Proper multi-type display calculi for classical and intuitionistic inquisitive logic

Transcription

1 1/18 Proper multi-type display calculi for classical and intuitionistic inquisitive logic Giuseppe Greco Delft University of Technology, The Netherlands TACL 2017, Prague Joint work with Sabine Frittella, Fan Yang and Alessandra Palmigiano

2 Outline 2/18 1 Inquisitive logic 2 A multi-type inquisitive logic 3 Intermezzo on proof theory 4 A multi-type sequent calculus for inquisitive logic

3 3/18 Inquisitive logic (Ciardelli, Groenendijk and Roelofsen 2009) Assertions Questions

4 3/18 Inquisitive logic (Ciardelli, Groenendijk and Roelofsen 2009) Assertions p: Moctezuma Xocoyotzin was the second Aztec emperor. q: Moctezuma defeated the Spanish invasion. Questions 10

5 3/18 Inquisitive logic (Ciardelli, Groenendijk and Roelofsen 2009) Assertions p: Moctezuma Xocoyotzin was the second Aztec emperor. q: Moctezuma defeated the Spanish invasion. Questions?p: Was Moctezuma Xocoyotzin the second Aztec emperor? 10

6 3/18 Inquisitive logic (Ciardelli, Groenendijk and Roelofsen 2009) Assertions p: Moctezuma Xocoyotzin was the second Aztec emperor. q: Moctezuma defeated the Spanish invasion. Questions?p: Was Moctezuma Xocoyotzin the second Aztec emperor??p := p p An information state (or team): a set of valuations

7 3/18 Inquisitive logic (Ciardelli, Groenendijk and Roelofsen 2009) Assertions p: Moctezuma Xocoyotzin was the second Aztec emperor. q: Moctezuma defeated the Spanish invasion. Questions?p: Was Moctezuma Xocoyotzin the second Aztec emperor??p := p p An information state (or team): a set of valuations Team Semantics (Hodges 1997)

8 3/18 Inquisitive logic (Ciardelli, Groenendijk and Roelofsen 2009) Assertions p: Moctezuma Xocoyotzin was the second Aztec emperor. q: Moctezuma defeated the Spanish invasion. Questions?p: Was Moctezuma Xocoyotzin the second Aztec emperor??p := p p An information state (or team): a set of valuations Team Semantics (Hodges 1997) Applied to Dependence logic (Väänänen 2007)

9 4/18 Inquisitive logic (InqL) Syntax φ ::= p φ φ φ φ φ φ ( φ ::= φ ) S = p iff for all v S, v(p) = 1 S = iff S = S = φ ψ iff S = φ and S = ψ S = φ ψ iff S = φ or S = ψ S = φ ψ iff for any T S: T = φ = T = ψ.

10 4/18 Inquisitive logic (InqL) Syntax φ ::= p φ φ φ φ φ φ ( φ ::= φ ) Team Semantics S = p iff for all v S, v(p) = 1 S = iff S = S = φ ψ iff S = φ and S = ψ S = φ ψ iff S = φ or S = ψ S = φ ψ iff for any T S: T = φ = T = ψ. Let S be a team (i.e., a set of valuations)

11 4/18 Inquisitive logic (InqL) Syntax φ ::= p φ φ φ φ φ φ ( φ ::= φ ) Team Semantics Let S be a team (i.e., a set of valuations). S = p iff for all v S, v(p) = 1 S = iff S = S = φ ψ iff S = φ and S = ψ S = φ ψ iff S = φ or S = ψ S = φ ψ iff for any T S: T = φ = T = ψ

12 4/18 Inquisitive logic (InqL) Syntax φ ::= p φ φ φ φ φ φ ( φ ::= φ ) Team Semantics Let S be a team (i.e., a set of valuations). S = p iff for all v S, v(p) = 1 S = iff S = S = φ ψ iff S = φ and S = ψ S = φ ψ iff S = φ or S = ψ S = φ ψ iff for any T S: T = φ = T = ψ

13 4/18 Inquisitive logic (InqL) Syntax φ ::= p φ φ φ φ φ φ ( φ ::= φ ) Team Semantics Let S be a team (i.e., a set of valuations). S = p iff for all v S, v(p) = 1 S = iff S = S = φ ψ iff S = φ and S = ψ S = φ ψ iff S = φ or S = ψ S = φ ψ iff for any T S: T = φ = T = ψ

