You lost me.

You know the derivation of Russell's paradox, right?

Assume EbAy(yeb <-> ~yey)
"There exists a set b such that for all sets y, y is a member of b if and only if y is not a member of y".

So beb <-> ~beb
"b is a member of b if and only if b is not a member of b".

So EbAy(yeb <-> ~yey) yields a contradiction.

But any sentence that yields a contradiction is false in every model, so EbAy(yeb <-> ~yey) is false in every model.

But a sentence that yields a contradiction is false in every model — TonesInDeepFreeze

I've never understood why this is so. It seems like an artificial definition of "false" Does it make sense to you?

I've never understood why this is so. — Tate

Per the valuation function for truth in models (the Tarski definition by recursion on formulas), every sentence is either true in the model or false in the model but not both. And per that function, the negation ~P of a sentence P is true in the model if and only if P is false in the model; and P is true in the model if and only if ~P is false in the model.

Now, suppose a contradiction P & ~P were true in a model. A conjunction is true in a model if and only if both conjuncts are true the model. So both P and ~P would be true in the model. But then P would be true in the model and false in the model, which is impossible.

Per the method of models (the Tarski method by induction on formulas), every sentence is either true in the model or false in the model but not both. And per that method of models, the negation ~P of a sentence P is true in the model if and only if P is false in the model; and P is true in the model if and only if ~P is false in the model.

Now, suppose a contradiction P & ~P were true in a model. Then P would be true in the model and false in the model, which is impossible. — TonesInDeepFreeze

This means that if we adopt the method of models, Russell's Paradox is impossible. What are the consequences of not adopting that method?

This means that if we adopt the method of models, Russell's Paradox is impossible. — Tate

That is incorrect. No matter about models, if you have inconsistent axioms, then you derive Russell's paradox. Then, it is merely an additional note, not confined to Russell's paradox or unrestricted comprehension, that any inconsistent axiom is perforce a non-logical axiom.

What are the consequences of not adopting that method? — Tate

The method of models is ubiquitous in mathematical logic. There are many theorems about models (the subject is called 'model theory') so I don't know how to say simply what the consequences are, since there are many consequences.

Three of the most famous consequences are the completeness (and soundness) theorem, the compactness theorem, and Lowenheim-Skolem. Those are sometimes considered to be the "Big Three Pillars" (my term) of first order logic.

[Edit: I overlooked that you said not adopting.]

The crucial consequence of not having a method of models is that we would need to find some other means of providing semantical interpretations for theories. The method of models is the way we say what the formal sentences mean.

That is incorrect. No matter about models, if you have inconsistent axioms, then you derive Russell's paradox. Then, it is merely an additional note, not confined to Russell's paradox or unrestricted comprehension, that any inconsistent axiom is perforce a non-logical axiom. — TonesInDeepFreeze

I understand, thanks for explaining that.

Also, I overlooked that you said "not adopting". So I added more response accordingly.

Also, aside from providing semantical interpretation, and myriad other result in model theory, we use models for consistency proofs, relative consistency proofs, and independence proofs (the independence of the axiom of choice and the independence of the continuum hypothesis most famously).

And Robinson's non-standard analysis was developed using model theory.

Anyway, languages (not just formal languages) have both syntax and semantics. Models are the ordinary semantics for languages in predicate logic. And logic itself is not just the study of proof but perhaps even more basically the study of entailment. And entailment is semantical in the sense that 'truth' is determined by the model theoretic semantics for a language.

Anyway, languages (not just formal languages) have both syntax and semantics. Models are the ordinary semantics for languages in predicate logic. And logic itself is not just the study of proof but perhaps even more basically the study of entailment. And entailment is semantical in the sense that 'truth' is determined by the model theoretic semantics for a language. — TonesInDeepFreeze

It's not intuitive to me that P & ~P is necessarily a false statement. I think it would be better to say it's a meaningless statement. Could it be that predicate logic is handling meaninglessness by calling it false?

