Standard First-order Logic Research Articles

Logical Constants and Arithmetical Forms

This paper reflects on the limits of logical form set by a novel criterion of logicality proposed in (Bonnay and Speitel, 2021). The interest stems from the fact that the delineation of logical terms according to the criterion exceeds the boundaries of standard first-order logic. Among ‘novel’ logical terms is the quantifier “there are infinitely many”. Since the structure of the natural numbers is categorically characterisable in a language including this quantifier we ask: does this imply that arithmetical forms have been reduced to logical forms? And, in general, what other conditions need to be satisfied for a form to qualify as “fully logical”? We survey answers to these questions.

Verifying Graph Programs with First-Order Logic

We consider Hoare-style verification for the graph programming language GP 2. In previous work, graph properties were specified by so-called E-conditions which extend nested graph conditions. However, this type of assertions is not easy to comprehend by programmers that are used to formal specifications in standard first-order logic. In this paper, we present an approach to verify GP 2 programs with a standard first-order logic. We show how to construct a strongest liberal postcondition with respect to a rule schema and a precondition. We then extend this construction to obtain strongest liberal postconditions for arbitrary loop-free programs. Compared with previous work, this allows to reason about a vastly generalised class of graph programs. In particular, many programs with nested loops can be verified with the new calculus.

Is Objectual Identity Really Dispensable?

Kai Wehmeier’s Wittgensteinian Predicate Logic (W-logic) is a formulation of first-order logic under the exclusive interpretation of the quantifiers. W-logic has a distinguished relation constant for co-reference but no sign for objectual identity. Wehmeier denies that objectual identity exists on the grounds that it cannot be a genuine binary relation. Fortunately W-logic is equi-expressive with standard first-order logic with identity and it appears that objectual identity is dispensable across the broader logical enterprise. This paper challenges the latter claim as objectual identity seems to be needed in the exclusive interpretation of quantified modal logic, specifically, for implementing certain kinds of de re quantification.

The Problem of Cross-world Predication

While standard first-order modal logic is quite powerful, it cannot express even very simple sentences like “I could have been taller than I actually am” or “Everyone could have been smarter than they actually are”. These are examples of cross-world predication, whereby objects in one world are related to (sometimes the same) objects in another world. Extending first-order modal logic to allow for cross-world predication in a motivated way has proven to be notoriously difficult. In this paper, I argue that the standard accounts of cross-world predication all leave something to be desired. I then propose an account of cross-world predication based on quantified hybrid logic and show how it overcomes the limitations of these previous accounts. I will conclude by discussing various philosophical consequences and applications of such an account.

A Logic Inspired by Natural Language: Quantifiers As Subnectors

Inspired by the grammar of natural language, the paper presents a variant of first-order logic, in which quantifiers are not sentential operators, but are used as subnectors (operators forming terms from formulas). A quantified term formed by a subnector is an argument of a predicate. The logic is defined by means of a meaning-conferring natural-deduction proof-system, according to the proof-theoretic semantics program. The harmony of the I/E-rules is shown. The paper then presents a translation, called the Frege translation, from the defined logic to standard first-order logic, and shows that the proof-theoretic meanings of both logics coincide. The paper criticizes Frege’s original regimentation of quantified sentences of natural language, and argues for advantages of the proposed variant.

Aristotle's Syllogistic and Core Logic

I use the Corcoran–Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen–Prawitz system of natural deduction, using the universal and existential quantifiers of standard first-order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic is a fragment of my (constructive and relevant) system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle.

The theory of form logic

We investigate a construction schema for first-order logical systems, called “form logic”. Form logic allows us to overcome the dualistic commitment of predicate logic to individual constants and predicates. Dualism is replaced by a pluralism of terms of different “logical forms”. Individual form-logical systems are generated by the determination of a range of logical forms and of the formbased syntax rules for combining terms into formulas. We develop a generic syntax and semantics for such systems and provide a completeness proof for them. To illustrate the idea of form logic, and the possibilities it facilitates, we discuss three particular systems, one of which is the form-logical reconstruction of standard first-order predicate logic.

Compactness in first-order Godel logics

Our aim in this article is twofold. First, for an arbitrary closed subset V of [0,1] containing both 0 and 1, the compactness theorem for the Godel logic GV is investigated. Next, following Cintula and Navara (2004, Fuzzy sets and systems, 143, 59–73) and Tavana et al. (2012, Logic J. IGPL, 20, 254–265), for any subset K of [0,1], the notions of K-satisfiability and K-compactness are introduced for the standard first-order Godel logic Gℝ. It is shown that whenever K is closed, the K-compactness fails for Gℝ if and only if K is infinitely countable and 1 ∉ K.

A semantic proof of strong cut-admissibility for first-order Godel logic

We provide a constructive direct semantic proof of the completeness of the cut-free part of the hypersequent calculus HIF for the standard first-order Godel logic (thereby proving both completeness of the calculus for its standard semantics, and the admissibility of the cut rule in the full calculus). The results also apply to derivations from assumptions (or ‘non-logical axioms’), showing in particular that when the set of assumptions is closed under substitutions, then cuts can be confined to formulas occurring in the assumptions. The methods and results are then extended to handle the (Baaz) Delta connective as well.

First-order logic and some existential sentences

Abstract ‘Quantified pure existentials’ are sentences (e.g., ‘Some things do not exist’) which meet these conditions: (i) the verb EXIST is contained in, and is, apart from quantificational BE, the only full (as against auxiliary) verb in the sentence; (ii) no (other) logical predicate features in the sentence; (iii) no name or other sub-sentential referring expression features in the sentence; (iv) the sentence contains a quantifier that is not an occurrence of EXIST. Colin McGinn and Rod Girle have alleged that standard first-order logic cannot adequately deal with some such existentials. The article defends the view that it can.

Propositional Quantification

Ramsey defined truth in the following way: x is true if and only if ∃p (x = [p] & p). This definition is ill-formed in standard first-order logic, so it is normally interpreted using substitutional or some kind of higher-order quantifier. I argue that these quantifiers fail to provide an adequate reading of the definition, but that, given certain adjustments, standard objectual quantification does provide an adequate reading.

Propositional Quantification

Ramsey defined truth in the following way: x is true if and only if ∃p(x = [p] & p). This definition is ill-formed in standard first-order logic, so it is normally interpreted using substitutional or some kind of higher-order quantifier. I argue that these quantifiers fail to provide an adequate reading of the definition, but that, given certain adjustments, standard objectual quantification does provide an adequate reading.

Some monkey devours every raisin

An elementary exercise in symbolizing an existential to universal relation reveals expressive limitations in standard first-order predicate-quantificational logic. Alternative translations of a sample some-every sentence are considered and rejected after criticism, leaving as the best choice a particular structure that demonstrably does not serve for all predicates available to the ordinary language to which the sample sentence belongs. We explain the difficulties encountered in trying to arrive at an adequate translation of the sentence in classical logic, as background to examining five alternative nonclassical logics to resolve the problem. We settle finally if only provisionally in the process on a version of intensional logic that permits reference and true predication of constitutive properties to nonexistent as well as existent objects.

Branching with Uncertain Semantics

Branching with Uncertain Semantics

Reply to Gregory Landini’s Review of Formal Ontology and Conceptual Realism

In our discussion of Greg Landini’s review, we should distinguish how the logical systems kHST and HST k that I have developed are to be understood in my reconstructions of Gottlob Frege’s and Bertrand Russell’s (1903) ontologies as opposed to how HST k is understood as a (proper) part of my ontology of conceptual realism. Both of these systems are type-free second-order predicate logics that allow predicate expressions (complex or simple) and formulas (propositional forms) to be nominalized and occur in formulas as abstract singular terms. The main logical difference between these systems, as Landini notes, is that whereas kHST contains standard first-order logic (with identity) as a proper part, the system HST k is free of existential presuppositions regarding singular terms, including nominalized predicates as abstract singular terms, which is essential to any argument for Russell’s paradox of predication. In particular, nominalization of the Russell predicate that is not predicable of itself turns out to be denotationless in HST k as an abstract singular term. The main ontological difference between Russell’s and Frege’s ontologies is that one is intensional and the other is extensional. Russell’s (1903) ontology is based on predication as the ontological nexus of propositions, whereas Frege’s is based on predication as a function from properties and relations to truth values. In conceptual realism, predication is based on the mutual saturation of referential and

