SC/MATH 1090

4- Theorem Calculation

Ref: G. Tourlakis, Mathematical Logic, John Wiley & Sons, 2008.

York University Department of Computer Science and Engineering

York University- MATH 1090 04_TheoremCalculation 1 Overview

• Logical axioms • Rules of inference • Theorem Calculations, or Proofs • Hilbert-style Proofs

York University- MATH 1090 04_TheoremCalculation 2 Logical axioms of Boolean Logic

York University- MATH 1090 04_TheoremCalculation 3 Axioms

• We will use the capital Greek letter "lambda" , , to denote the set of all logical axioms. • Note that since the logical axioms (shown in previous slide) are schemata,  is infinite. • All assumptions or hypotheses for a specific problem, are called special axioms or nonlogical axioms and are denoted by "gamma", . • Note that  is not fixed.

York University- MATH 1090 04_TheoremCalculation 4 Primary Rules of Inference

• The numerator shows the premises, hypotheses, or assumptions. • The denominator shows the conclusion or result of the rule. • The first rule is the rule of Equanimity or Eqn. • The second rule is the Leibniz rule or Leib.

York University- MATH 1090 04_TheoremCalculation 5 Theorem Calculations, or -Proofs

• Let be a given set of formulae (our assumptions) • A theorem-calculation (or proof) from  is any finite (ordered) sequence of formulae that can be written following these rules: 1. We may write a formula from  or  at any step 2. We may write the denominator of an instance of an inference rule, provided all formulae in the numerator (of the same instance) have been written in a previous step.

York University- MATH 1090 04_TheoremCalculation 6 Theorem

• Definition. (Theorems) Any formula A that appears in a -proof is called a -theorem. This is denoted by ⊢ A. – The above proof is said to prove A from . – If = (empty set), we write ⊢ A, and call A just a theorem or an absolute theorem, or logical theorem.

York University- MATH 1090 04_TheoremCalculation 7 Hilbert-Style Proof - framework

• To Prove ⊢ A: (1) ...... (2) ...... Steps in a theorem calculation (n) A

• Annotations explain the step written in a proof. • In a Hilbert style proof, conclusion appears at the last step (although by definition, it is not wrong to have more (unnecessary!) steps).

York University- MATH 1090 04_TheoremCalculation 8 Some simple theorems a) ⊢A  A b) A ⊢A c) A, A  B ⊢ B d) A  B ⊢ C[p:=A]  C[p:=B] e) A  B, B  C ⊢ A  C Transitivity f) ⊢ A  A

York University- MATH 1090 04_TheoremCalculation 9 Strengthening metatheorems!

• Metatheorem. (Hypothesis Strengthening) If ⊢ A and , then also  ⊢ A. – If ⊢ A, then also ⊢ A for any set of formulae . • Metatheorem. (Transitivity of ⊢) Assume we have

⊢ B1, ⊢ B2, ..., ⊢ Bn Then ⊢ A. and B1, B2, ..., Bn ⊢ A • Corollary. If  ∪ {A}⊢ B and also ⊢ A, then ⊢ B. • Corollary. If  ∪ {A}⊢ B and also ⊢ A, then ⊢ B.

York University- MATH 1090 04_TheoremCalculation 10 More tools for our toolbox a) B, A  B ⊢ A The other Eqn! b) ⊢    c) ⊢ ┬ d) C[p:=A], A  B ⊢ C[p:=B] Eqn + Leib merged e) ⊢ (A  (B  C))  ((A  B)  C) f) ⊢ A  A  B  B – ⊢     B  B – ⊢ A  A    

York University- MATH 1090 04_TheoremCalculation 11 Redundant True

• Redundant True Theorem: ⊢ ┬  A  A and ⊢ A  A  ┬

• (Redundant True) Metatheorem. For any  and A, ⊢ A iff ⊢ A  ┬. – Special case: A⊢ A  ┬ • Metatheorem. For any , A, and B, if ⊢ A and ⊢ B, then ⊢ A  B.

York University- MATH 1090 04_TheoremCalculation 12