You propose that there are closed well formed formulas that are meaningless (have no interpretation or the valuation function also has meaningless in its range). Mathematical logic does not have that presupposition, so it is not, in its own terms, taking meaninglessness as falsehood. Indeed, the notion that contradictions are false is the ordinary notion through the centuries of the subject of logic. And it facilitates the ordinary notion of entailment.

Of course, one is free to develop a logic in which contradictions are not false but instead valuations include meaningless in addition to true and false, or whatever multi-valued logic one wants to have. I'm only telling you how it happens to work in ordinary predicate logic in which the domain of the valuation function is the set of sentences and the range is {true false}. So every sentence is assigned truth or falsehood and not both per any given model. This is desirable in ordinary predicate logic, especially, as I mentioned, for facilitating the ordinary notions including the ordinary notion of entailment.

Anyway, the context of discussion was the mathematical paradoxes, especially Russell's paradox, and especially the other poster's wildly mistaken notion that classical logic is trivial (proves every formula) because it is inconsistent, unless it eschews the principle of explosion*. In that context, one would ordinarily take it that contradictions are false.

* Contrary to the other poster's foolishness, classical logic is not trivial (it does not prove every formula) and it is not inconsistent, and therefore it does not need to eschew the principle of explosion.

It appears that, from a cursory reading of the Wikipedia article on fuzzy logic, that there are some/many aspects of reality that are spectral in character in terms of how we interact with them. The spectrum in question is divided up into manageable chunks, the boundaries between them are not precise values. This vagueness is what fuzzy logic was developed to tackle. I suppose we could say that fuzzy logic is grey zone logic.

As for the law of noncontradiction (the LNC) and fuzzy logic, my view is that the former holds in the latter. Yes that truth value can be somewhere in between 0 (false) and 1 (true) but it can never be 1 & 0 at the same time, nor is it that a truth value is (say) 0.7 and not 0.7.

Paraconsistent logics, on the other hand, are, as you rightly pointed out, systems in which a proposition has the combined truth value of 0 and 1 (at the same time and in the same respect) i.e. that proposition is a bona fide contradiction.

So, as per my analysis, fuzzy logic isn't a paraconsitent logic. I could be wrong of course, I do hope I'm not though.

Your BothAnd system feels more like paraconsistent logic than fuzzy logic to me. Maybe it's the words "both" and "and" which indicates you want to reconcile thesis (yes/1) AND antithesis (no/0) by approving BOTH.

It could be that I'm getting mixed up between the principle of bivalence and the law of noncontradiction.

It could be that I'm getting mixed up between the principle of bivalence and the law of noncontradiction. — Agent Smith

It is not difficult.

excluded middle: P or not-P

non-contradiction: not(P and not-P)

bivalence: (P or not-P) and not(P and not-P)

So bivalence is just the conjunction of excluded middle and non-contradiction.

:smile:

By chance or design, in this month's Philosophy Now:

Paradox Lost

Bev: Right. Russell’s Paradox is not a paradox. The apparent paradox is merely the result of following through a form of words which purport to describe an actuality; but actually nothing is picked out from mathematical reality by the phrase ‘the set of all sets that are not members of themselves being self-membered or non-self-membered’.

Indeed, the notion that contradictions are false is the ordinary notion through the centuries of the subject of logic. And it facilitates the ordinary notion of entailment. — TonesInDeepFreeze

If you're talking about the LNC, only one interpretation of it says contradictions are false statements. Another interpretation is that it's impossible to believe a contradiction. But I think we're off topic now. Again, thanks for the explanations.

The question of all questions is "is the imprecision a bug in language or a feature of reality?" — Agent Smith

Is not the superposition of an elementary particle within quantum mechanics an existential paradox?

Is not quantum computing already underway?

Is not Schrödinger's Cat a thought experiment in paradox at the human scale of sensory experience?

Does Schrödinger's Cat have no impact upon the LNC?


Anton Zeilinger, referring to the prototypical example of the double-slit experiment, has elaborated regarding the creation and destruction of quantum superposition:

"[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ -- The Apple Dictionary

The absence of any such information is the essential criterion for quantum interference to appear.

