A Appendix

# The First-order Logical Environment

## Abstract

This paper describes the first-order logical environment FOLE. Institutions in general (Goguen and Burstall [4]), and logical environments in particular, give equivalent heterogeneous and homogeneous representations for logical systems. As such, they offer a rigorous and principled approach to distributed interoperable information systems via system consequence (Kent [6]). Since FOLE is a particular logical environment, this provides a rigorous and principled approach to distributed interoperable first-order information systems. The FOLE represents the formalism and semantics of first-order logic in a classification form. By using an interpretation form, a companion approach (Kent [7]) defines the formalism and semantics of first-order logical/relational database systems. In a strict sense, the two forms have transformational passages (generalized inverses) between one another. The classification form of first-order logic in the FOLE corresponds to ideas discussed in the Information Flow Framework (IFF [12]). The FOLE representation follows a conceptual structures approach, that is completely compatible with formal concept analysis (Ganter and Wille [2]) and information flow (Barwise and Seligman [1]).

###### Keywords:
schema, specification, structure, logical environment.

## 1 Introduction

The paper “System Consequence” (Kent [6]) gave a general and abstract solution to the interoperation of information systems via the channel theory of information flow (Barwise and Seligman [1]). These can be expressed either formally, semantically or in a combined form. This general solution closely follows the theories of institutions (Goguen and Burstall [4]), 1 information flow and formal concept analysis (Ganter and Wille [2]). By following the approach of the “System Consequence” paper, this paper offers a solution to the interoperation of distributed systems expressed in terms of the formalism and semantics of first-order logic. It does this be defining FOLE, the first-order logical environment. 2 Since this paper develops a classification form of first order logic as a logical environment, the interaction of information systems expressed in first order logic have a firm foundation. Section 2 surveys the architecture of the first-order logical environment FOLE. Section 3 discusses the linguistic/formal and semantic components of FOLE; detailed discussions of the functional base and relational superstructure are given in Appendix A.1 and Appendix A.2, respectively. Section 4 explains how FOLE is a logical environment; a proof of this fact is given in Appendix A.4. Section 5 discusses FOLE information systems. Finally, section 6 summarizes and states future plans for work on these topics.

## 2 Architecture

Figure 1 is a 3-dimensional visualization of the fibered architecture of the first-order logical environment FOLE. Each node of this figure is a mathematical context, whereas each edge is a passage between two contexts. There is a projection from the 2-D prism below representing the relational superstructure (subsec. A.2) to the 2-D prism below representing the functional base (subsec. A.1). The front diamond below represents the linguistics/formalism, whereas the back diamond below represents the semantics. The projective passages from semantics to linguistics/formalism represent the fibration left-to-right and the indexing right-to-left. The vee-shape at the top of each diamond states that the top mathematical context is a product of the side contexts modulo the bottom context. The mathematical contexts on the left side of each diamond form the relational aspect, whereas the mathematical contexts on the right side form the functional aspect that lifts the relational to the (first-order) logical aspect. The 2-D prism below represents the institutional architecture.

## 3 Components

The architectural components (Fig.1) divide up according to kind and aspect. The outer level describes the kind of component. The indexing kind is a language (type set, relational schema, operator domain, etc.) (front diamond Fig.1), whereas the indexed kind is either a formalism or a semantics (classification, relational structure, algebra, etc.) (back diamond Fig.1). The inner level describes the aspect of component. There are basic, relational, functional and logical aspects (bottom, left, right or top node in either Fig.1 diamond).

Fig.2 illustrates an analogy between the top-level ontological categories discussed in (Sowa [9]) and the components of the first-order logical environment FOLE (the relational aspect or 2-D prism below ). The pair ‘physical-abstract’, which corresponds to the Heraclitus distinction physis-logos, is represented in the FOLE by a classification between instances and types of various kinds. The triples (triads) ‘actuality-prehension-nexus’ and ‘form-proposition-intention’ correspond to Whitehead’s categories of existence. The latter triple, which is analogous to the ‘entity type-signature-relation type’ triple, is represented in the FOLE by a relational language (schema) (Appendix A.2.1). The former triple, which is analogous to the ‘entity instance-tuple-relation instance’ triple, is represented in the FOLE by the tuple function (part of a FOLE structure). The firstness category of ‘independent(actuality,form)’ is represented in the FOLE by an entity classification (Appendix A.2.2). The thirdness category of ‘mediating(nexus,intention)’ is represented in the FOLE by a relation classification between relational instances (keys) and relational types (or a classification between relational instances and logical formula, more generally) (Appendix A.2.2). The secondness category of ‘relative(prehension,proposition)’ is represented in the FOLE by the list construction of an entity classification between tuples and signatures (Appendix A.2.2). Finally, the entire graph of the top-level ontological categories is represented in the FOLE by a (model-theoretic) structure (classification form) , where the relation and entity classifications are connected by a list designation (Appendix A.2.2). This is appropriate, since a (model-theoretic) structure represents the knowledge in the local world of a community of discourse.

