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) ...... <annotation> (2) ...... <annotation> Steps in a theorem calculation (n) A <annotation> • 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.