14 4/18 Inquisitive logic (InqL) Syntax φ ::= p φ φ φ φ φ φ ( φ ::= φ ) Team Semantics Let S be a team (i.e., a set of valuations). S = p iff for all v S, v(p) = 1 S = iff S = S = φ ψ iff S = φ and S = ψ S = φ ψ iff S = φ or S = ψ S = φ ψ iff for any T S: T = φ = T = ψ. (Downward Closure) For every formula φ of InqL, T S = φ = T = φ

15 5/18 inquisitive logic (InqL) Syntax φ ::= p φ φ φ φ φ φ ( φ ::= φ ) Team Semantics Let S be a team (i.e., a set of valuations). S = p iff for all v S, v(p) = 1 S = iff S = S = φ ψ iff S = φ and S = ψ S = φ ψ iff S = φ or S = ψ S = φ ψ iff for any T S: T = φ = T = ψ. Def. A formula φ is said to be flat iff for all teams S, S = φ v S, v(φ) =

16 inquisitive logic (InqL) Syntax φ ::= p φ φ φ φ φ φ ( φ ::= φ ) Team Semantics Let S be a team (i.e., a set of valuations). S = p iff for all v S, v(p) = 1 S = iff S = S = φ ψ iff S = φ and S = ψ S = φ ψ iff S = φ or S = ψ S = φ ψ iff for any T S: T = φ = T = ψ. Def. A formula φ is said to be flat iff for all teams S, S = φ v S, v(φ) = Fact: A formula α is flat iff it is equivalent to some -free formula. iff α α 5/18

17 6/18 Theorem (Ciardelli, Roelofsen, 2009) The following Hilbert-style system of InqL is sound and complete: Axioms: 1 IPC axiom schemata 2 Kreisel-Putnam axiom schemata: 3 p p Rule: Modus Ponens ( φ (ψ χ)) ( φ ψ) ( φ χ) InqL = KP p p InqL is NOT closed under uniform substitution.

18 6/18 Theorem (Ciardelli, Roelofsen, 2009) The following Hilbert-style system of InqL is sound and complete: Axioms: 1 IPC axiom schemata 2 Kreisel-Putnam axiom schemata: 3 p p Rule: Modus Ponens ( φ (ψ χ)) ( φ ψ) ( φ χ) InqL = KP p p InqL is NOT closed under uniform substitution.

19 6/18 Theorem (Ciardelli, Roelofsen, 2009) The following Hilbert-style system of InqL is sound and complete: Axioms: 1 IPC axiom schemata 2 Kreisel-Putnam axiom: For any flat formula α, 3 α α for any flat formula α Rule: Modus Ponens (α (ψ χ)) (α ψ) (α χ) InqL = KP p p InqL is NOT closed under uniform substitution.

20 A multi-type inquisitive logic 7/18

21 8/18 Flat Type General Type

22 Fix a set V of propositional variables. 8/18 Flat Type B = ( (2 V ),,, ( ) c,, 2 V ) General Type

23 Fix a set V of propositional variables. 8/18 Flat Type General Type B = ( (2 V ),,, ( ) c,, 2 V ) A = ( (B),,,,, (2 V ))

24 Fix a set V of propositional variables. 8/18 Flat Type General Type B = ( (2 V ),,, ( ) c,, 2 V ) A = ( (B),,,,, (2 V )) S

25 Fix a set V of propositional variables. 8/18 Flat Type General Type B = ( (2 V ),,, ( ) c,, 2 V ) A = ( (B),,,,, (2 V )) S S := {T T S}

26 Fix a set V of propositional variables. 8/18 Flat Type General Type B = ( (2 V ),,, ( ) c,, 2 V ) A = ( (B),,,,, (2 V )) S S S S := {T T S} f f

27 Fix a set V of propositional variables. 8/18 Flat Type General Type B = ( (2 V ),,, ( ) c,, 2 V ) A = ( (B),,,,, (2 V )) S S S S := {T T S} f f f f {{v} v S} { }

28 Fix a set V of propositional variables. Flat Type B = ( (2 V ),,, ( ) c,, 2 V ) General Type A = ( (B),,,,, (2 V )) S S S S := {T T S} f f f f {{v} v S} { } Multi-type inquisitive logic: Flat α ::= p 0 α α α α General A ::= α A A A A A A 8/18