## 4 Logical Environment

The institution (logical system) (Kent [6]) has at its core the mathematical context of first-order logic (FOL) languages . For any language , there is a set of constraints representing the formalism at location , and there is a mathematical context of structures representing the semantics at location . For any first-order logic (FOL) language morphism , there is a constraint function (Appendix A.2.1) representing flow of formalism in the forward direction, and there is a structure passage (Appendix A.2.2) representing flow of semantics in the reverse direction. This structure passage has a relational component and a functional (algebraic) component .

is an institution, since the satisfaction relation is preserved during information flow along any first-order logic (FOL) language morphism : iff In short, “satisfaction is invariant under change of notation”. The institution is a logical environment, since for any language , if is a -vertical structure morphism over , then we have the intent order ; that is, implies for any -sequent . In short, “satisfaction respects structure morphisms”. (See Appendix A.4 for a proof of this in the relational aspect.)

## 5 Information Systems

Following the theory of general systems, an information system consists of a collection of interconnected parts called information resources and a collection of part-part relationships between pairs of information resources called constraints. Semantic information systems have logics 3 as their information resources. Just as every logic has an underlying structure, so also every information system has an underlying distributed system. As such, distributed systems have structures for their component parts.

A FOLE distributed system is a passage pictured as a diagram of shape within the ambient mathematical context of first-order structures. As such, it consists of an indexed family of structures together with an indexed family of structure morphisms. A FOLE (semantic) information system is a diagram within the mathematical context of first-order logics. This consists of an indexed family of logics and an indexed family of logic morphisms . An information system has an underlying distributed system of the same shape with for all . An information channel consists of an indexed family of structure morphisms with a common target structure called the core of the channel. Information flows along channels. We are mainly interested in channels that cover a distributed system , where the part-whole relationships respect the system constraints (are consistent with the part-part relationships). In this case, there exist optimal channels. An optimal core is called the sum of the distributed system, and the optimal channel components (structure morphisms) are flow links.

System interoperability is defined by moving formalism over semantics. The fusion (unification) of the information system represents the whole system in a centralized fashion. The fusion logic is defined by direct system flow: (i) direct logic flow of the component parts of the information system along the optimal channel over the underlying distributed system to a centralized location (the mathematical context of structures at the optimal channel core), and (ii) product combining the contributions of the parts into a whole. The consequence of the information system represents the whole system in a distributed fashion. This is an information system defined by inverse system flow: (i) consequence of the fusion logic, and (ii) inverse logic flow of this consequence back along the same optimal channel, transfering the constraints of the whole system (the fusion logic) to the distributed locations (structures) of the component parts. See Kent [6] for further details. 4

## 6 Summary and Future Work

In this paper we have described the first-order logical environment FOLE in classification form. This gives a holistic treatment of first-order logic, by the use of several novel elements: the use of signatures (type lists) for relational arities, in place of ordinal numbers; the use of abstract tuples (relational instances, keys), thus making FOLE compatible with relational databases; the use of classifications for both entities and relations; and the use of relational constraints for the sentences of the FOLE institution. FOLE also has an interpretation form (Kent [7]) that represents the formalism and semantics of logical/relational databases, including relational algebra. There are transformational passages between the classification form and a strict version of the interpretation form. Appendix A.5.2 briefly discusses the transformation from sound logics to logical/relational databases.

FOLE has advantages over other approaches to first-order logic: in FOLE the formalism is completely integrated into the semantics; the classification form of FOLE has a natural extension to relational/logical databases, as represented by the interpretation form of FOLE; and FOLE is a logical environment, thus allowing practitioners a rigorously defined approach towards the interoperation of online semantic systems of information resources that include relational databases.

Future work includes: finishing work on the interpretation form of FOLE; further work on defining the transformational passages between the classification and interpretation forms; developing a linearization process from FOLE to sketch-like forms of logic such as Ologs (Spivak and Kent [11]); and linking FOLE with the Common Logic standard.

## Appendix A Appendix

### a.1 Functional Base.

#### Linguistics/Formalism.

##### Base Linguistics: \mathrmbfSet.

A set (of entity types) defines a mathematical context of type lists (signatures) . The FOLE uses type lists for relational arities, instead of ordinal numbers.

The first subcomponent of any linguistic component is a set of entity types (sorts) . Examples of entity types are ‘human’ representing the set of all human beings, ‘blue’ representing the set of all objects of color blue, etc. A type list (signature) consists of an arity set and a type map mapping elements of the arity to entity types. This can be denoted by the list notation or the type declaration notation for and . For example, the type list ’(make:String,model:String,year:Number,color:Color)’ is a type list for cars with valence 4, arity set , and type map . A type list morphism is an arity function that satisfies the commutative diagram . We say that is at least as general as .