Does not quantum mechanics declare the imprecision is a feature of reality?

@ucarr

Good that you broached the quantum mechanics (QM) topic. QM is frequently used as a source of examples of weirdness (logical hanky panky) in the world.

However, methinks this is misguided because the mathematical descriptions seem not to exhibit any inconsistencies whatsoever.

It's as if a book in language x (math) is perfectly coherent but the moment it's translated into another language (any natural language) it becomes incoherent (full of paradoxes).

That's my analysis of course; I'm neither a mathematician nor a physicist. All I have to offer is what I cobbled together from info I gathered from the internet.

There are true paradoxes e.g. The Grelling-Nelson Paradox and that means the LNC has to go, which in turn implies we need to switch from classical logic to some strain of paraconsistent logic.

The other alternative is to isolate true paradoxes like we do to Covid + people in the ongoing pandemic - damage control that is (vide ex falso quodlibet).

Am I making any sense?

If I'm not mistaken, there is work in combining formal paraconsistent logic with formal fuzzy logic. But fuzzy logic itself is not a formalization of paraconsistent logic. — TonesInDeepFreeze

Perhaps, I should have prefaced that personal opinion with "it seems to me, that . . .". Before Agent mentioned it, I had never heard of "paraconsistent logic". But a quick Wiki review sounded like a description of Fuzzy Logic, which I was already familiar with. For my general purposes, I prefer the more colloquial and less technical-sounding term. From my layman's perspective, both terms seem to reflect the Uncertainty Principle of Quantum Theory, as applied to other fields of investigation. :smile:

I had never heard of "paraconsistent logic". — Gnomon

My understanding of paraconsistent logic, from Graham Priest, is that things can contradict each other and still be true

My understanding of paraconsistent logic, from Graham Priest, is that things can contradict each other and still be true — Jackson

Yes. That is the implication of my personal BothAnd Philosophy. Some apparent "paradoxes" result from viewing only one side of the same coin. :smile:


Both/And Principle :
My coinage for the holistic principle of Complementarity, as illustrated in the Yin/Yang symbol. Opposing or contrasting concepts are always part of a greater whole. Conflicts between parts can be reconciled or harmonized by putting them into the context of a whole system.
BothAnd Blog Glossary

Yes. That is the implication of my personal BothAnd Philosophy. Some apparent "paradoxes" result from viewing only one side of the same coin. — Gnomon

This is also Hegel's dialectic. In simple terms, all opposites are defined by a shared property.

Niiiice! :grin:

What's paradoxical about the twin paradox? Wouldn't it be a paradox if the twin astronaut returned with the same age to his brother on Earth? Is the paradox time dependent?

What's paradoxical about the twin paradox? Wouldn't it be a paradox if the twin astronaut returned with the same age to his brother on Earth? — Hillary

:up:

I wanna become a "Disciple of Gill"...

I don't think anyone has mentioned dialetheism:

A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true. If falsity is assumed to be the truth of negation, a dialetheia is a sentence which is both true and false.

Dialetheism is the view that there are dialetheias...dialetheism amounts to the claim that there are true contradictions.

Examples include: Russell's paradox, and the liar paradox.

However, methinks this is misguided because the mathematical descriptions seem not to exhibit any inconsistencies whatsoever. — Agent Smith

If I present a proof that a certain claim is false, would you say my proof is invalid because it's not also false?

"[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ -- The Apple Dictionary — ucarr

Is not the above statement telling us that no existential paradox can be experienced empirically (i.e. isolated) because even the linguistic concept of paradox collapses the existential expression of paradox?

I have the feeling, mr. Gill, that you and I, despite a considerable difference in age, we're quite alike. You try (tried...) to pull yourself to the top of the boulders by your arms (!), I try a kind of same thing with the boulders of physics. I have the chalk and technique. I know what's on the top. It's a beautiful view. But there are little other people. They rather stay safe down in town. So to speak.

we're quite alike — Hillary

:cool: