Minimal Logic from the Equalizer of Classical Logic

Minimal Logic from the Equalizer of Classical Logic and Intuitionistic Logic C´esar Jesus Rodrigues CIDMA, Department of Mathematics Aveiro University Aveiro, Portugal

1 Introduction Sometime someone asked me what was my logic? I answered it was classical logic without proofs by contadiction. This logic is in fact minimal logic, classical logic minus the intuitionistic absurdity rule and the classical absurdity rule, cf. natural deduction presentation of these logics [20]. But I was thinking in constructive proofs, the type of proofs I try usually. But when no constructive proof exists I try to use nonconstructive reasonig. The nonconstrutive trend already appeared in architecture. Thus, in this case I use classical logic. Nowadays we know that calassical logic is constructive [7] and benefits from a Curry-Howard isomorphism [3, 10], properties that first were an exclusive of intuitionistic logic. The Curry-Howard isomorphism for classical logic involves a generalization of the λ-calculus, the so called λµ-calculus [16]. This calculus is the proof that type theory also appears in the context of classical logic. Thre motum of this paper is to define minimal logic from classical logic and intuitionistic logic, by appealing to the categorical concept of an equalizer [12]. In this context the definition of minimal logic from classsical logic is similar to its definition from intuitionistic logic. Just replace one logic by the other. It appeared that the equalizer of intuitionistic logic and classical logic is used in the definition of minimal logic, when these logics are presented as abstract logics in category Set. An abstract logic comes after the investigations of Tarski, and is a consequence operator in the context of a signature [19]. There is a similar concept in abstract algebraic logic, a deductive system, which is a consequence relation plus a signtature [2]. A generalization of an abstract logic is a π-institution [4, 5], which is the deductive component of a general logic [14], which has an instititution[8, 9] as the semantic component of the respective general logic.

2 Minimal Logic We took contact with minimal logic in a book on proof theory [20]. It didn’t present the mentioned logic. We thought minimal logic was intodiuced by pedagogical reasons. 1

Later we realized that it exites on their own. Minimal Logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson [11]. It is a variant of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does), but also the principle of explosion (ex falso quodlibet), rejected by the logic of inconsistent mathematics. Just like intuitionistic logic, minimal logic can be formulated in a language using →,∧,vee,⊥ (implication, conjunction, disjunction and falsum) as the basic connectives, treating ¬A as abbreviation for (A → ⊥). In this language it is axiomatized by the positive fragment (i.e. formulas using only →, ∧, ∨) of intuitionistic logic, with no additional axioms or rules about ⊥. Thus minimal logic is a subsystem of intuitionistic logic. Adding the ex falso axiom ¬A → (A → B) to minimal logic results in intuitionistic logic, and adding the double negation law ¬¬A → A to minimal logic results in classical logic. Minimal logic is closely related to simply typed lambda calculus via the CurryHoward isomorphism, i.e. the typing derivations of simply typed lambda terms are isomorphic to natural deduction proofs in minimal logic.

3 Equalizers in Logic The logics we shall address in category SET are taken as corresponding to propositional abstract logics (dialgebras) in the category of abstract logics, i.e. category of P, P dialgebras. The reason is that an equalizer of logics is defined over arrows. We sall define the equalizer of classical logic (C) and intuitionistic logic (I), which is monic (injective function) k, over set K, used in the definition of Minimal Logic: C

k

K → PX

→ I

→

PX

M = C ·k =I ·k Set K is defined as the kernel of logics C and I. K = kernel(C, I) = {ΓǫPX|C(Γ) = I(Γ)}

4 Dialgebra theory αS Given endofunctors G, F over Set a G, F-dialgebra is a function F S o GS . Set S is referred to as carrier of the dialgebra; function αS is the G, F-transition structure of the system. Dialgebras are related by homomorphisms, as defined below:

Definition 4.1 Let (X, p : GX −→ F X) and (Y, q : GY −→ F Y ) be dialgebras for functors G, F. A morphism connecting p and q is a function h between their carriers

2

such that the following diagram commutes: GX

p

/ FX

Gh

 GY

i.e.

