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"P or not P" is a tautology of classical logic, but not of all logics. It is not a tautology of intuitionistic logic, for example. So, one approach would be to say that classical logic does not apply to unprovable propositions in mathematics. Indeed, intuitionists maintain that it does not apply to mathematics at all, since they hold that ...
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Wittgenstein was reviving Kant's old view that logical deduction only brings out what is implicitly thought in the premises. Of course, Kant had in mind Aristotle's term logic, which is roughly equivalent to the logic of one place predicates (monadic predicate calculus) in modern terms, and Wittgenstein had in mind Boole's logic of propositions (classes), ...
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As a matter of terminology, some logicians use 'tautology' as a synonym for a logical truth, while others restrict it to logical truths of the propositional calculus. I shall use the more general term logical truth.
For a given logic, such as classical logic, a logical truth is a proposition that comes out true under all circumstances, or all interpretations,...
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This scene seems to imply that Russell didn't view logic as tautologies.
Correct. Wittgenstein's view about "logic=tautologies" was grounded on propositional logic and truth table. Unfortunately, truth table is not applicable to predicate logic and thus valid predicate logic formulas are not tautologies in the propositional sense.
In addition, ...
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An interpretation applicable for classical logic is...
The system in which CH is (provably) undecidable is under specified with respect to statements involving the truth of CH.
If I set up a formal system:
Socrates is a man
All men are mortal
Cassius is a cat
Then the proposition "Cassius is mortal" is undecidable in this formalism. If we want a ...
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The classical discussion is in Aristotle where he points out the LEM (Law of the Excluded Middle) doesn't appear to hold for future statements. Mathematically, this began as a new current in mathematics by Brouwers intuitionistic logic. You might find it intetesting to look at Heyting algebras which is the analogue of Boolean algebras in this context.
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@jkusin
I empathize with you. I do not join sites like these because beliefs about mathematics and science are so incommensurable that almost anything one can say can be answered with a skeptical retort. I lost patience with it all long ago. I am grateful to the posters on this site like JR, conifold, Dcleve, and doubleknot. They, and a few others have given ...
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You are correct that formal logic provides a link between mathematics, and tangible things we can operate on mechanically (formal proofs).
The concept of a "formal proof" is an abstract, mathematical one, which is no more or less real than perfect circles or the integers. However, the concept of a formal proof is also designed to be something we ...
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In the example you give, exactly one of those things is true. Either CH is true or it isn't. The fact that we don't know and can't prove whether or not CH is true doesn't mean that it's neither true nor untrue, but rather just that we don't know and can't prove it. Something being unprovable or even unknowable does not change that it's either true or it isn'...
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It is true that one or the other holds. Given a specific model of ZFC, it is true that either CH or not CH hold in it. It being undecidable merely means that you can't prove which one is true from inside the model. You just don't have enough information to be able to prove one or the other. Undecidability has more to do with provability than it has with &...
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You are essentially stating the position of Intuionist mathematics. Intuitionism doesn't consider the Law of the Excluded Middle (p or not p) to be valid.
What Intuitionism amounts to is the claim that nothing in mathematics is either true or false, but merely derivable or not derivable from the axioms. This idea comes from the formalization and ...
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