Subminimal Logics and Relativistic Negation

Transcription

1 School of Information Science, JAIST March 2, 2018

2 Outline 1 Background Minimal Logic Subminimal Logics 2 Some More 3

3 Minimal Logic Subminimal Logics Outline 1 Background Minimal Logic Subminimal Logics 2 Some More 3

4 Minimal Logic Subminimal Logics Languages Definition (L +, L, L ) We shall use the following propositional languages: L + ::= p A B A B A B L ::= p A B A B A B L ::= p A B A B A B A In L, we take A to be the abbreviation for A.

5 Minimal Logic Subminimal Logics Minimal/Intuitionistic Logic Definition (MPC, IPC ) MPC is the smallest set of formulas of L containing the axioms below. Plus: If A, A B MPC then B MPC (MP). Axioms A (B A); (A (B C)) ((A B) (A C)); A (A B); B (A B); (A C) ((B C) (A B C)); A B A; A B B; A (B (A B)). IPC in addition contains the axiom EFQ: A. in MPC behaves like a propositional variable.

6 Minimal Logic Subminimal Logics Negation and Contradiction Definition (MPC ) MPC is the smallest set of formulas of L containing the axioms below. Plus: If A, A B MPC then B MPC. Axioms A (B A); (A (B C)) ((A B) (A C)); A (A B); B (A B); (A C) ((B C) (A B C)); A B A; A B B; A (B (A B)); M: [(A B) (A B)] A Call the negation-less(l + ) fragment of MPC as PPC.

7 Minimal Logic Subminimal Logics Counter-intuitive Inferences Involving Negation Definition (EFQ, NeF) EFQ: (A A) B [for MPC ] NeF: (A A) B EFQ: holds in intuitionistic logic. NeF: holds in minimal and intuitionistic logic. They are seen as unsatisfactory from the criteria of: (Relevance) Premises and the conclusions are related. (Paraconsistency) Contradictions do not trivialise the logic.

8 Minimal Logic Subminimal Logics Paths to Subminimality This motivates the study of logics with a weaker negation. We can weaken MPC or MPC. MPC : no axiom for difficult to weaken MPC : has the axiom M amendable with weaker negation axioms Such axioms are called subminimal axioms, and the logics with them (defined over PPC) subminimal logics.

9 Minimal Logic Subminimal Logics Outline 1 Background Minimal Logic Subminimal Logics 2 Some More 3

10 Minimal Logic Subminimal Logics Known Subminimal Axioms Definition (Co, An, NeF, N) Colacito, de Jongh and Vargas (2017) studied the following subminimal axioms. Co: (A B) ( B A); An: (A A) A; NeF: (A A) B; N: (A B) ( A B); Proposition (Colacito (2016), Colacito et al.(2017)) (i) Co NeF, Co N (ii) An+N M (iii) Co A A Call PPC+N (Co) as NPC (CoPC); NPC+NeF as NeFPC. NPC is taken as the basic subminimal logic.

11 Minimal Logic Subminimal Logics Graphical Representation Logic Negation Axiom(s) MPC N + An: (A A) A CoPC Co: (A B) ( B A) NeFPC N + NeF: (A A) B NPC N: (A B) ( A B) Question Is there a logic between MPC and CoPC?

12 Outline Background Some More 1 Background Minimal Logic Subminimal Logics 2 Some More 3

13 An : A Weaker Version of An Some More Definition (An ) An : (A A) ( B A) We define An PC as NPC + An. Proposition (separating An PC from CoPC [N.]) (i) An PC Co; CoPC An. (ii) An PC CoPC. Hence CoPC is not maximal. Proposition (some properties of An PC [N.]) An PC A A; An PC A A.

14 Sequent Calculus for An PC Some More Definition (Sequent Calculus GAn for An PC) Axioms: Ax: p p (R : Γ ) Rules for positive connectives: Γ, A, B C Γ A Γ B L : R : Γ, A B C Γ A B Γ, A C Γ, B C Γ A i L : R : (i {1, 2}) Γ, A B C Γ A 1 A 2 Γ, A B A Γ, B C Γ, A B L : R : Γ, A B C Γ A B Rules for negation: Γ, A, A B Γ, A, B A N: Γ, A B An : Γ, B, A A Γ, B A

15 Some More Cut and Equivalence with Hilbert-system We will in addition consider the following rule. Definition (Cut) Γ A Γ, A B Cut: Γ, Γ B It is straightforward to establish the following equivalence: Proposition (equivalence with An PC [N.]) Γ An A if and only if GAn +Cut Γ A

16 A Characterisation of An PC Some More Definition (classes F + /F ) F + ::= p P 1 P 2 P A A P A P N A F ::= A N A A N N 1 N 2 P N (P F +, N F, A F + F ) Proposition (separating An PC from MPC [N.]) (i) If GAn +Cut Γ A and A F, then Γ has a formula in F. (ii) GAn +Cut A for any A; hence MPC An PC. To see the last part, recall e.g. M (p p). Negation in An PC is relativistic, in the sense of (ii).

