A Judgmental Formulation of Modal Logic

Transcription

1 A Judgmental Formulation of Modal Logic Sungwoo Park Pohang University of Science and Technology South Korea Estonian Theory Days Jan 30, 2009

2 Outline Study of logic Model theory vs Proof theory Classical logic vs Constructive logic Judgmental analysis of propositional logic Modal logic Summary 2

3 Model vs. Proof Model I Model theory Eg. assignment of truth values Semantic consequence A 1,, A n I C A 1,, A n C Proof theory Inference rules use premises to obtain the conclusion Syntactic entailment A 1,, A n C 3

4 Classical Logic Every proposition is either true or false. Concerned with: "whether a given proposition is true or not." Tautologies in classical logic 4

5 Constructive Logic We know only what we can prove. Concerned with: "how a given proposition becomes true." Not provable in constructive logic 5

6 This talk is about Constructive Proof Theory. Per Martin-Löf. On the meaning of the logical constants and the justifications of the logical laws, Nordic Journal of Philosophical Logic, 1(1):11-60, Frank Pfenning and Rowan Davies. A judgmental reconstruction of modal logic, Mathematical Structures in Computer Science, 11(4) , 2001.

7 Outline Study of logic Model theory vs Proof theory Classical logic vs Constructive logic Judgmental analysis of propositional logic Judgmental analysis of modal logic Summary 7

8 Judgments and Proofs A judgment = an object of knowledge that may or may not be provable. If there exists a proof, the judgment becomes evident. we know the judgment. Examples "1-1 is equal to 0" is true. "1 + 1 is equal to 0" is false. "It is snowing" is true. "1-1 is equal to 0" is false. 8

9 Inference Rules and Axioms A proof consists of applications of inference rules. J i are premises (1 i n). J is a conclusion. "If J 1 through J n (premises) hold, then J (conclusion) holds." If n = 0 (no premise), the inference rule is an axiom. 9

10 Proposition A statement such that we know what counts as a verification of it. If A is a proposition, we know how to check the validity of the proof of its truth. Example: "It is raining." Secondary notion 10

11 Proposition Without arithmetic rules, what is the meaning of "1-1 is equal to 0"? 11

12 Propositions Propositional Logic Judgments: 12

13 Natural Deduction System Introduced by Gentzen, 1934 For each connective,,,... introduction rule: how to establish a proof elimination rule: how to exploit an existing proof 13

14 Implication 14

15 Disjunction 15

16 Truth and Falsehood 16

17 What if Elimination Rules were Too strong Too weak 17

18 Elimination Rules are OK Local soundness Elimination rules are not too strong. Local completeness Elimination rules are not too weak 18

19 Local Soundness and Completeness 19

20 Definition Hypothetical Judgments Substitution principle 20

21 Inference Rules 21

22 Study of logic Outline Judgmental analysis of propositional logic Judgmental analysis of modal logic Modal necessity Modal possibility Lax modality O Summary 22

23 POSTECH 23

24 Modalities and A : necessarily A A : possibly A Spatial interpretation: A : everywhere A A : somewhere A Temporal interpretation: A : always A A : sometime A Modal Logic 24

25 Modal necessity First Judgments, Then Propositions.

26 Validity Judgment A valid A is valid if A is true at a world about which we know nothing, or at any world. Modal proposition A Introduction rule 26

27 New Forms of Hypothetical Judgments Definition Substitution principle 27

28 Modal necessity 28

29 Local Soundness and Completeness 29

30 Axiomatic Characterization (S4) 30

31 Modal possibility Again, First Judgments, Then Propositions.

32 Possibility Judgment A poss A is possibly true if A is true at a certain world. 32

33 Modal possibility 33

34 Local Soundness and Completeness 34

35 Axiomatic Characterization 35

36 Lax modality O Yet again, First Judgments, Then Propositions.

37 Lax Judgment A lax A is true under a certain constraint. 37

38 Lax Modality O 38

39 Local Soundness and Completeness 39

40 Axiomatic Characterization 40

41 Study of logic Outline Judgmental analysis of propositional logic Judgmental analysis of modal logic Modal necessity Modal possibility Lax modality O Summary 41

42 Applications, Type system for staged computation Type system for distributed computation O Type system for effectful computation Monad in functional language Haskell 42

43 Internalizing Normal Proofs Normal proofs Internalizing normal proofs using a modality Introduction and elimination rules 43

44 Uses two judgments Sequent Calculus Satisfies cut-elimination 44

45 Thank you.