But a sentence that yields a contradiction is false in every model — TonesInDeepFreeze
I've never understood why this is so. — Tate
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. — Tate
What are the consequences of not adopting that method? — 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. — TonesInDeepFreeze
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 could be that I'm getting mixed up between the principle of bivalence and the law of noncontradiction. — Agent Smith
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
The question of all questions is "is the imprecision a bug in language or a feature of reality?" — Agent Smith
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: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
Yes. That is the implication of my personal BothAnd Philosophy. Some apparent "paradoxes" result from viewing only one side of the same coin. :smile:My understanding of paraconsistent logic, from Graham Priest, is that things can contradict each other and still be true — Jackson
However, methinks this is misguided because the mathematical descriptions seem not to exhibit any inconsistencies whatsoever. — Agent Smith
"[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
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