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**1 - 5**of**5**### The institution-theoretic scope of logic theorems

"... Abstract In this essay we analyse and clarify the method to establish and clarify the scope of logic theorems offered within the theory of institutions. The method presented pervades a lot of abstract model theoretic developments carried out within institution theory. The power of the proposed gene ..."

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Abstract In this essay we analyse and clarify the method to establish and clarify the scope of logic theorems offered within the theory of institutions. The method presented pervades a lot of abstract model theoretic developments carried out within institution theory. The power of the proposed general method is illustrated with the examples of (Craig) interpolation and (Beth) definability, as they appear in the literature of institutional model theory. Both case studies illustrate a considerable extension of the original scopes of the two classical theorems. Our presentation is rather narrative with the relevant logic and institution theory concepts introduced and explained gradually to the non-expert reader. Institution theory -a very brief introduction Institution theory is a categorical abstract model theory that arose about three decades ago The starting concept of institution theory is the formal definition of a logical system; this includes the syntax, the semantics and the satisfaction relation between them. It plays the same role as for example the definition of group plays for group theory. Although the definition of group is very simple, group theory is a vast sophisticated mathematical area. The same with the definition of institution and institution theory. An institution is a tuple (Sign, Sen, Mod, (| = Σ ) Σ∈|Sign| |) that consists of • a category Sign whose objects are called signatures, • a functor Sen : Sign → Set (to the category of sets) giving for each signature a set whose elements are called sentences over that signature, • a (contravariant) functor Mod : (Sign) op → CAT (to the 'category' of categories), giving for each signature Σ a category whose objects are called Σ-models, and whose arrows are called Σ-(model) homomorphisms, and • a relation | = Σ ⊆ |Mod(Σ)| × Sen(Σ) for each Σ ∈ |Sign|, called the satisfaction relation, such that for each morphism ϕ : Σ → Σ ∈ Sign, the Satisfaction Condition The literature (e.g. Given a signature Σ in an institution, for any sets E and E of Σ-sentences by E | = E we denote the situation that for each The method to clarify the scope of logic theorems Among mathematicians it is common to think that there cannot be an in-depth understanding of a result in the absence of the understanding of the proof of this result. The understanding of the scope of logic theorems may be therefore considered at two different levels. A coarse level refers to the actual results (statements of logic theorems), and a subtle level refers to the causalities leading to the results (i.e. dependencies among logic theorems and methods to prove these theorems). Clearly, answers at the subtle level determine answers at the coarse level. Institution theory and its abstract approach to model theory develop a distinctive and clear way to determine the scope of logic theorems, with emphasis on revealing bare causalities that are stripped off irrelevant details. In many situations this led not only to quite unexpected extensions of the scopes of logic theorems, but even to a reformed and a more realistic understanding of fundamental logic concepts (these points will be illustrated also by the case studies discussed in this essay). The following scheme captures a method to determine the scope of logic theorems that proved effective in numerous developments within institution theory: 1. Choose a proof of a logic theorem. 2. Extract its essence by leaving out the irrelevant details and by identifying the conceptual structure and the causalities underlying the result. 3. Formulate the conceptual structure at the level of an abstract institution. 4. Lift the proof considered to the level of an abstract institution, shaping an abstract, generic scope of the result. 5. Determine the actual scope by analysing the abstract conditions used in the proof. In the following we illustrate this scheme with a few representative cases from institutional model theory. First case study: interpolation by axiomatisability Because of its many applications in logic and computer science, interpolation is one of the most desired and studied properties of logical systems. Although it has a strikingly simple and elementary formulation, in general it is very difficult to establish. The famous result of Craig [9] marks perhaps the birth of the study of interpolation, proving it for first-order logic. The actual scope of Craig's result has been gradually extended to many other logical systems (for example in the world of modal logics, see It has been widely believed that equational logic lacks interpolation; likewise for Horn-clause logic and other such fragments of first-order logic. As far as we know, Piet Rodenburg was the first to point out that this is a misconception due to a basic misunderstanding of interpolation, rooted in the heavy dependency of logic culture on classical first-order logic with all its distinctive features taken for granted. Then it follows the grave fault of exporting a coarse understanding of concepts dependent on details of a particular logical system to other logical systems of a possibly very different nature, where some detailed features may not be available. In the case of interpolation, the gross confusion has to do with looking for an interpolant as a single sentence. In first-order logic, which has conjunction, looking for interpolants as finite sets of sentences ({ρ 1 , . . . , ρ n }) is just the same as looking for interpolants as single sentences (ρ 1 ∧ · · · ∧ ρ n ). Hence, the common formulation of interpolation requires a single-sentence interpolant. However, this is not an adequate formulation for equational logic which lacks conjunction, i.e., conjunction ρ 1 ∧ρ 2 of universally quantified equations ρ 1 and ρ 2 cannot be captured as a universally quantified equation in general. Rodenburg At item 2. of the scheme, we go through a process of understanding that the essence of the proof in [28] is in fact independent of equational logic and of Birkhoff's variety theorem. The key is a causal relationship between interpolation and a 'Birkhoff-style' axiomatisability property of the logic, both of them considered in a rather general sense. In other words, the proof in At the next step (item 3.) we work towards formalising the above understanding of the proof in [28] • by defining the concept of interpolation at the level of abstract institutions, and • by formulating a general axiomatisability concept at the level of abstract institutions that captures the essence of the use of Birkhoff's variety theorem in the proof of [28] as abstractly as possible. Interpolation in abstract institutions. The standard formulation of (Craig) interpolation property for first-order logic is as follows. Given signatures Σ 1 , Σ 2 , Σ 1 -sentence ρ 1 and Σ 2 -sentence ρ 2 , if ρ 2 is a consequence of ρ 1 (written ρ 1 ρ 2 ) then there exists an 'interpolant' (Σ 1 ∩ Σ 2 )-sentence ρ such that ρ 1 ρ and ρ ρ 2 . It is by far not straightforward how to express this property at the level of abstract institutions. First, we have to interpret the consequence relation between (sets of) sentences as the semantic consequence | =, which is naturally defined in any institution. Then, in order to free our discussion from the existence of conjunctions, we replace single sentences by finite sets of sentences. Finally, we have to capture the relationship between signatures Σ 1 , Σ 2 and their union Σ 1 ∪ Σ 2 (where the consequence ρ 1 ρ 2 happens) 3 and intersection Σ 1 ∩ Σ 2 (the signature of the interpolant), depicted by the following diagram where arrows indicate the obvious inclusions: While intersections ∩ and unions ∪ are more of less obvious for signatures as used in first-order logic and in many other standard logics, they are not so in some other logical systems, and certainly not at the level of abstract institutions where signatures are just objects of an arbitrary category. One immediate response to this problem would be to add an infrastructure to the abstract category of signatures that would support concepts of ∩ and ∪; in fact this is already available in the institution theoretic literature and is called inclusion system and any finite sets of sentences E 1 ⊆ Sen(Σ 1 ) and E 2 ⊆ Sen(Σ 2 ), if θ 1 (E 1 ) | = θ 2 (E 2 ) then there exists a finite set E of Σ-sentences such that E 1 | = ϕ 1 (E) and ϕ 2 (E) | = E 2 . However, this concept proves a bit too strong (for example many-sorted first-order logic does not support this The use of pushout squares above meets so-called model amalgamation property, which is a crucial technical property pervading most of the developments in institutional model theory. This requires that for any pushout of signatures as above, Σ 1 -model M 1 and Σ 2 -model M 2 with common reduct to Σ, i.e., Mod(ϕ 1 )(M 1 ) = Mod(ϕ 2 )(M 2 ), admit a unique common expansion M to Σ , i.e., Mod(θ 1 )(M ) = M 1 and Mod(θ 2 )(M ) = M 2 . This property is evident in most institutions of interest, and is tacitly assumed in many model-theoretic developments. Often its weaker variant, that does not require the uniqueness of M , suffices; this is called weak model amalgamation. Abstract Birkhoff institutions. The other concept that plays a crucial role in our analysis of the scope of interpolation by axiomatisability is that of Birkhoff(-style) axiomatisability. Let us start from the classical result of Birkhoff [6] about the equationally defined classes of algebras, which gives the following algebraic characterisation of the 1 Given a signature morphism ϕ : Σ → Σ , we abbreviate Sen(ϕ) as ϕ, and so for a set of sentences E ⊆ Sen(Σ), ϕ(E) is the image of E under Sen(ϕ).

