Notes on First Order Logic
... Induction Step Suppose that ϕ is (∀y)ψ. Since τ is substitutable for x in ϕ we have two cases: 1. x does not occur free in ψ. Then ((∀y)ψ)[x/τ ] is the same as (∀y)ψ. Furthermore s and s[x/τ ] agree on all free variables in (∀y)ψ. By Theorem ??, we have A, s |= (∀y)ψ[x/τ ] iff A, s |= (∀y)ψ iff A, ...
... Induction Step Suppose that ϕ is (∀y)ψ. Since τ is substitutable for x in ϕ we have two cases: 1. x does not occur free in ψ. Then ((∀y)ψ)[x/τ ] is the same as (∀y)ψ. Furthermore s and s[x/τ ] agree on all free variables in (∀y)ψ. By Theorem ??, we have A, s |= (∀y)ψ[x/τ ] iff A, s |= (∀y)ψ iff A, ...
Document
... An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true. ...
... An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true. ...
Unit 1
... There is another important fact about the mathematical language which should be noticed. For all numbers a, b, and c a(b + c) = ab + ac and for all numbers a, r, and x a(r +x) = ar+ax state precisely the same fact that is stated by (2)^. That is, the particular letters that are used in a statement o ...
... There is another important fact about the mathematical language which should be noticed. For all numbers a, b, and c a(b + c) = ab + ac and for all numbers a, r, and x a(r +x) = ar+ax state precisely the same fact that is stated by (2)^. That is, the particular letters that are used in a statement o ...
The Gödelian inferences - University of Notre Dame
... other formulas that also do, so that it makes sense to speak of ‘the SENTENTIAL FORMULA’ with this feature? The next clause invites further thought, still. Why, from the unprovability of a (or ‘the’) formula that states (in some way yet to be specified) that P is consistent, did Gödel infer that th ...
... other formulas that also do, so that it makes sense to speak of ‘the SENTENTIAL FORMULA’ with this feature? The next clause invites further thought, still. Why, from the unprovability of a (or ‘the’) formula that states (in some way yet to be specified) that P is consistent, did Gödel infer that th ...
Completeness in modal logic - Lund University Publications
... necessity and ◊ for possibility. Φ is true at a point p iff Φ is true at every point p can see. ◊Φ is true at p iff Φ is true at some such point. The points may be called possible worlds or just worlds, since that is the most illustrative way to think of them in modal applications. We can now define ...
... necessity and ◊ for possibility. Φ is true at a point p iff Φ is true at every point p can see. ◊Φ is true at p iff Φ is true at some such point. The points may be called possible worlds or just worlds, since that is the most illustrative way to think of them in modal applications. We can now define ...