29 9/18 Multi-type inquisitive logic: Flat α ::= p 0 α α α α General A ::= α A A A A A A Axioms: (A1) CPC axiom schemata for Flat-formulas (A2) IPC axiom schemata for General-formulas (A3) ( α (A B)) ( α A) ( α B) (A4) α α Rule: Modus Ponens for formulas of both types

30 9/18 Multi-type inquisitive logic: Flat α ::= p 0 α α α α General A ::= α A A A A A A Axioms: (A1) CPC axiom schemata for Flat-formulas (A2) IPC axiom schemata for General-formulas (A3) ( α (A B)) ( α A) ( α B) Rule: (A4) α α Modus Ponens for formulas of both types

31 9/18 Multi-type inquisitive logic: Flat α ::= p 0 α α α α General A ::= α A A A A A A Axioms: (A1) CPC axiom schemata for Flat-formulas (A2) IPC axiom schemata for General-formulas (A3) ( α (A B)) ( α A) ( α B) Rule: (A4) α α Modus Ponens for formulas of both types (f f )

32 Intermezzo on proof theory 10/18

33 11/18 Canonical cut elimination, 1/2 Definition A sequent x y is type-uniform if x and y are of the same type. A (cut) rule is strongly type-uniform if its premises and conclusion are of the same type. Theorem (Canonical cut elimination) If a calculus satisfies the properties below, then it enjoys cut elimination.

34 12/18 Canonical cut elimination, 2/2 1 structures can disappear, formulas are forever; 2 tree-traceable formula-occurrences, via suitably defined congruence: same shape, same position, same type, non-proliferation; 3 principal = displayed 4 rules are closed under uniform substitution of congruent parameters within each type; 5 reduction strategy exists when cut formulas are both principal. Specific to multi-type setting: 6 type-uniformity of derivable sequents; 7 strongly uniform cuts in each/some type(s).

35 A multi-type sequent calculus for inquisitive logic 13/18

36 14/18 Structural and operational languages Flat α ::= p 0 α α α α Γ ::= α Φ Γ, Γ Γ Γ FX General A ::= α A A A A A A X ::= A Γ F Γ X ; X X > X

37 Structural and operational languages Flat α ::= p 0 α α α α Γ ::= α Φ Γ, Γ Γ Γ FX General A ::= α A A A A A A X ::= A Γ F Γ X ; X X > X Interpretation of structural connectives Flat connectives: Structural symbols Φ, Operational symbols (1) 0 ( ) ( ) General connectives: Structural symbols ; > Operational symbols ( ) Multi-type connectives: (f f ) Structural symbols F F Operational symbols (f ) (f) (f) 14/18

38 15/18 Cut rules Γ α α Cut Γ X A A Y Cut X Y Structural rules Flat type: Id p p Π Γ (, Σ) CG Π (Γ ), Σ Interaction between the two types: Π (Γ ), Σ Π Γ (, Σ) IG Γ F Γ bal X Y FX FY f mon F Γ Γ F f adj FX Γ X Γ d adj X FY X Y d-f elim X (Γ ) X F Γ > d dis FX, FY Z F(X ; Y ) Z f dis X F Γ > (Y ; Z ) X F Γ > (Y ; Z ) KP X (F Γ > Y ) ; (F Γ > Z )

39 Completeness 1/2 16/18 α α bal F α α F α α B B C C B C B ; C α (B C) F α > (B ; C) α α bal F α α F α α α (B C) (F α > B) ; (F α > C) α (B C) ( α B) ( α C) B B C C B C B ; C α (B C) F α > (B ; C) f adj, d-f elim KP

40 Completeness 2/2 17/18 α α α 0, α α, Φ 0, α Φ α (0, α) Φ (α 0), α Φ α, (α 0) CG α Φ α 0 bal F (α Φ) (α 0) 0 Φ d dis, f adj, d-f elim (α Φ) α 0 Φ def α 0 (α Φ) > Φ α (α Φ) > Φ α ((α Φ) Φ) F α (α Φ) Φ IG F α α d adj α α α α d dis d adj

41 18/18 Future work: intuitionistic inquisitive logic (Ciardelli, Iemhoff and Yang 2017) Fix an set V of propositional variables. Flat Type B = ( (2 V ),,,,, 2 V ) General Type A = ( (B ),,,,, (2 V )) S S S S := {T T S} f f f f {T T J, T S}