### From Universal Logic to Computer Science, and back

"... Abstract. Computer Science has been long viewed as a consumer of mathematics in general, and of logic in particular, with few and minor contributions back. In this article we are challenging this view with the case of the relationship between specification theory and the universal trend in logic. 1 ..."

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Abstract. Computer Science has been long viewed as a consumer of mathematics in general, and of logic in particular, with few and minor contributions back. In this article we are challenging this view with the case of the relationship between specification theory and the universal trend in logic. 1 From Universal Logic... Although universal logic has been clearly recognised as a trend in mathematical logic since about one decade only, mainly due to the efforts of Jean-Yves Béziau and his colleagues, it had a presence here and there since much longer. For example the anthology [9] traces universal logic ideas back to the work of Paul Herz in 1922. In fact there is a whole string of famous names in logic that have been involved with universal logic in the last century, including Paul Bernays,

### Foundations for Structuring Behavioural Specifications

"... We develop foundations for structuring behavioural specifications based on the logic tradition of hidden algebra. This includes an analysis of a number of important technical compositional properties for behavioural signatures, such as pushouts, inclusions and unions, as well as an investigation of ..."

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We develop foundations for structuring behavioural specifications based on the logic tradition of hidden algebra. This includes an analysis of a number of important technical compositional properties for behavioural signatures, such as pushouts, inclusions and unions, as well as an investigation of algebraic rules for behavioural module composition. As a particularity of behavioural specifications, some of the constructions and results arise in a partial algebraic form. This partiality aspect is one of the distinguishing features of our approach to behavioural specification modules. In addition, our study does not commit to any actual choice of structuring constructs, thus being applicable to a wide variety of structuring situations. 1.

### Graded consequence: an institution theoretic study

"... We develop a general study of graded consequence (of many-valued logic) in an institution theoretic (in the sense of Goguen and Burstall) style. This means both syntax and semantics are considered fully abstract, as well as the satisfaction between them. Our approach contrasts to other approaches on ..."

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We develop a general study of graded consequence (of many-valued logic) in an institution theoretic (in the sense of Goguen and Burstall) style. This means both syntax and semantics are considered fully abstract, as well as the satisfaction between them. Our approach contrasts to other approaches on many-valued logic in that it is a multi-signature one, in the spirit of institution theory. We consider graded consequence at three different conceptual levels: entailment, semantic, and closure operators, and explore several interpretations between them. We also study logical connectors and quantifiers both at the entailment and semantic level, compactness and soundness properties. 1.