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Nonclassical First Order Logics: Semantics and Proof Theory

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Nonclassical First Order Logics: Semantics and Proof Theory Nonclassical first order logics: Semantics and proof theory Apostolos Tzimoulis joint work with G. Greco, P. Jipsen, A. Kurz, M. A. Moshier, A. Palmigiano TACL Nice, 20 June 2019 Starting point Question How can we define general relational semantics for arbitrary non-classical first-order logics? I What are the models? I What do quantifiers mean in those models? I What does completeness mean? 2 / 29 Methodology: dual characterizations q Non − classical algebraic Non − classical proof th: semantics first order models Classical algebraic Classical proof th: semantics first order models i 3 / 29 A brief recap on classical first-order logic: Language I Set of relation symbols (Ri)i2I each of finite arity ni. I Set of function symbols ( f j) j2J each of finite arity n j. I Set of constant symbols (ck)k2K (0-ary functions). I Set of variables Var = fv1;:::; vn;:::g. The first-order language L = ((Ri)i2I; ( f j) j2J; (ck)k2K) over Var is built up from terms defined recursively as follows: Trm 3 t ::= vm j ck j f j(t;:::; t): The formulas of first-order logic are defined recursively as follows: L 3 A ::= Ri(t) j t1 = t2 j > j ? j A ^ A j A _ A j :A j 8vmA j 9vmA 4 / 29 A brief recap on classical first-order logic: Meaning The models of a first-order logic L are tuples D D D M = (D; (Ri )i2I; ( f j ) j2J; (ck )k2K) D D D where D is a non-empty set and Ri ; f j ; ck are concrete ni-ary relations over D, n j-ary functions on D and elements of D resp. interpreting the symbols of the language in the model M. M j= > Always M j= ? Never M j= A ^ B () M j= A and M j= B M j= A _ B () M j= A or M j= A M j= :A () M 2 A M j= 8xA(x) () M j= A(d) for all d 2 D M j= 9xA(x) () M j= A(d) for some d 2 D. For a set of sentences Σ, we write Σ j= A if M j= A for every model M such that M j= B for all B 2 Σ. 5 / 29 Display Calculi I Natural generalization of Gentzen’s sequent calculi; I sequents X ` Y, where X and Y are structures: - formulas are atomic structures - built-up: structural connectives (generalizing meta-linguistic comma in sequents φ1; : : : ; φn ` 1; : : : ; m) - generation trees (generalizing sets, multisets, sequences) I Display property: Y ` X > Z X ; Y ` Z Y ; X ` Z X ` Y > Z display rules semantically justified by adjunction/residuation I Canonical proof of cut elimination (via metatheorem) 6 / 29 Quantifiers as adjoints Consider 8x : }(D × Dn) ! }(Dn), 9x : }(D × Dn) ! }(Dn) and π−1 : }(Dn) ! }(D × Dn) defined as: I 8 T f 2 j 2 g x(A) = d02D d D (d0; d) A I 9 S f 2 n j 2 g x(A) = d02D d D (d0; d) A −1 I πx (B) = D × B We have: −1 πx (B) ⊆ A () B ⊆ 8x(A) −1 9x(A) ⊆ B () A ⊆ πx (B) I Existential and universal quantification are the left and right adjoints respectively of the inverse projection map (Lawvere). 7 / 29 Algebraic properties of inverse projection maps Let M be a model of L. For every finite set F ( Var such that x < F we define maps (x)F; 9xF; 8xF such that (x)F is the inverse projection map, 8xF > # F[fxg F }(D ) o (x)F }(D ) > < 9xF then the following properties hold: (x)(A \ B) = (x)(A) \ (x)(B)(x)(A [ B) = (x)(A) [ (x)(B) (x)(DF n A) = DF[fxg n (x)(A)(x)(y)(A) = (y)(x)(A) (x)8y(A) = 8y(x)(A)(x)9y(A) = 9y(x)(A) 8 / 29 Algebraic semantics for first-order logic An heterogeneous L-algebra is a tuple H = (A; Q), such that I A = fAF j F 2 P!(Var)g; I Q = f(x)F; 9xF; 8xF; j x < F ( Varg; where for every F 2 P!(Var), AF is a complete Boolean algebra 8xF[fxg > AF[fxg o (x)F o AF > > 9xF[fxg such that (x)F is an order embedding and the following hold: (x)(a ^ b) = (x)(a) ^ (x)(b)(x)(a _ b) = (x)(a) _ (x)(b) (x)(−a) = −(x)(a)(x)(y)(a) = (y)(x)(a) (x)8y(a) = 8y(x)(a)(x)9y(a) = 9y(x)(a) 9 / 29 Logical connectives and types I Types will be named after the elements F 2 }!