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

York University- MATH 1090

1

04_TheoremCalculation

Overview

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

York University- MATH 1090 2 04_TheoremCalculation

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> (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

Steps in a theorem calculation

Some simple theorems

York University- MATH 1090 04_TheoremCalculation 9

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

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 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

Then ⊢ A.

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