4- Theorem Calculation Ref: G. Tourlakis, Mathematical Logic , John - PowerPoint PPT Presentation

SC/MATH 1090 4- Theorem Calculation Ref: G. Tourlakis, Mathematical Logic , John Wiley & Sons, 2008. York University Department of Computer Science and Engineering 1 York University- MATH 1090 04_TheoremCalculation

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 ⊢ A  A f) 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  ⊢ B 1 ,  ⊢ B 2 , ...,  ⊢ B n Then  ⊢ A. and B 1 , B 2 , ..., B n ⊢ 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) ⊢ A  A  B  B f) ⊢     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