(Var). I A type LF contains a formula A iff FV(') = F. I A 2 LF[fyg () 8yA 2 LF I A 2 LFnfxg () (x)A 2 LF I Symbols for quantifiers and cylindrification for each x 2 Var: Structural symbols Qx ((x)) Operational symbols 9x 8x (x) (x) 10 / 29 Display Calculus Introduction rules for quantifiers and their adjoint: QxA ` F X X ` F A 9L 9R 9xA ` F X QxX ` Fnfxg9xA A ` F X X ` FQxA 8L 8R 8xA ` FnfxgQxA X ` F8xA X ` FnfxgY ◦M ((x))X ` F[fxg((x))Y ((x))A ` F X X ` F((x))A ·L ·R (x)A ` F X X ` F(x)A 11 / 29 Display Calculus Display postulates for quantifiers and cylindrification: QxX ` FnfxgY Y ` FnfxgQxX X ` F[fxg((x))Y ((x))Y ` F[fxgX Further adjunction rules: ((x))QxX ` FY X ` F((x))QxY X ` FY X ` FY Interaction rules: ((x))X; ((x))Y ` Z Z ` ((x))X; ((x))Y ((x))(X; Y) ` Z Z ` ((x))(X; Y) ((x))QyX ` Y Y ` ((x))QyX Qy((x))X ` Y Y ` Qy((x))X 12 / 29 Completeness (Canonical model) If Σ is consistent then it has a model. 13 / 29 Σ can be extended to a set Σ0 such that I Σ0 is a maximal consistent theory (an ultrafilter) I If 9xA 2 Σ0 then A(t) 2 Σ0 for some term t Define t ≡ s if and only if t = s 2 Σ0: I Let D = Trm= ≡ I Let RD(t) iff R(t) 2 Σ0 I Let M = (D; RD; f D; cD) Then M j= A if and only if A 2 Σ0. 14 / 29 Witnesses Problem f9xP(x); :P(t1; );:::; :P(tn);:::g. Solution(s) I Add infinite constants in the language, and construct the ultrafilter by “carefully” using the constants for witnesses. I Rename the variables in your set so that you have enough (infinite) variables unused. 15 / 29 Algebraic predicate semantics A heterogeneous L-algebra is a tuple H = (A; Q) where for every F 2 P!(Var), AF is a complete Heyting algebra/ distributive lattice/DLE/LE/etc::: 8xF[fxg > AF[fxg o (x)F o AF > > 9xF[fxg such that (x)F is an order embedding and a Heyting algebra/distributive lattice/etc::: homomorphism and the following hold: (x)(y)(a) = (y)(x)(a) (x)8y(a) = 8y(x)(a)(x)9y(a) = 9y(x)(a) 16 / 29 Completeness revisited Classical completeness If Σ is consistent then it is satisfiable. Completeness in weaker logics I If Σ is consistent then it is satisfiable I If ∆ is not provable then it is falsifiable I If Σ does not imply ∆ then there is a model that satisfies Σ and falsifies ∆. Filter-ideal pairs DL: Every disjoint filter ideal pair can be extended to a prime filter-ideal pair. Lattices: Every disjoint filter ideal pair can be extended to a maximal one. 17 / 29 A case in point: Intuitionistic logic I We let M := (W; ≤), where each element of W is a classical first-order model I u ≤ w implies that f : Dw ! Du is a homomorphism of models. I w j= 9xA(x) if and only if there is some d 2 Dw, w j= A(d) I w j= 8xA(x) if and only if for all w 2 W such that u ≤ w and for all d 2 Du, u j= A(d). Categorically A model is a functor from a poset W to the category of f.o. models and homomorphisms. The meaning of a formula (with one free variable) is a subobject of the model in this category of functors. 18 / 29 Canonical model: A story in multi-type I A is the algebra of the logic excluding infinite free variables I F is a prime filter of A I If 9xA 2 F then A(t) 2 F for some term t Define t ≡ s if and only if t = s 2 F : I Let D = Trm= ≡ I Let RD(t) iff R(t) 2 Σ0 D D D I Let MF = (D; R ; f ; c ) Define M = (W; ≤) I W = fMF j F is a prime filter of some Ag I MF ≤ MG if and only if G ⊆ F 19 / 29 A curiosity? (x)(A ! B) = (x)A ! (x)B if and only if (9xA(x))^B = 9x(A(x)^B) (x)(A n B) = (x)A n (x)B if and only if 8x(A(x) _ B) = (8xA(x)) _ B 20 / 29 Witnesses, counterexamples and completeness Question Why witnesses and not counterexamples? Why not require that if 8xA(x) is falsified then A(t) is falsified for some t? Lemma 1. Assume that (9xA(x)) ^ B = 9x(A(x) ^ B) and let (F; I) be a disjoint filter-ideal pair. Then (F; I) can be expanded to a filter ideal pair (F ; I) such that for all 9xA(x) < I there exists some A(xn) < I for some n.
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