The identity connective is usually interpreted in non-Fregean logic as an operator representing the identity of situations. This interpretation is related to the modal criterion of the identity of sentence correlates, characteristic of the WT system and some stronger systems. However, this connective can also be interpreted in a different way – as an operator representing the identity of propositions. The “propositional” interpretation is in turn associated with the modal-contents criterion of the identity of sentence correlates. This begs (...) the question of whether there is a system of non-Fregean logic, providing an adequate formalization of this criterion. The aim of the paper is to systematize the metalogical and philosophical context of the issue and to point to a system that provides its solution. (shrink)

The paper is an extended version of a talk given to the XXXth Conference on the History of Logic devoted to the work of Professor Roman Suszko . Its aim is to present Sentential Calculus with Identity in comparison with other formalizations of propositional identity.

This paper presents the main assumptions of Andrzej Grzegorczyk’s last research project concerning the logic of synonymity. It shows that the basis of logic of analytic equivalence, presented in the first part of the work, fully corresponds with these assumptions.

We present a new logic -based approach to the reasoning about knowledge which is independent of possible worlds semantics.? k is a non- Fregean logic whose models consist of propositional universes with subsets for true, false and known propositions. Knowledge is, in general, not closed under rules of inference; the only valid epistemic principles are the knowledge axiom K i??? and some minimal conditions concerning common knowledge in a group. Knowledge is explicit and all forms of the (...) logical omniscience problem are avoided. Various stronger epistemic properties such as positive and/or negative introspection, the K-axiom, closure under logical connectives, etc, can be restored by imposing additional semantic constraints. This yields corresponding sublogics for which we present sound and complete axiomatizations. As a useful tool for general model constructions we study abstract versions of some 3-valued logics in which we interpret truth as knowledge. We establish a connection between? k and the well-known syntactic approach to explicit knowledge proving a result concerning equi-expressiveness. Furthermore, we discuss some self - referential epistemic statements, such as the knower paradox, as relaxations of variants of the liar paradox and show how these epistemic "paradoxes" can be solved in? k. Every specific? k - logic is defined as a certain extension of some underlying classical abstract logic. (shrink)

We present a new logic-based approach to the reasoning about knowledge which is independent of possible worlds semantics. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\in_K}$$\end{document} is a non-Fregean logic whose models consist of propositional universes with subsets for true, false and known propositions. Knowledge is, in general, not closed under rules of inference; the only valid epistemic principles are the knowledge axiom Kiφ → φ and some minimal conditions concerning common knowledge in a group. (...) Knowledge is explicit and all forms of the logical omniscience problem are avoided. Various stronger epistemic properties such as positive and/or negative introspection, the K-axiom, closure under logical connectives, etc. can be restored by imposing additional semantic constraints. This yields corresponding sublogics for which we present sound and complete axiomatizations. As a useful tool for general model constructions we study abstract versions of some 3-valued logics in which we interpret truth as knowledge. We establish a connection between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\in_K}$$\end{document} and the well-known syntactic approach to explicit knowledge proving a result concerning equi-expressiveness. Furthermore, we discuss some self-referential epistemic statements, such as the knower paradox, as relaxations of variants of the liar paradox and show how these epistemic “paradoxes” can be solved in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\in_K}$$\end{document}. Every specific \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\in_K}$$\end{document}-logic is defined as a certain extension of some underlying classical abstract logic. (shrink)

We show that there are continuum many different extensions of SCI (the basic theory of non-Fregean propositional logic) that lie below WF (the Fregean extension) and are closed under substitution. Moreover, continuum many of them are independent from WB (the Boolean extension), continuum many lie above WB and are independent from WH (the Boolean extension with only two values for the equality relation), and only countably many lie between WH and WF.

