Subminimal Logics and Relativistic Negation Satoru Niki School of - - PowerPoint PPT Presentation

Background Logics with Relativistic Negation Outlook

Subminimal Logics and Relativistic Negation

Satoru Niki

School of Information Science, JAIST

March 2, 2018

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook

Outline

1

Background Minimal Logic Subminimal Logics

2

Logics with Relativistic Negation Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

3

Outlook

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Minimal Logic Subminimal Logics

Outline

1

Background Minimal Logic Subminimal Logics

2

Logics with Relativistic Negation Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

3

Outlook

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook 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 → ⊥.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook 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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook 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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook 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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook 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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Minimal Logic Subminimal Logics

Outline

1

Background Minimal Logic Subminimal Logics

2

Logics with Relativistic Negation Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

3

Outlook

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook 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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Minimal Logic Subminimal Logics

Graphical Representation

NPC N: (A ↔ B) → (¬A ↔ ¬B) NeFPC N + NeF: (A ∧ ¬A) → ¬B CoPC Co: (A → B) → (¬B → ¬A) MPC¬ N + An: (A → ¬A) → ¬A Logic Negation Axiom(s) Question Is there a logic between MPC¬ and CoPC?

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

Outline

1

Background Minimal Logic Subminimal Logics

2

Logics with Relativistic Negation Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

3

Outlook

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

An−: A Weaker Version of An

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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

Sequent Calculus for An−PC

Definition (Sequent Calculus GAn− for An−PC) Axioms: Ax: p ⇒ p (R⊤: Γ ⇒ ⊤) Rules for positive connectives: Γ, A, B ⇒ C L∧: Γ, A ∧ B ⇒ C Γ ⇒ A Γ ⇒ B R∧: Γ ⇒ A ∧ B Γ, A ⇒ C Γ, B ⇒ C L∨: Γ, A ∨ B ⇒ C Γ ⇒ Ai R∨: (i ∈ {1, 2}) Γ ⇒ A1 ∨ A2 Γ, A → B ⇒ A Γ, B ⇒ C L→: Γ, A → B ⇒ C Γ, A ⇒ B R→: Γ ⇒ A → B Rules for negation: Γ, ¬A, A ⇒ B Γ, ¬A, B ⇒ A N: Γ, ¬A ⇒ ¬B Γ, ¬B, A ⇒ ¬A An− : Γ, ¬B ⇒ ¬A

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

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

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

A Characterisation of An−PC

Definition (classes F +/F −) F + ::= p|P1 ∧ P2|P ∨ A|A ∨ P|A → P|N → A F − ::= ¬A|N ∧ A|A ∧ N|N1 ∨ N2|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).

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

Graphical Representation

NPC N: (A ↔ B) → (¬A ↔ ¬B) NeFPC N + NeF: (A ∧ ¬A) → ¬B CoPC Co: (A → B) → (¬B → ¬A) An−PC N + An−: (A → ¬A) → (¬B → ¬A) MPC¬ N + An: (A → ¬A) → ¬A Logic Negation Axiom(s)

• All subminimal extensions of An−PC have relativistic negation.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

Outline

1

Background Minimal Logic Subminimal Logics

2

Logics with Relativistic Negation Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

3

Outlook

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

MPC¬ and Kripke Semantics

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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

An−PC and Kripke Semantics

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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

Completeness of An−PC

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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

Outline

1

Background Minimal Logic Subminimal Logics

2

Logics with Relativistic Negation Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

3

Outlook

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

Axiom LP

• 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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

LPPC

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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

Graphical Representation

NPC N: (A ↔ B) → (¬A ↔ ¬B) NeFPC N + NeF: (A ∧ ¬A) → ¬B CoPC Co: (A → B) → (¬B → ¬A) An−PC N + An−: (A → ¬A) → (¬B → ¬A) LPPC N + An− + LP: (A ↔ ¬A) → ¬A MPC¬ N + An: (A → ¬A) → ¬A Logic Negation Axiom(s)

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

Outline

1

Background Minimal Logic Subminimal Logics

2

Logics with Relativistic Negation Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

3

Outlook

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook Axiom An− and An−PC Semantics of An−PC Axiom LP and LPPC Some More Logics with Relativistic Negation

Countably Many Logics with Relativistic Negation

Formulas below set the maximal length of intuitionistic frames. Proposition (a result from intermediate logics) Let bd1 := p1 ∨ ¬p1, bdn+1 := pn+1 ∨ (pn+1 → bdn). Then F I bdi ⇔ 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 Gd1 := (p1 → ¬p1) → ¬p1, Gdn+1 := pn+1 ∨ (pn+1 → Gdn). Then F LP Gdi ⇔ W\G does not have chains of ≥ i worlds. With this it is easy to verify (via soundness), MPC¬ = LPPC + Gd1 LPPC + Gd2 . . . LPPC.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook

Future Directions

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)?

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook

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: 165-174.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook

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.

Satoru Niki Subminimal Logics and Relativistic Negation

Background Logics with Relativistic Negation Outlook

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.

Satoru Niki Subminimal Logics and Relativistic Negation