Combinations

of Logics and

Combinations

of Theories

Vladimir L. VasyukovInstitute of PhilosophyRussian Academy of SciencesMoscowRussia vasyukov4@gmail.com

Abstract Universal Logic would be treated as a general theory of logical

systems considered as a specific kind of mathematical structures the same manner Universal Algebra treats algebraic systems.

A category-theoretical approach where logical systems are combined in a category of the special sort provides us with some basis for inquiring the Universe of Universal Logic.

In the framework of such an approach some categorical constructions are introduced which describe the internal structure of the category of logical systems.

Signatures

A signature is an indexed set = {n }nN, where each n is the set of n-ary constructors.

We consider that the set of propositional variables is included in 0.

LanguageThe language over a given signature , which we denote by L, isbuilt inductively in the usual way:• 0 L;• If nN, 1, . . . , n L and cn, then c(1, . . . , n )L .

We call -formulas to the elements of L, or simply formulas when is clear from the context.

Logical SystemsA logical system is a pair = ,⊢, where is a signature and is a consequence operator on ⊢ L (in the sense of Tarski), that is, : 2⊢ L 2L is a function that satisfies the following properties, for every , L :Extensiveness: ⊢ ;Monotonicity: If then ⊢ ⊢ ;Idempotence: (⊢) ⊢ ⊢ .

Here ⊢ is a set of consequences of . For the sake of generality, we do not require that the consequence operator to be finitary, or even structural.

Combinations of Logics

Coproducts would be characterized as• Σ1,1Σ2,2 = Σ1Σ2, 12

where 12 is a consequence operator such that Г i implies Г 12 for every Г {}L Σi (i = 1, 2)

Combinations of Logics

Products would be characterized as• Σ1, 1Σ2, 2 = Σ1Σ2, 12

where 12 is a consequence operator such that

Г1,Г2 12 1, 2 implies Гi i i for every Г i { i} Σi

(i = 1, 2)

Combinations of Logics

Coexponentials

would be characterized as

21 = Σ1, 12

where 12 is a consequence operator such that Г12 iff g[Г]2 g()

for all Log-morphisms g:1 2

Combinations of Logics

Exponentials would be characterized as

• 12 = Σ1, 12

where 12 is a consequence operator such that Г12 iff there exist Log-morphisms g:1 2 and h:2 1 such that h(g[Г])1 h(g()).

Universe of Universal Logics

A structure of the Universe of Universal Logics can be described as a threefold construction consisting of :

1. A category Sig of signatures and their morphisms (N-indexed families of functions h = {hn : 1

n DC2n }nN,

where DC k is the set of all derived

connectives of arity k over and a derived connective of arity kN is a -term d = 1. . .k. where L

k)

C.Caleiro and R.Gonçalves

Universe of Universal Logics

2. A category Log of logical systems and their morphisms (a signature morphism h : Σ1 Σ2 such that h[Φ1] h[Φ]2 for every Φ LΣ1

)C.Caleiro and R.Gonçalves

Universe of Universal Logics

3. A category Tsp of theory spaces (complete lattices tsp = Th, where Th is the set of all theories of given logic) and their morphisms (functions h : Th1 Th2 such that

h(T) = h[T] for every T Th).

C.Caleiro and R.Gonçalves

Universe of Universal Logics

Equipollency Relation

Two formulas , are said to be logically equivalent in , , if both {}⊢ and {}⊢ , or equivalently if {}⊢ = {}⊢.1 = Σ1, 1 and 2 = Σ2, 2 are equipollent if and only if there exists Log-morphisms h: 1 2 and g: 2 1 such that the following conditions hold:• 1

g(h()) for every L Σ1 ;

• 2 h(g()) for every LΣ2

.

Universe of Universal Logics

The equality of Log-morphisms is the smallest equivalence relation between morphisms such that

• f g if and only if dom(f) is equipollent to dom(g) and codom(f) is equipollent to codom(g), i.e. 1 and 2 are equipollent together with 1 and 2 are equipollent for logical system morphisms f : 1 2 , g : 1 2 ; • gf = h implies gf h; • f f and g g implies gf g f ; • f id1 f id1f;• (hg)f h(gf) for all f,f : 1 2 , g,g: 2 3, h,h: 3 4.

Universe of Universal Logics

Log is both a topos

and a complement

topos but all respective constructions works just up to the equivalence which is based

exactly on the equipollence

Structure of Tsp

• The last fact means that if we transfer categorical constructions from Log to Tsp then all equipollences will transform into usual categorical isomorphisms.

• We obtain in Tsp coproducts, products, coexponentials and exponentials with the respective diagrams.

• All of them will be the complete lattices according to the properties of Th.

Structure of Tsp

Tsp is both a topos

and a complement topos

The End

Thank you for your attention