Generalised Geometric Logic

. This paper introduces a notion of generalised geometric logic. Connections of generalised geometric logic with L-topological system and L-topological space are established.


Introduction
This work is motivated by S. Vickers's work on topology via logic [17]. To show the connection of topology with geometric logic, the notion of topological system played a crucial role. A topological system is a triple (X, |=, A), consisting of a non-empty set X, a frame A and a binary relation |= (known as satisfaction relation) between X and A satisfying certain conditions. The notion of topological system was introduced by S. Vickers in 1989. Topological system is an interesting mathematical structure, which unifies the concepts of topology, algebra, logic in a single framework. In our earlier work [1], we had introduced a notion of fuzzy geometric logic to answer the question viz. "From which logic can fuzzy topology be studied?". For this purpose first of all we introduced the notion of fuzzy topological system [6] which is a triple (X, |=, A) consisting of a non-empty set X, a frame A and a fuzzy relation |= (i.e. [0, 1] valued relation) from X to A. J. Denniston et al. introduced the notion of lattice valued topological system (L-topological system) by considering frame valued relation between X and A. In [3], categorical relationship of Lattice valued topological space (Ltopological space) with frame was established using the categorical relationships of them with L-topological system. Moreover categorical equivalence between spatial L-topological system with L-topological space was shown. In this paper P. Jana the main focus is to answer the question viz. "From which logic can L-topology be studied?". From [1], it is clear that the satisfaction relation |= of fuzzy topological system reflects the notion of satisfiability (sat) of a geometric formula by a sequence over the domain of interpretation of the corresponding logic. Hence we considered the grade of satisfiabilty from [0, 1]. As for L-topological system the satisfaction relation is an L (frame)-valued relation, the natural tendency is to consider the grade of satisfiability from L. Keeping this in mind, generalised geometric logic (c.f. Section 3) is proposed to provide the answer of the raised question successfully.
The paper is organised as follows. Section 2, includes some of the preliminary definitions and results which are used in the sequel. Generalised geometric logic is proposed and studied in details in Section 3. Section 4, explains the connection of the proposed logic with L-topological system whereas Section 5, contains the study of the connection of the proposed logic with L-topological space. Section 6, concludes the work presented in this article and provides some of the future directions.

Preliminaries
In this section we include a brief outline of relevant notions to develop our proposed mathematical structures and results. In [1,2,3,5,7,8,13,17,18] one may found the details of the notions stated here.
i.e., the binary meet distributes over arbitrary join.
Definition 2 (Fuzzy topological space). Let X be a set, and τ be a collection of fuzzy subsets of X s.t.
Elements of τ are called fuzzy open sets of fuzzy topological space (X, τ ).
Definition 3 (L-topological space). Let X be a set, and τ be a collection of L-fuzzy subsets of X i.e.,Ã : X → L, where L is a frame, s.t.
Elements of τ are called L-open sets of L-topological space (X, τ ).

Definition 4. [17]
A topological system is a triple, (X, |=, A), consisting of a non empty set X, a frame A and a binary relation |=⊆ X × A from X to A such that: 1. for any finite subset S of A, x |= S if and only if x |= a for all a ∈ S; 2. for any subset S of A, x |= S if and only if x |= a for some a ∈ S.
Definition 5 (L-topological system). An L-topological system is a triple (X, |=, A), where X is a non-empty set, A is a frame and |= is an L-valued relation from X to A (|=: Note that if L = [0, 1] then the triple is known as fuzzy topological system. Similarly considering L = {0, 1}, we arrive at the notion of a topological system. Definition 6 (Spatial). An L-topological system (X, |=, A) is said to be spatial if and only if (for any x ∈ X, |= (x, a) =|= (x, b)) imply (a = b), for any a, b ∈ A. Theorem 1. Category of spatial L-topological systems, for a fixed L, is equivalent to the category of L-topological spaces.

Generalised Geometric Logic
In this section we will introduce the notion of generalised geometric logic which may be considered as a generalisation of fuzzy geometric logic and consequently of so called geometric logic. Detailed studies on fuzzy logic, geometric logic and fuzzy geometric logic may be found in [1,4,9,10,11,12,14,15,16,17].
The alphabet of the language L of generalised geometric logic comprises of the connectives ∧, , the existential quantifier ∃, parentheses ) and ( as well as: • countably many individual constants c 1 , c 2 , . . . ; • denumerably many individual variables x 1 , x 2 , . . . ; • propositional constants ⊤, ⊥; • for each i > 0, countably many i-place predicate symbols p i j 's, including at least the 2-place symbol "=" for identity; • for each i > 0, countably many i-place function symbols f i j 's. Definition 7 (Term). Terms are recursively defined in the usual way.
• every constant symbol c i is a term; • every variable x i is a term; • if f j is an i-place function symbol, and t 1 , t 2 , . . . , t i are terms then • nothing else is a term.
Definition 8 (Geometric formula). Geometric formulae are recursively defined as follows: • ⊤, ⊥ are geometric formulae; • if p j is an i-place predicate symbol, and t 1 , t 2 , . . . , t i are terms then p i j t 1 t 2 . . . t i is a geometric formula; • if t i , t j are terms then (t i = t j ) is a geometric formula; • if φ and ψ are geometric formulae then (φ ∧ ψ) is a geometric formula; • if φ i 's (i ∈ I) are geometric formulae then {φ i } i∈I is a geometric formula, when I = {1, 2} then the above formula is written as φ 1 ∨ φ 2 ; • if φ is a geometric formula and x i is a variable then ∃x i φ is a geometric formula; • nothing else is a geometric formula.

