Logical Paradoxes — Stephen Read

Historian and Philosopher of Logic Stephen Read on the history of paradoxes, semantic paradoxes, and its direct connection to the foundations of mathematics

[Pimp-Master]

He does a good job keeping all the explanations straight without backtracking too much, stuttering, etc...

[Lordssodier]

It would be nice if he reached the discussion till Gödel... That would be serious science to me...

[jamestagge]

A MORE SPECIFIC RESONSE (to the card paradox of the cards assumes that each statement’s claim is by definition true or valid. This is where the paradox fails. In its form of presentation, it requires that the law of non-contradiction be violated in that the each statement must be both true and false at once. Also, if the first statement claims that the second statement is true, from what does the second statement derive its validity, but from the assumed truth of the first statement? In other words, the second statement’s truth is contingent upon the assumption that the first statement is valid. Given that, it cannot then turn around and deny the source of its own validation for in that it would destroy the very structure by which the paradoxical function manifests. This would be akin to claiming, “I think I am not thinking”, i.e., to deny within the scope of a proposition, by that proposition, the terms employed in its very expression. This is FUBAR logic and just more sophistry, though the cleverest I have yet seen.

[IndestructibleMandelbrot]

Another interesting thing about these self-referential paradoxes is, that they have a kind of duality to them. An expression like _"This sentence is false."_ can't really be pinned down to a truth value (everytime it seems to evaluate to _true_ it shifts to being _false_ and vice versa). It seems to be neither _true_ nor _false._

But then there's _"this sentence is true"_ which on first glance, seems less impressive, because you can just "try" if it's _true_ and you see: No contradiction with assigning _true_ to the sentence. But then, when you do the same thing with _false_ you will see that there's no contradiction as well. The sentence seems to be both _true_ and _false._

[jamestagge]

There are no conceptual contradictions, not true paradoxes. Materiality does not permit them nor does the realm of abstraction, itself a direct reflection of the structure of the material realm. Just as a physical entity cannot be here and there at once, no abstraction or concept can contradict itself in the ways suggested in the video. For example, one cannot appeal to truths to formulate a position which denies truth, i.e., one cannot employ terms to form a statement, which are then denied in its very expression. One cannot claim, “I think I am not thinking” and expect that it could ever be true.

Before proceeding, note that the speaker in this video and in all others who seem naïve enough to believe that these paradoxes are something other than sophistry, proceed in full respect of the architecture of the language used to define them and the structures they impose, then look away when a challenge to them is presented.

The original liars paradox from which that offered by Quine, “all Cretans are liars”, was like that derived from it, not a paradox at all. That all Cretans are liars is the premise of the paradox and must be assumed to be true for it to work. It then only appears that the Cretan who claimed all Cretans are liars could not be telling the truth because all other Cretans are liars as well as he. But if all Cretans are liars, it must be true, but again, he is a liar so it cannot be, thus the paradox. However, this only works “if” it is possible that each and every thing a Cretan “could” say could be a lie. But there are some things which can be said which by definition “cannot” be lies, such as “I am, or I am speaking, etc. So, if we retain the original assumption, which we are obliged to do, that in fact all Cretans are liars, which now we understand can only mean that they lie when it is possible, then we know that the Cretan calling all Cretans liars is telling that which is necessarily true (that which is by definition true)(that all Cretans lie when they can) thus the paradox is exposed for the sophistry it is.

Move then to Quines version, “this statement is false”. This is piffle. This statement is false conveys no information which can be judged as such. the terms statement in this paradox is a noun and the subject but a set definition of which there are no members. It is not a noun like rabbit which is self-contained with regard to meaning and requires no reference object. What is it about “this statement is false” which is false? In order to create the appearance of paradox, content by which it might be judged must be omitted. The consequence is that the statement defies the non-contradiction law of logic by making it both true and false at once. Additionally, the construction forced the use of the sentence object, “false” to be the cause and the effect at the same time, of the paradoxical function. This is one of the most sophomoric attempts at such deception I have ever seen. Quine claimed that this represented a “crisis of thought”. It is a crisis of nothing but perhaps his credibility.

Russell’s paradox was no less silly. In the semantic version, there is a town in which all the men must be shaved and there are two sets of men, those that shave themselves and those who do not but rather visit the barber who put up a sign which read that he only shaved those who do not shave themselves. The paradox arises when trying to decide whether the barber shaves himself or that he can’t, given the rule of his sign but that he has no choice but to shave himself, there being no barber for him to visit (but were there another barber, who would shave him, etc.) The resolution is so simple, i.e., that the two sets of men on which the paradox’s function depends are defined by the same three criteria. They are all men, they live in town and they must be shaved. These then cannot define them as members of two separates sets. That which does separate them is the one criterion peculiar to each, the relationship to shaving, i.e., that one shaves themselves and the other does not. So, “if” the logic by which these two sets of men were defined as such to begin with, is to be maintained throughout the paradox, which it must be, the barber must be found to “not be” a member of either of these sets, but rather of a separate third set of men of which he is the only member, those who shave others. Thus, were he to shave himself there would be no paradox. “If” you deny this, you deny the logic by which the first two sets of men are defined not as a single set but two, that upon which the paradox depends, and the paradox fails. If you accept this, the paradox fails. That is because it is not a paradox. It is sophistry.

There is no supposed paradox which cannot be similarly resolved to the satisfaction of all. This does “not” show any failing of the concepts of truth, but rather only the limited understanding of those, purported to be such authorities of philosophy and mathematics. I am beginning to doubt (I don’t quite fully understand it yet) even Goedel’s incompleteness theorem for his having deliberately brought self-reference into his field to try to demonstrate that there are some formulae which cannot be proved.

I can assure you, once you hear the terms infinity and/or self-reference employed in the “sciences”, you know you are being lied to or instructed by someone ill-equipped.

[shanejohns]

"This sentence is false." -- This happens with lots of self-referential thinking.

I actually had an advanced honors algebra teacher who gave quizzes every Friday. And he told the class on Thursday that he decided that if ANYONE gets a 10/10 100% on the Quiz, he'd give the entire class A+ for the semester. The quiz, properly answered, was all "True" for the first 9 true/false questions. Question #10 was: "At least one of the answers on this quiz is correctly answered 'False'?" I saw the problem, and I invented a new letter. It was essentially a T overlaid atop an F. Still didn't get it correct...

[mhc]

It seems that at the heart of logic there is a switch.

[abyisac]

@1:48 if my memory is correct (i read the book many years ago when i was a teenager, and could not quite meet the dictum that "one should read it atleast thrice: in youth, middle age, and in old age..."), it was not this, that was Sancho"s judgement. Sancho told that it is indeed a puzzling paradox, and in either case there is an element of error. However since saving a life is far greater/magnanimous than taking it away, his judgement was to allow him to go alive.

[arttu_hietala]

I'm glad I found this channel. I love your content!