q · Gh = F h · p

(1)

Fh q

 / FY

 A important concept in the theory of dialgebraic systems is that of bisimulation. We will adopt the formulation of bisimulation in [21]: Definition 4.2 (Bisimulation) A subset R ⊆ S × T of the cartesian product of S and T is called an G, F-bisimulation if there exists an G, F-transition structure αR : GR → FR such that the projections π1 and π2 from R to S and T are G, F-homomorphisms (the projections are a tabulation [6] in allegory and category Rel): G(S) o

G(π1 )

G(R)

G(π2 ) / G(T )

αS αR αT    / F(T ) F (S) o F(R) F(π1 ) F(π2 )  This definition generalizes the one for dialgebras and bisimulation for traditional algebras is a substitutive relation. The definition of a transition relation corresponding to a dialgebra δ is as follows: ←δ = (ǫF · δ) · ǫ◦G So, a dialgebra (X, δ : GX → FX), corresponds to the pair (X, ←δ : X → X), a pair formed by a set and a relation on it, which we can call an unlabelled transition system. In the sequel we define the arrow operator for any function f with appropriate type, as a generalization of the transition relation for dialgebras: f

← =

(ǫF · f ) · ǫ◦G

5 Studying Minimal Logic The minimal logic can be defined using intuitionistic logic as: M =I·k Going pointwise we have: 3

(2)

M φ = (I · k)φ = I(kφ) As φ ∈ K, Cφ = Iφ and φ in P X. We can translate this reasoning as the following logical formula: ∀φ ∈ P X.φ ∈ K ⇒ Cφ = Iφ If Cφ = Iφ then going pointfree C =I Minimal logic can be defined as, M φ = (I · k)φ = (C · K)φ Going poinfree we get the definition of minimal logic using classical logic: M =C ·k Set K is the set of formulas in P X, such that classical logic and intuitionistic logic agree. In practice, minimal logic is defined either as intuitionistic logic or classical logic over set of formulas K.

6 Transposition for Dialgebras Power transposition was defined in [1] using the folowing universal property f = ΛR ≡ ǫ · f = R and was generalized to transposition [15, 17], as given by the following more general universal property,

f = ΓF R ≡

∈F · f = R

cf. diagram

BO o

R

A

(3)

|| || ∈F | | |} | f FA which has some connections with universal coalgebra, e.g. f may be seen as a coalgebra if A = B and R as the corresponding transition relation. As a final generalization we consider transposition as given by 4

f = ∆F,G R

ǫF · f · ǫ◦G = R

≡

cf. diagram

BO o

R

ǫF FA o

f

AO ǫG

(4)

GB

which has some connections with universal dialgebra, e.g. f may be seen as a dialgebra if A = B and R as the corresponding transition relation. We stress that a coalgebra is a particular dialgebra, where functor G is such that: G = Id. First we shall prove the injectivity law of dialgebraic transposition, which depends on the cancellation law of dialgebraic transposition. Cancellation arises from (4) by substitution f := ∆G,F R: ǫF · ∆G,F R · ǫ◦G = R

(5)

Substitution f := ∆G,F S in (4) and cancellation (5) lead to the injectivity law, ∆G,F S = ∆G,F R ≡ S = R

(6)

The injectivity law implies that the transposition operator ∆G,F is an injective function between all transition relations and dialgebras. The transposition operator is an isomorphism since the arrow operator is defined for any dialgebra. We will present and prove these two more laws, involving the arrow operator: Lemma 6.1 (arrow laws) f ·Gh

f

← =← ·h

Fh·f

f

← = h· ←

(7) (8)

Proof: Membership ǫF is a natural transformation, as it is a lax natural transformation. We shall prove that converse of membership ǫ◦F is also a natural transformation: Fh · ǫ◦F = ǫ◦F · h as proved in the sequel. T RUE ≡

{ nat. transf. ǫF } f · ǫF = ǫF · Ff

≡

{ substitution f := h−1 }

5

h−1 · ǫF = ǫF · Fh−1 ≡

{ Equality of relations: R = S ≡ R ⊆ S ∧ S ⊆ R} h−1 · ǫF ⊆ ǫF · Fh−1 ∧ ǫF · Fh−1 ⊆ h−1 · ǫF

≡

{ monotonicity of converse: R ⊆ S ≡ R◦ ⊆ S ◦ } ǫ◦F · (h−1 )◦ ⊆ (Fh−1 )◦ · ǫ◦F ∧ (Fh−1 )◦ · ǫ◦F ⊆ ǫ◦F · (h−1 )◦

≡

{ Equality of relations: R = S ≡ R ⊆ S ∧ S ⊆ R} ǫF ◦ · (h−1 )◦ = (Fh−1 )◦ · ǫ◦F

≡

{ Fh−1 = (Fh)−1 , cf. [13]} ǫF ◦ · (h−1 )◦ = ((Fh)−1 )◦ · ǫ◦F

⇒

{ f = h−1 ⇒ f = h◦ } ǫF ◦ · (h◦ )◦ = ((Fh)◦ )◦ · ǫ◦F

≡

{ (R◦ )◦ = R} ǫ◦F · h = Fh · ǫ◦F

≡

{ trivial} Fh · ǫ◦F = ǫ◦F · h

Law (7): f ·Gh

≡

f

← =← ·h { arrow operator}

ǫF · f · Gh · ǫ◦G = ǫF · f · ǫ◦G · h { ǫ◦G is a natural transformation}

≡

ǫF · f · ǫ◦G · h = ǫF · f · ǫ◦G · h ≡

{ trivial} T RUE

Law (8): Fh·f

f

← = h· ←

≡

{ arrow operator} ǫF · Fh · f · ǫ◦G = h · ǫF · f · ǫ◦G

≡

{ ǫF is a natural transformation} h · ǫF · f · ǫ◦G = h · ǫF · f · ǫ◦G 6

≡

{ trivial} T RUE



7 Characterizing Minimal Logic by Characterizing Classical/Intuitionistic Logics Recall that we can define an homomorphism in the category of abstract logics, category of P, P -dialgebras, involving intuitionistic and classical logics [19]: CIP C · P h = P h.CCP C By Leibniz: ←−(CIP C ·P h) =←−(P h.CCP C ) Applying the arrow laws: ←−CIP C ·h = h· ←−CCP C

(9)

Function h is a functional bisimulation (as h is an homomorphism, a classical result in universal dialgebra, as it was in universal coalgebra) that involves intuitionistic logic and classical logic, up to isomorphism. The transposition operator is an isomorphism between dialgebras and corresponding transition relations: ∆(←−CIP C ) = CIP C ∆(←−CCP C ) = CCP C The functional bisimulation h is a witness of the relation of bisimilarity, which is an equivalence relation (9). An equivalence relation is the kernel of a (refinement) pre-order, which is tested by a meaningful corresponding equivalence. ←−CIP C ·h = h· ←−CCP C =←−CIP C ·h ⊆ h· ←−CCP C ∧h· ←−CCP C ⊆←−CIP C ·h (10) We can define a pre-order  as, cf. [18]: def

C  C ′ = ←−C ·h ⊆ h· ←−C′ We can rewrite (10) as: ←−CIP C ·h = h· ←−CCP C =←−CIP C h ←−CCP C ∧ ←−CCP C h−1 ←−CIP C 7

Aop AS o AS h

 CS o

⊇

h

Cop

 CS

Functions h and h−1 are downward simulations. A downward simulation is a single complete relational rule for dialgebraic refinement, cf. [18]. This means that downward simulation h−1 is an upward simulation h: o Aop AS AS O O h

CS o

⊆

h

Cop

CS

8 Conclusion We know that intuitionistic logic is a sublogic of classical logic. Minimal logic is also a sublogic of intuitionistic logic and of classical logic. So the set of theorems of minimal logic is a subset of the set of theorems of intuitionstic logic which is a subset of the set of theorems of classical logic. By transitivity of inclusion, the set of theorems of minimal logic is also a subset of the set of theorems of classical logic. We explored the application of equalizers in logic, which involves a monic morphism. That was the starting point for the definition of minimal logic using either classical logic or intuitionistic logic. Defining the set K as the kernel of intuitionitic logic and classical logic, then minimal logic is equivalent to intuitionistic logic and classical logic over K. It is defined a bisimulation [18] involving classical logic and intuitionistic logic, up to isomorphism, in the context of an exercice over the single complete rule (ie. (downward) simulation) for dialgebraic refinement.