17 Graphical Representation Some More Logic Negation Axiom(s) MPC N + An: (A A) A An PC CoPC N + An : (A A) ( B A) Co: (A B) ( B A) NeFPC N + NeF: (A A) B NPC N: (A B) ( A B) -All subminimal extensions of An PC have relativistic negation.

18 Some More Further Proof-theoretic Properties of An PC Cut turns out to be admissible in GAn : Proposition (N.) (i) If GAn +Cut Γ A then GAn Γ A (ii) An PC is decidable. As further consequences of cut-admissibility, We can show the disjunction property of An PC; The interpolation theorem holds for An PC, extending the result of Colacito (2016) on NPC.

19 Outline Background Some More 1 Background Minimal Logic Subminimal Logics 2 Some More 3

20 MPC and Kripke Semantics Some More Definition (Kripke semantics for MPC ) A minimal frame is a triple (W,, F). (W, ) is a poset. F W is an upward closed set; i.e. w F and w w implies w F. We have the following valuation of negation. - M, w A w w[m, w A w F] F denotes the set of worlds where all negations hold.

21 An PC and Kripke Semantics Some More Definition (Kripke semantics for An PC) An An -frame is a quadruple (W,, F, G). (W, ) is a poset. F, G W are upward closed subsets s.t. F G; We have the following valuation of negation. - M, w A w w[m, w A w F] w G G denotes the set of worlds where some negations hold. Thus G is a natural counterpart of F. G\F is the area where negations hold untrivially.

22 Completeness of An PC Some More The following properties hold with repect to the semantics. Proposition (completeness of An PC [N.]) Γ An A Γ An A Proposition (finite model property for An PC [N.]) An PC is weakly complete with respect to the class of finite An -frames.

23 Outline Background Some More 1 Background Minimal Logic Subminimal Logics 2 Some More 3

24 Axiom LP Background Some More Q. Is there a logic between MPC and An PC? Definition (LP) LP: (A A) A LP can be seen as expressing the liar s paradox. Proposition (class of LP-frames [N.]) Let F be an An -frame. Then: F LP w W [w G w w(w G\F)] The frame property says you will eventually arrive in G. If we take G =, then LP is not valid in the frame. Thus by soundness, LP is not a theorem of An PC.

25 LPPC Background Some More Definition (LPPC) We define LPPC := An PC + LP By the previous proposition, LPPC An PC. Proposition (N.) LPPC is sound and complete with the class of LP-frames. Any LP-frame with G W can refute An; so MPC LPPC. LPPC satisfies the disjunction property and the finite model property.

26 Graphical Representation Some More Logic Negation Axiom(s) MPC N + An: (A A) A LPPC N + An + LP: (A A) A An PC CoPC N + An : (A A) ( B A) Co: (A B) ( B A) NeFPC N + NeF: (A A) B NPC N: (A B) ( A B)

27 Outline Background Some More 1 Background Minimal Logic Subminimal Logics 2 Some More 3

28 Some More Countably Many Formulas below set the maximal length of intuitionistic frames. Proposition (a result from intermediate logics) Let bd 1 := p 1 p 1, bd n+1 := p n+1 (p n+1 bd n ). Then F I bd i W does not have chains of > i worlds. We can apply this to the length of chains in W \G in LP-frame. Proposition (N.) Let Gd 1 := (p 1 p 1 ) p 1, Gd n+1 := p n+1 (p n+1 Gd n ). Then F LP Gd i W \G does not have chains of i worlds. With this it is easy to verify (via soundness), MPC = LPPC + Gd 1 LPPC + Gd 2... LPPC.

29 Future Directions Background Is there a maximal subminimal logic with relativistic negation? How many logics are there between MPC and An PC? (Bezhanishvili, Colacito and de Jongh (2017) showed uncountably many exist between MPC and NPC.) How does our semantics correspond with the Kripke semantics of Colacito et al.(2017)?

30 Reference N. Bezhanishvili, A. Colacito and D. de Jongh (2017). A lattice of subminimal logics of negation. TbiLLC 2017: Twelfth International Tbilisi Symposium on Language, Logic and Computation. A. Colacito (2016). Minimal and Subminimal Logic of Negation. Master s Thesis. University of Amsterdam. A. Colacito, D. de Jongh and A.L. Vargas (2017). Subminimal negation. Soft Computing 21:

31 Reference J. van Benthem (1984). Correspondence Theory. in D. Gabbay and F. Guenthner (eds.), Handbook of Philosophical Logic, Volume II: Extensions of Classical Logic. Kluwer Academic Publishers. A. Chagrov and M. Zakharyaschev (1997). Modal Logic. Oxford University Press. D. van Dalen (2013). Logic and Structure. Fifth edition. Springer.

32 Reference K. Sano, T. Kurahashi, T. Usuba, H. Kurokawa and M. Kikuchi (2016). Proof and Truth in Mathematics: Modal Logic and the Foundation of Mathematics (in Japanese). Kyoritsu Shuppan. A.S. Troelstra and H. Schwichtenberg (2000). Basic Proof Theory. Second edition. Cambridge University Press. A.S. Troelstra and D. van Dalen (1988). Constructivism in Mathematics: An Introduction. Volume I, Elsevier.