"I. Roman Suszko (9.11.1919, Podobora – 3.06.1979, Warsaw) was one of the most fascinating personalities in Polish academic community after the Second World War and one of the most outstanding logicians of the time. He was above all a scientist but he also participated in academic life. He was Dean of the Faculty of Philosophy at Warsaw University for two terms of office. He studied abstract problems of logic, but also played a part in the satirical film Rejs [The (...) Cruise] directed by M. Piwowski. Suszko was involved in various scientific problems, for example: logical syntax of natural language, liar antynomy, logical probability; but two of his achievements have the greatest value for philosophy of science, i.e., diachronic logic and non-Fregean logic. Like Ferdinand de Saussure, who, in his monograph Cours de Linguistique Générale.. (shrink)

We study the non-Fregean propositional logic with propositional quantifiers, denoted by $\mathsf{SCI}_{\mathsf{Q}}$. We prove that $\mathsf{SCI}_{\mathsf{Q}}$ does not have the finite model property and that it is undecidable. We also present examples of how to interpret in $\mathsf{SCI}_{\mathsf{Q}}$ various mathematical theories, such as the theory of groups, rings, and fields, and we characterize the spectra of $\mathsf{SCI}_{\mathsf{Q}}$-sentences. Finally, we present a translation of $\mathsf{SCI}_{\mathsf{Q}}$ into a classical two-sorted first-order logic, and we use the translation to prove some (...) model-theoretic properties of $\mathsf{SCI}_{\mathsf{Q}}$. (shrink)

The paper concerns Grzegorczyk’s non-Fregean logics that are intended to be a formal representation of the equimeaning relation defined on descriptions. We argue that the main Grzegorczyk logics discussed in the literature are too strong and we propose a new logical system, \, which satisfies Grzegorczyk’s fundamental requirements. We present a sound and complete semantics for \ and we prove that it is decidable. Finally, we show that many non-classical logics are extensions of \, which makes it a generic (...) non-Fregean logic. (shrink)

We show that there are continuum many different non-Fregean sentential logics that have adequate models. The proof is based on the construction of a special class of models of the power of the continuum.

The non-Fregean logic SCI is obtained from the classical sentential calculus by adding a new identity connective = and axioms which say ?a = ß' means ?a is identical to ß'. We present complete and sound proof system for SCI in the style of Rasiowa-Sikorski. It provides a natural deduction-style method of reasoning for the non-Fregean sentential logic SCI.

According to Frege's principle the denotation of a sentence coincides with its truth-value. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense.A 2nd-order matrix is an algebra together with an algebraic closed set system on its universe. A deductive system is a second-order matrix over the formula algebra of some fixed but arbitrary language. A second-order matrix (...) A is Fregean if, for any subset X of A, the set of all pairs a,b such that X{a} and X{b} have the same closure is a congruence relation on A. Hence a deductive system is Fregean if interderivability is compositional. The logics intermediate between the classical and intuitionistic propositional calculi are the paradigms for Fregean logics. Normal modal logics are non-Fregean while quasi-normal modal logics are generally Fregean.The main results of the paper: Fregean deductive systems that either have the deduction theorem, or are protoalgebraic and have conjunction, are completely characterized. They are essentially the intermediate logics, possibly with additional connectives. All the full matrix models of a protoalgebraic Fregean deductive system are Fregean, and, conversely, the deductive system determined by any class of Fregean second-order matrices is Fregean. The latter result is used to construct an example of a protoalgebraic Fregean deductive system that is not strongly algebraizable. (shrink)

We present $\in_I$-Logic (Epsilon-I-Logic), a non-Fregean intuitionistic logic with a truth predicate and a falsity predicate as intuitionistic negation. $\in_I$ is an extension and intuitionistic generalization of the classical logic $\in_T$ (without quantifiers) designed by Sträter as a theory of truth with propositional self-reference. The intensional semantics of $\in_T$ offers a new solution to semantic paradoxes. In the present paper we introduce an intuitionistic semantics and study some semantic notions in this broader context. Also we (...) enrich the quantifier-free language by the new connective < that expresses reference between statements and yields a finer characterization of intensional models. Our results in the intuitionistic setting lead to a clear distinction between the notion of denotation of a sentence and the here-proposed notion of extension of a sentence (both concepts are equivalent in the classical context). We generalize the Fregean Axiom to an intuitionistic version not valid in $\in_I$. A main result of the paper is the development of several model constructions. We construct intensional models and present a method for the construction of standard models which contain specific (self-)referential propositions. (shrink)

