Fundamentals of Logic

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1 Fundamentals of Logic No.4 Proof Tatsuya Hagino Faculty of Environment and Information Studies Keio University 2015/5/11 Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 1 / 17

2 So Far Propositional Logic Logical connectives Truth table Tautology Normal form Disjunctive normal form Conjunctive normal form Restricting logical connectives Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 2 / 17

3 Inference (Deduction) Using truth table to show the correctness of propositions Calculate the truth value from the truth value of propositional variables. Inference Infer new correct proposition from correct propositions already known Apply inference rules to propositions Infer A from premises B 1,..., B n Inference Rule Rule to infer correct proposition from correct premise propositions Example: From A and A B, infer B modus ponens or syllogium All men are mortal and Socrates is a man, therefore Socrates is mortal. Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 3 / 17

4 Axiom and Theorem Axiom Premises which we believe correct. There is only one straight line which goes through two different points. Parallel straight lines never meet. Theorem Propositions which are inferred from axioms using inference rules Proof is the inference steps of theorem The sum of internal angles of any triangle is 180 degrees. Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 4 / 17

5 Formal Logical Framework Framework for handling logic formally Framework for handling logical formulae Consist of axioms and inference rules Frameworks for Classical Propositional Logic: Hilbert framework (Hilbert style) Axiomatic framework Only one inference rule: modus ponens Natural Deduction by Gentzen NK framework (NK system) Close to ordinary (human) inference Sequent Calculus by Gentzen LK framework (LK system) Easy to formalize Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 5 / 17

6 LK Sequent LK system use sequent: A 1,..., A m B 1,..., B n Antecedent: A 1,..., A m Succedent: B 1,..., B n If A 1 to A m are true, at least one of B 1 to B n is also true. m and/or n may be zero. B 1,..., B n No antecedent At least one of B 1 to B n is true. A 1,..., A m No succedent If A 1 to A m are true, contradicts. At least one of A 1 to A m is not true. No antecedent, no succedent Contradiction. Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 6 / 17

7 LK Axiom and Inference Rules Axiom: Initial Sequent A A Inference rules for structure: (weakening-l) Γ A, Γ (weakening-r) Γ Γ, A (contraction-l) A, A, Γ A, Γ (contraction-r) Γ, A, A Γ, A (exchange-l) Γ, A, B, Π Γ, B, A, Π (exchange-r) Γ, A, B, Σ Γ, B, A, Σ Γ, A A, Π Σ (cut) Γ, Π, Σ (Γ,, Π, Σ are sequences of logical formulae) Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 7 / 17

8 Inference rules Inference rules for logical connectives: ( L1) A, Γ A B, Γ ( L2) B, Γ A B, Γ ( R) Γ, A Γ, B Γ, A B ( L) A, Γ B, Γ A B, Γ ( R1) Γ, A Γ, A B ( R2) Γ, B Γ, A B ( L) Γ, A B, Π Σ A B, Γ, Π, Σ ( R) A, Γ, B Γ, A B ( L) Γ, A A, Γ Initial sequent for propositional constants: ( R) A, Γ Γ, A Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 8 / 17

9 LK Proof Figure LK Proof Figure: Start from initial sequent and apply inference rules The bottom sequent is called end sequent of the proof figure. Example: Proof Figure A A A, A A, A A A A, A A A, A A A A End Sequent: A A When there is a proof figure of which end sequent is S, S is provable in LK. Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 9 / 17

10 Extending Inference Rules Applying inference rules to formula in the sequent other than left-most or right-most one. Using exchange rules, formula at any position can be moved to left-most or right-most position. Extend inference rules of contraction, weakening and logical connectives to formula at any position. When S can be inferred from S 1, S 2,..., S n using LK inference rules, we write: S 1 S 2 S n S The above inference is inferred in LK. Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 10 / 17

11 Meaning of Sequent Theorem: The followings are equivalent: 1 A sequent A 1,..., A m B 1,..., B n is provable in LK. 2 A sequent A 1 A m B 1 B n is provable in LK. 3 A formula A 1 A m B 1 B n is provable in LK. Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 11 / 17

12 Exercise (1) Show the proof figure of (A B) A B Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 12 / 17

13 Exercise (2) Show the proof figure of A B, A B A Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 13 / 17

14 Exercise (3) Show the proof figure of A B (A B) Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 14 / 17

15 Exercise (4) Show the proof figure of A A Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 15 / 17

16 Exercise (5) Show the proof figure of A A Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 16 / 17

17 Summary Inference Axiom Theorem LK System Sequent Initial and End Sequent LK Inference Rules Proof Proof Figure Tatsuya Hagino (Faculty of Environment and InformationFundamentals Studies Keio of University) Logic 2015/5/11 17 / 17

0.1 Equivalence between Natural Deduction and Axiomatic Systems

0.1 Equivalence between Natural Deduction and Axiomatic Systems Theorem 0.1.1. Γ ND P iff Γ AS P ( ) it is enough to prove that all axioms are theorems in ND, as MP corresponds to ( e). ( ) by induction