12-logic - First Order Logic A logic is a formal language...

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First Order Logic A logic is a formal language with strings over the alphabet Σ : Σ= {∧ , , ¬ , ( , ) , , ,x,R 1 ,...,R k } A well-formed formula (WFF) is a string over this alphabet and can be defined inductively as follows. A string φ is a WFF if it is an atomic formula of the form R i ( x 1 ,...,x n ) with R i called a relation symbol, or has one of the following forms: φ 1 φ 2 , φ 1 φ 2 , ¬ φ 1 , where φ 1 and φ 2 are smaller formulas, or has the form x i [ φ 1 ] or x i [ φ 1 ] , where φ 1 is a smaller formula. Notes: Quantifiers bind variables within their scope (square brackets). If a variable is not bound in a formula then we call it a free variable . Formulas with no free variables are called sentences or statements . –p
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First Order Logic WFFs: R 1 ( x 1 ) R 2 ( x 1 ,x 2 ,x 3 ) x 1 [ R 1 ( x 1 ) R ( x 1 ,x 2 ,x 3 )] x 1 x 2 x 3 [ R 1 ( x 1 ) R 2 ( x 1 ,x 2 ,x 3 )] Observation: Only the last WFF above is a sentence. We say, for all x 1 there exist x 2 and x 3 such that R 1 ( x 1 ) and R 2 ( x 1 ,x 2 ,x 3 ) are true. –p
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We set this system up so that we can reason about sentences such as, 1. q, x, y [ p>q ( x, y > 1 xy 6 = p )] . (finitely many primes) 2. a, b, c, n [( a, b, c > 0 n> 2) a n + b n 6 = c n ] . (Fermat’s Last Theorem) 3. q p x, y [ p>q ( x, y > 1 ( xy 6 = p xy 6 = p + 2))] . (Twin Prime Conjecture) First order logic is a language rich enough to formalize mathematics, but as we will see, it has its limitations (c.f. Principia Mathematica , Whitehead and Russell, 1913). And it is rich enough to serve as a Turing-complete programming language. –p
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12-logic - First Order Logic A logic is a formal language...

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