This paper asks which free logic a Fregean should adopt. It examines options within the tradition including Carnap’s (1956) chosen object theory, Lehmann’s (1994, 2002) strict Fregean free logic, Woodruff’s (1970) strong table about Boolean operators and Bencivenga’s (1986, 1991) supervaluational semantics. It argues for a neutral free logic in view of its proximity towards explaining natural languages. However, disagreeing with Lehmann, it claims a Fregean should adopt the strong table based on Frege’s discussion (...) on generality. Supervaluation uses strong table and aims to give it a semantic justification. However, supervaluation is in turn justified by convention or mental experiments, which Lehmann argues as inadequate. The paper proposes a new justification of supervaluation based on sense and two-dimensional semantics. The resulting model, coined Supervaluational Neutral Free Logic (SNFL), resolves many conflicts between Lehmann and Bencivenga while staying close with Frege’s discussions about non-denotation. It also provides new insights into the relations among truth, logical truth, and supervaluated truth (or supertruth, for short). (shrink)

One of the most prominent myths in analytic philosophy is the so- called “Fregean Axiom”, according to which the reference of a sentence is a truth value. In contrast to this referential semantics, a use-based formal semantics will be constructed in which the logical value of a sentence is not its putative referent but the information it conveys. Let us call by “Question Answer Semantics” (thereafter: QAS) the corresponding formal semantics: a non-Fregean many-valued logic, where the meaning (...) of any sentence is an ordered n-tupled of yes-no answers to corresponding questions. A sample of philosophical problems will be approached in order to justify the relevance of QAS. These include: (1) illocutionary forces, and the logical analysis of speech-acts; (2) the variety of logical negations, and their characterization in terms of restricted ranges of logical values; (3) change in meaning, and the use of dynamic oppositions for belief sets. (shrink)

George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation (...) fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership. (shrink)

This paper provides a new argument for a position of Crispin Wright's: given that ethical statements can be embedded within all sorts of sentential operators and are subject to definite standards of warrantedness, they must have truth conditions. Allan Gibbard's normative logic' is the only noncognitivist logic that stands a chance of avoiding Geach's Fregean objection. But what, according to Gibbard, is the point of avoiding inconsistency in one's ethical statements? He must say that it is to (...) ensure that one's ethical statements have a property that has certain features. But the only property that has those features is truth. (shrink)

This paper offers a framework for extending Arnon Avron and Iddo Lev’s non-deterministic semantics to quantified predicate logic with the intent of resolving several problems and limitations of Avron and Anna Zamansky’s approach. By employing a broadly Fregean picture of logic, the framework described in this paper has the benefits of permitting quantifiers more general than Walter Carnielli’s distribution quantifiers and yielding a well-behaved model theory. This approach is purely objectual and yields the semantical equivalence of both (...) α-equivalent formulae and formulae differing only by codenotative terms. Finally, we make a brief excursion into non-deterministic model theory, proving a strong Łoś’ Theorem and compactness for all finitely-valued, non-deterministic logics whose quantifiers have intensions describable in a first order metalanguage. (shrink)

This paper defines provably non-trivial theories that characterize Frege’s notion of a set, taking into account that the notion is inconsistent. By choosing an adaptive underlying logic, consistent sets behave classically notwithstanding the presence of inconsistent sets. Some of the theories have a full-blown presumably consistent set theory T as a subtheory, provided T is indeed consistent. An unexpected feature is the presence of classical negation within the language.

There are logics where necessity is defined by means of a given identity connective: \ is a tautology). On the other hand, in many standard modal logics the concept of propositional identity \ can be defined by strict equivalence \}\). All these approaches to modality involve a principle that we call the Collapse Axiom : “There is only one necessary proposition.” In this paper, we consider a notion of PI which relies on the identity axioms of Suszko’s non-Fregean (...) class='Hi'>logic SCI. Then S3 proves to be the smallest Lewis modal system where PI can be defined as SE. We extend S3 to a non-Fregean logic with propositional quantifiers such that necessity and PI are integrated as non-interdefinable concepts. CA is not valid and PI refines SE. Models are expansions of SCI-models. We show that SCI-models are Boolean prealgebras, and vice-versa. This associates non-Fregean logic with research on Hyperintensional Semantics. PI equals SE iff models are Boolean algebras and CA holds. A representation result establishes a connection to Fine’s approach to propositional quantifiers and shows that our theories are conservative extensions of S3–S5, respectively. If we exclude the Barcan formula and a related axiom, then the resulting systems are still complete w.r.t. a simpler denotational semantics. (shrink)