Designing Visual Languages for Description Logics

Semantic networks were developed in cognitive science and artificial intelligence studies as graphical knowledge representation and inference tools emulating human thought processes. Formal analysis of the representation and inference capabilities of the networks modeled them as subsets of standard first-order logic (FOL), restricted in the operations allowed in order to ensure the tractability that seemed to characterize human reasoning capabilities. The graphical network representations were modeled as providing a visual language for the logic. Sub-sets of FOL targeted on knowledge representation came to be called description logics, and research on these logics has focused on issues of tractability of subsets with differing representation capabilities, and on the implementation of practical inference systems achieving the best possible performance. Semantic network research has kept pace with these developments, providing visual languages for knowledge entry, editing, and presenting the results of inference, that translate unambiguously to the underlying description logics. This paper discusses the design issues for such semantic network formalisms, and illustrates them through detailed examples of significant generic knowledge structures analyzed in the literature, including determinables, contrast sets, genus/differentiae, taxonomies, faceted taxonomies, cluster concepts, family resemblances, graded concepts, frames, definitions, rules, rules with exceptions, essence and state assertions, opposites and contraries, relevance, and so on. Such examples provide important test material for any visual language formalism for logic.

Standard Quantification Theory in the Analysis of English

Standard first-order logic plus quantifiers of all finite orders (“SFOLω”) faces four well-known difficulties when used to characterize the behavior of certain English quantifier phrases. All four difficulties seem to stem from the typed structure of SFOLω models. The typed structure of SFOLω models is in turn a product of an asymmetry between the meaning of names and the meaning of predicates, the element-set asymmetry. In this paper we examine a class of models in which this asymmetry of meaning is removed. The models of this class permit definitions of the quantifiers which allow a desirable flexibility in fixing the domain of quantification. Certain SFOLω type restrictions are thereby avoided. The resulting models of English validate all of the standard first-order logical truths and are free of the four deficiencies of SFOLω models.

A Formal Semantics for Plural Quantification, Intersentential Binding and Anaphoric Pronouns as Rigid Designators

It is well known that certain kinds of plural quantification as well as intersentential pronominal cross-reference in natural language cannot be formalizable by the devices of standard first-order logic, i.e., by a formal language containing variables x,y,z,..., ranging over individuals, the quantifiers V and 3, the truth functional connectives and the equal sign. The most famous example of the former is perhaps the Geach-Kaplan sentence, (1) Some critics admire only one another. An example of the latter is (2) Some woman will land on Mars in the year 2010. (3) She will be an American. (2) and (3) cannot be represented as, (2') ∃x(x is a woman & x lands on Mars in the year 2010) (3') x is an American (2') and (3'), respectively, since the variable in (3'), representing the anaphoric pronoun 'she', is free. But for it to refer to the woman who lands on Mars in the year 2010, it should be bound by the quantifier ∃x' in (2'). In this paper, we shall present a formal language and a model-theoretic semantics (hereafter, a formal semantical system) for such plural quantification and for anaphoric expressions requiring intersentential.binding.

Is Hintikka's Logic First-Order?

Jaakko Hintikka has argued that ordinary first-order logic should be replaced byindependence-friendly first-order logic, where essentially branching quantificationcan be represented. One recurring criticism of Hintikka has been that Hintikka'ssupposedly new logic is equivalent to a system of second-order logic, and henceis neither novel nor first-order. A standard reply to this criticism by Hintikka andhis defenders has been to show that given game-theoretic semantics, Hintikka'sbranching quantifiers receive the exact same treatment as the regular first-orderones. We develop a different reply, based around considerations concerning thenature of logic. In particular, we argue that Hintikka's logic is the logic that bestrepresents the language fragment standard first-order logic is meantto represent. Therefore it earns its keep, and is also properly regarded as first-order.

Argument deletion, thematic roles, and Leibniz's logico-grammatical analysis of relations

I present a formal framework historically faithful to Leibniz's analysis of relational sentences, which: (i) engrafts thematic roles and the non-truth-functional connective insofar as (quatenus) into the monadic fragment of first-order logic; (ii) suggests a plausible ontological picture of thematic roles and relational facts; (iii) supports argument deletion and related inferential patterns that are not taken into account by standard first-order logic.