value such as pi or e or phi etc, and we have the reals, R, thus also the continuum. Where, from Z on, we have has entities with magnitude and direction, vectors.
21: Now, propose an operation i*, rotation pivoting on 0 through a right angle. This gives us i*R, an orthogonal axis with continuum, and where for any r in R+, i*r is on the new [y] axis.
22: Now too, go i*i*r, and we find -r. That is we have that i = sqrt(-1), which here has a natural sense as a vector rotation. Any coordinate in the xy plane as described is now seen as a position vector relative to the origin.
23: We have abstract planar space, thus room for algebraic and geometrical contemplation of abstract, mathematically perfect figures. For instance consider the circle r^2 = x^2 + y^2, centred on o.
24: In its upper half let us ponder a triangle standing at -r [A] and r [B] with third vertex at C on the upper arc. This is a right angle triangle with all associated spatial properties, starting with angle sum triangle and Pythagorean relationships, trig identities etc. Between these two figures and extensions, the world of planar figures opens up.
25: Extend rotations to ijk unit vectors and we are at 3-d abstract “flat” space. All of this, tracing to distinct identity.
26: We may bring in Quaternions and Octonions, the latter now being explored as a context for particle physics.
27: The Wigner Math-Physics gap is bridged, at world-root level.
28: Similarly, we have established a large body of intelligible, rational entities and principles of structure and quantity implicit in distinct identity. Such are the substance we discover by exploration (which is culturally influenced) rather than invent.
29: Where it is an obvious characteristic of invention, that it is temporally bound past-wards, Until some time t, entity e did not exist. Then, after t, having been created, it now exists.
30: The above abstracta are implicit in the distinct identity of a world and so have existed so long as reality has. That is, without past bound. (It can readily be shown that if a world now is, some reality always was.)
I again point out that once a demonstration is on the table, only a counter-demonstration suffices to answer it. Unsubstantiated opinions to the contrary avail nothing. Perhaps, it is helpful to note Aristotle on pathos, ethos, logos:
Of the modes of persuasion furnished by the spoken word there are three kinds. The first kind depends on the personal character of the speaker [ethos]; the second on putting the audience into a certain frame of mind [pathos]; the third on the proof, or apparent proof, provided by the words of the speech itself [logos]. Persuasion is achieved by the speaker’s personal character when the speech is so spoken as to make us think him credible . . . Secondly, persuasion may come through the hearers, when the speech stirs their emotions. Our judgements when we are pleased and friendly are not the same as when we are pained and hostile . . . Thirdly, persuasion is effected through the speech itself when we have proved a truth or an apparent truth by means of the persuasive arguments suitable to the case in question . . . . [Aristotle, The Rhetoric, Book I, Ch. 2. Cf. summary with scholarly observations at http://plato.stanford.edu/entr…..-rhetoric/ and http://www.public.iastate.edu/…..index.html for a hypertext version of the book]
For, appeals to our passions, perceptions and felt reactions are of no more weight than the soundness of underlying judgements. Those to the credibility of an authority or presenter hold no more weight than the merits of the underlying case. It is therefore to the merits of fact and logic that we must always go. This should not be controversial, but that is where we now have reached.