Definition 9 (Interpretation). An interpretation I consists of
• a set D, called the domain of interpretation; • an element I(c i ) ∈ D for each constant c i ; • an L-fuzzy relation I(p i j ) : D i −→ L, where L is a frame, for each predicate symbol p i j i.e. it is an L-fuzzy subset of D i .
Now we define grade of satisfiability of φ by s written as gr(s sat φ), where φ is a geometric formula, as follows: Throughout this article ∧ and ∨ in L will stand for the meet and join of the frame L respectively. The expression φ ⊢ ψ, where φ, ψ are wffs, is called a sequent. We now define satisfiability of a sequent. 1. Let t and t ′ be terms. For every sequence s over D, 2. Let φ be a geometric formula and t be a term. For every sequence s over D, gr(s sat φ[t/x k ]) = gr(s(s(t)/x k ) sat φ).
Proof. By induction on t and φ respectively.

Rules of Inference
The rules of inference for generalised geometric logic are as follows.
Theorem 3. The rules of inference for generalised geometric logic are universally valid.
Proof. To show the universal validity of the rules of inference is kind of a routine check but for the sake of clarity we would like to provide the proof in details upto certain extent.

Generalised logic of finite observations
In this subsection we will consider the propositional fragment of the proposed generalised geometric logic and call it generalised logic of finite observations. Throughout this part the justification to choose the name will be provided. In [17], one may notice that the logic of finite observations is nothing but the logic of affirmative assertions. Recall that an assertion is affirmative if and only if the assertion is true precisely in the circumstances where it can be affirmed. Note that we need to do the job in finite time, finite amount of work and on the basis of what we can actually observe. In [17], it is nicely explained why logic of affirmative assertions allows the connectives ∧, , ⊤, ⊥ but not ¬ and →. If we wish to make the idea of logic of affirmative assertions more close to real life situations then discussing the validity (truth value) of affirmative assertions upto some extent instead of, whether affirmative assertions are valid (true) or not valid (false), is a better idea. To address this issue we need to concentrate on the notion of valuation function. In this stage it is better to quickly recapture the propositional part of our proposed generalised logic for better understanding. Let Φ be a set of propositional variables. The language GGL(Φ) of generalised logic of finite observations of propositional generalised geometric formula is given by where p ∈ Φ and I is some index set. The rules are given by The valuation function v : Φ → L can be extended tov : GGL(Φ) → L defined by Now notice that if we consider the range of the valuation function a frame (L) instead of {0, 1} then mathematically we can reach our goal. Consideration of the range as any frame allows us to think about the incomparable truth values of affirmative assertions, which is a natural phenomenon to arise in our daily life situations. If we think in this line then it is not very hard to understand how the definition of the extended valuation function considered here is the expected one. In this sense we will be able to generalise the notion of the so-called logic of affirmative assertions or logic of finite observations to address real life situations in a better way. Moreover this generalised version of the logic connects the desired mathematical structures (L-topological space, L-topological system and frame) as well. The proposition stated above implies that {φ ∧ ψ | ψ ∈ S} ⊢ φ ∧ S is valid. We will use this piece of information in the next section.

L-Topological System via Generalised Geometric Logic
In this section the way to get an L-topological system from generalised geometric logic is provided. In this respect it is to be noted that the propositional fragment of the proposed generalised geometric logic is enough to serve our purpose. Let us consider the triplet (X, |=, A) where X is the non empty set of extended valuation functions, A is the set of geometric formulae and |= defined as |= (v, φ) =v(φ).
The above defined "≈" is an equivalence relation. Thus we get A/ ≈ .
Consequently we arrive at the conclusion that finite meet distributes over arbitrary join, i.e., the frame distributive property holds good. Proposition 3 and Proposition 4 provides the following theorem.

L-Topology via Generalised Geometric Logic
We first construct the L-topological system (X, |= ′ , A/ ≈ ) from generalised geometric logic. Then (X, ext(A/ ≈ )) is constructed as follows: It can be shown that ext(A/ ≈ ) forms an L-topology on X as follows.
Similarly it can be shown that ext(A/ ≈ ) is closed under arbitrary union. Hence (X, ext(A/ ≈ )) is an L-topological space obtained via generalised geometric logic.
Proposition 5 indicates that (X, |= ′ , A/ ≈ ) is a spatial L-topological system and hence from Theorem 1 we arrive at the conclusion that (X, |= ′ , A/ ≈ ), (A, ext(A/ ≈ )) are equivalent to each other. That is, (X, |= ′ , A/ ≈ ) and (X, ∈, ext(A/ ≈ )) represent the same L-topological system. The following diagram summarize all the facts what we have proved till this stage: Generalised logic of finite observations Spatial L − Topological system L − Topological space Let X be an L-topological space, τ is its L-topology. Then the corresponding generalised geometric theory can be defined as follows: All other axioms for the (propositional) generalised geometric logic will follow from the above clauses. If x ∈ X, then x gives a model of the theory in which the truth value of the interpretation of PT will beT (x).
Hence one may study L-topology via generalised geometric logic.

Concluding Remarks
In this paper the notion of generalised geometric logic is introduced and studied in details. Using the connection between L-topological system and Ltopological space, the strong connection between the proposed logic and Ltopological space is established. The interpretation of the predicate symbols for the generalised geometric logic are L (frame)-valued relations and so the proposed logic is more expressible. That is, the proposed logic has the capacity to interpret the situation where the truth values are incomparable. Generalising the proposed logic considering graded consequence relation is in future goal.
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