Given the natural numbers , let denote the mathematical context of finite ordinals (number sets) and functions between them. This is the skeleton of the mathematical context of finite sets and functions. Both represent the single-sorted case where . We have the following inclusion of base language mathematical contexts. 5

 ℵ––\shortstack{\rule{0.0pt}{10.0pt}skeleton}⊆\mathrmbfFin\shortstack{\rule[2.0pt]{0.0pt}{10.0pt}% single-sorted}⊆∗\mathrmbfList(X)\rule{0.0pt}{% 10.0pt}many-sorted

Traditional first-order systems use the natural numbers for indexing relations. More flexible first-order systems, such as FOLE or relational database systems, use finite sets when single-sorted or type lists when many-sorted.

Algebraic Linguistics: . A functional language (operator domain) is a pair , where is a set of entity types (sorts) and is an -sorted operator domain; that is, is a collection of sets of function (operator) symbols, where is a function symbol of entity type (sort) and finite arity , 6 symbolized by . An element is called a constant symbol of sort . Any operator domain defines a mathematical context of terms , whose objects are -signatures and whose morphisms are term vectors , where is an indexed collection (vector) of -ary terms. Terms and term vectors are defined by mutual induction.

A morphism of functional languages is a pair , where is a function of entity types (sorts) and is a collection of maps between function symbol sets: maps a function symbol in to a function symbol in . Given any morphism of functional languages , there is a term passage defined by induction. Let denote the mathematical context of functional languages (operator domains).

Algebraic Formalism. Let be an operator domain. An -equation is a parallel pair of term vectors . We represent an equation using the traditional notation . An equational presentation consists of an operator domain and a set of -equations . A congruence is any equational presentation closed under left and right term composition. Any equational presentation generates a congruence , which defines a quotient mathematical context of terms with a morphism being an equivalence class of terms. There is a canonical passage . A morphism of equational presentations is a morphism of functional languages that preserves equations: an -equation in is mapped to an -equation in the congruence . Hence, there is a term passage that commutes with canons.

#### Semantics.

Base Semantics: . For any entity classification , there is a tuple passage defined as the extent of the list classification . It maps a type list (signature) to its extent . An entity infomorphism defines a bridge between tuple passages. For any source signature , the tuple function is define by composition.

Algebraic Semantics: . A many-sorted algebra consists of an entity classification , an operator domain , and an -algebra compatible with , where is an -sorted set and assigns an -ary -sorted function (operation) to each function symbol with the product set. A many-sorted algebra defines (by induction) an algebraic interpretation passage , which extends the tuple passage by compatibility. An algebra satisfies an equation , symbolized by , when the interpretation maps the terms to the same function . A many-sorted algebraic homomorphism consists of an entity infomorphism , a morphism of many-sorted operator domains , and an -algebra morphism compatible with . A many-sorted algebraic homomorphism defines an algebraic bridge between algebraic interpretations, which extends the tuple bridge by compatibility. Let denote the mathematical context of many-sorted algebras. (The base semantics embeds into the functional semantics Fig. 3.)

### a.2 Relational Superstructure.

#### Linguistics/Formalism.

##### Relational Linguistics: \mathrmbfSch.

Schemas. A relational language (schema) has two components: a base and a superstructure built upon the base. The base consists of a set of entity types (sorts) , which defines the type list mathematical context . The superstructure consists of a set of relation types (symbols) and a (discrete) type list passage mapping a relation symbol to its type list . A relational language (schema) morphism also has two components: a base and a superstructure built upon the base. The base consists of an entity type (sort) function , which defines the type list passage mapping a type list to the type list . The superstructure consists of a relation type function which preserves type lists, satisfying the condition . Let symbolize the mathematical context of relational languages (schemas) with type set projection passage .

Formulas. For any type list , let denote the set of all relation types with this type list. These are called -ary relation symbols. Formulas form a schema that extends : with inductive definitions, the set of relation types is extended to a set of logical formulas and the relational type list function is extended to a type list function . For any type list , let denote the set of all formulas with this type list. These are called -ary formulas. Formulas are constructed by using logical connectives within a fiber and logical flow between fibers.

• Let be any type list. Any -ary relation symbol is an (atomic) -ary formula; that is, . For any pair of -ary formulas and , there are the following -ary formulas: meet , join , implication and difference . For any -ary formula , there is an -ary negation formula .

• Let be any type list morphism. For any -ary formula , there are -ary existentially/universally quantified formulas and . For any -ary formula , there is a -ary substitution formula .

Formula Fiber Passage. A schema morphism can be extended to a formula schema morphism . The formula function , which satisfies the condition , is recursively defined in Table 2.

###### Proposition 1

There is an idempotent formula passage that forms a monad with embedding.

Constraints. Let