References [1] R. Bird and O. de Moor. Algebra of Programming. Series in Computer Science. Prentice-Hall International, 1997. C.A.R. Hoare, series editor. [2] W. Blok and J. Rebagliato. Algebraic semantics for deductive systems. Studia Logica, 74:153–180, 2003. Special Issue on Abstrct Algebrac Logic - Part 2, Edited by Josep Maria Font, Ramon Jansana, and Don Pigozzi. [3] H. Curry. Functionality in combinatory logic. Proceedings of the National Academy of Sciences, 20:584–590, 1934. 8

[4] J. Fiadeiro and A. Sernadas. Structuring theories on consequence. In D. Sanella and A. Tarlecki, editors, Recent Trends in Data Type Specification. Specification of Abstract Data Types (Papers from the Fifth Workshop on Specification of Abstract Datac Types, Gullane, 1987), volume 332 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1988. [5] J. Font, R. Jansana, and D. Pigozzi. A survey of abstract algebraic logic. Technical Report 329, Mathematics Preprint Series, University of Barcelona, April 2003. ˇ cedrov. Categories, Allegories, volume 39 of Mathematical [6] P. J. Freyd and A. Sˇ Library. North-Holland, 1990. [7] Jean-Yves Girard. A new constuctive logic: Classical logic. Mathematical Structures in Computer Science, 1:32–40, 1991. [8] J. Goguen and R. Burstall. Institutions: Abstract model theory for computer science. Technical Report CSLI-85-30, Center for the Study of Language and Information, Stanford University, August 1985. [9] J. Goguen and R. Burstall. Institutions: Abstract model theory for specification and programming. JACM (39)95-146:, January 1992. [10] W. Howard. The formulae-as-types notion of construction. In J. Roger Seldin, Jonathan P. Hindley, editor, To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Boston, MA, pages 479–490. Academic Press, 1980. original paper manuscript from 1969. [11] I. Johansson. Der minimalkalkul, ein reduzierter intuitionistischer formalismus. Compositio Mathematica, 4:119–136, 1936. [12] S. MacLane. Categories for the Working Mathematician. Springer-Verlag, NewYork, 1971. [13] C. McLarty. Elementary Categories, Elementary Toposes, volume 21 of Oxford Logic Guides. Clarendon Press, 1992. [14] J. Meseguer. General logics. In H.-D. Ebbinghaus et al. editor, Logic Colloqium’87, pages 275–329, 1992. [15] J.N. Oliveira and C.J. Rodrigues. Transposing relations: from Maybe functions to hash tables. In D. Kozen, editor, MPC’07: Seventh International Conference on Mathematics of Program Construction, 12-14 July, 2004, Stirling, Scotland, UK, volume 3125 of LNCS, pages 334–356. Springer, July 2004. [16] M. Parigot. Lambda-mu-calculus: An algorithmic intyerpretation of classical natural deduction. In Logic Programming and Automated Reasoning, International Conference LPAR’92 St. Petersburg (LNCS 624), pages 190–201, 1992. [17] C.J. Rodrigues. Foundations of Program Refinement by Calculation. PhD thesis, University of Minho, School of Engineering, Informatics Department, July 2008. 9

[18] C.J. Rodrigues. A single complete relational rule for dialgebraic refinement of (co)functional dialgebras, September 2011. [19] C.J. Rodrigues. The category of abstract logics as category of P, P -dialgebras. Technical report, Departamento de Inform´atica, Universidade do Minho, 2013. [20] A.S. Troelstra and H. Schwichtenberg. Basic Proof Theory. Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1996. [21] G. Voutsadakis. Universal dialgebra: Unifying universal algebra and coalgebra. Far East Journal of Mathematical Sciences, 44(1):1–53, 2010.

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