Artificial Intelligence

An Introduction to Mathematical Logic and Type Theory: To by Peter B. Andrews

By Peter B. Andrews

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This creation to mathematical good judgment starts off with propositional calculus and first-order good judgment. issues coated contain syntax, semantics, soundness, completeness, independence, general varieties, vertical paths via negation common formulation, compactness, Smullyan's Unifying precept, common deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability.

The final 3 chapters of the e-book supply an creation to kind thought (higher-order logic). it's proven how numerous mathematical options may be formalized during this very expressive formal language. This expressive notation allows proofs of the classical incompleteness and undecidability theorems that are very stylish and simple to appreciate. The dialogue of semantics makes transparent the $64000 contrast among general and nonstandard versions that's so vital in realizing perplexing phenomena akin to the incompleteness theorems and Skolem's Paradox approximately countable versions of set theory.

Some of the various workouts require giving formal proofs. a working laptop or computer application referred to as ETPS that's on hand from the net allows doing and checking such exercises.

Audience: This quantity can be of curiosity to mathematicians, laptop scientists, and philosophers in universities, in addition to to computing device scientists in who desire to use higher-order good judgment for and software program specification and verification.

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Additional resources for An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Applied Logic Series)

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XpnAJ. _, p 1 . Otherwise, for each assignment cp let A'P = pf 1\ ... 1\ p~, where Pr = Pi if cppi = T = ,. _, Pi if cppi = F. CHAPTER 1. PROPOSITIONAL CALCULUS 50 Thus V'PA'P = T, and if 1/J is any assignment such that V,pA'P = T, then 1/Jpi = rppi for each i :::; n. Let A = V

Case 2a. A is pf, ... ,p~ 1- B pf, ... ,p~ 1- B :) rv Band VcpB = T. rvrv B by inductive hypothesis Sub: 1104 §12. SEMANTICS, CONSISTENCY, AND COMPLETENESS OF P pf, ... ,Ph f- "'"' B 37 MP Since A"' is "'"'B, this is the desired conclusion. Case 2b. A is "' B and VrpB = F. pf, ... ,Ph f- "' B But A'~' by inductive hypothesis is "' B, so this is the desired conclusion. Case 3a. A is [B V C] and Vrp B pf, .. ·, Ph f- B pf, · · . , Ph f- B :::> • B pf, ... , Ph f- B V C V = T. by inductive hypothesis 1112: 1106 C MP Case 3b.

This contradiction shows that there is no assignment which falsifies the w:ff, so it must be a tautology. Of course, when one applies the method above to certain examples, one may eventually reach a point where one must consider several possible cases, but one still avoids writing out the entire truth table. If one applies the §12. SEMANTICS, CONSISTENCY, AND COMPLETENESS OF 'P 35 method to a wff which is actually not a tautology, one will obtain an assignment which falsifies the wff, and thus see that it is not a tautology.

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