The First International Workshop brings together researchers from the theoretical ends of the logic programming and artificial intelligence communities to discuss their mutual interests. Logic programming deals with the use of models of mathematical logic as a way of programming computers, where theoretical AI deals with abstract issues in modeling and representing human knowledge and beliefs. One common ground is nonmonotonic reasoning, a family of logics that includes room for the kinds of variations that can be found (...) in human reasoning. Topics covered: Stable Semantics. Default Logic. AutoEpistemic Logic. Truth Maintenance Systems. Implementation Issues. Diagnosis. Applications. Inheritance Reasoning. Logics of Belief. Inconsistency and Non-Monotonicity. (shrink)

For decades Ryszard Wójcicki has been a highly influential scholar in the community of logicians and philosophers. Our aim is to outline and comment on some essential issues on logic, methodology of science and semantics as seen from the perspective of distinguished contributions of Wójcicki to these areas of philosophical investigations.

Contemporary logicians have expanded upon the old notions of psychologism in logic and proposed new, weakened versions of it. Those weakened versions postulate that psychologistic logic does not have to inform about the ontology or metaphysics of reasoning. Instead, logic applied in cognitive science could serve as one of many paradigms for making empirical predictions about the observable process of human reasoning. The purpose of this article is to entertain this notion and answer the question: what properties (...) should a logical system formally representing actual human reasoning have? Based on the existing evidence from cognitive science and neuroscience we identified three potential candidates: context-sensitivity (satisfied for example by adaptive logics), content-sensitivity (satisfied by non-Fregean logics) and probabilism (satisfied for example by fuzzy logics). (shrink)

This paper focuses on the relation between logic and ontology. In particular, it demonstrates how classical logical theory can clarify the ontological part of Ludwig Wittgenstein’s Tractatus Logico-Philosophicus. To this end, the work examines the adequacy of a formal system that was devised by the Polish logician, mathematician and philosopher Roman Suszko (1919–1979) as a model for the Tractatus. Following a brief explanation of the Tractarian ontology, the main ideas of Suszko’s system and its philosophical significance will be considered. (...) The latter will be illustrated in the context of central Tractarian concepts. Finally, two implications for a better understanding of the Tractarian ontology will be pointed out. (shrink)

Logic and psychology overlap in judgment, inference and proof. The problems raised by this commonality are notoriously difficult, both from a historical and from a philosophical point of view. Sundholm has for a long time addressed these issues. His beautiful piece of work [A Century of Inference: 1837-1936] begins by summarizing the main difficulty in the usual provocative manner of the author: one can start, he says, by the act of knowledge to go to the object, as the Idealist (...) does; one can also start by the object to go to the act, in the Realist mood; never the two shall meet. He is himself inclined to accept the first perspective as the right one and he has eventually developed an original version of antirealism which starts, not from considerations about the publicity of meaning, in the manner of Dummett, but from an epistemic standpoint, trying to search in a non-Fregean tradition of analysis of judgement and cognate notions a way of founding constructivist semantics. The present paper ploughes the same field. We concentrate on the significance, for Sundholm’s program, of the perspective that has been opened by Twardowski in his important essay on acts and products (1912. (shrink)

An alternative semantic framework is proposed in the following to reconstruct and make sense of “Eastern logics”: a Question-Answer Semantics (thereafter: QAS), including a set of questions-answers and a finite number of ensuing non-Fregean logical values. Thus, meaning is provided by yes-no answers to corresponding questions about relevant properties. These logical values help to show that the saptabhaṅgī (and its dual, viz., the Buddhist Mādhyamaka catuṣkoṭi) is not a many-valued paraconsistent logic but, rather, a one-valued proto-logic: a (...) constructive machinery that serves as a formal theory of judgment, rather than a Tarskian-like theory of consequence. Such an explanatory model of contradiction assumes a deep redefinition of logical values. (shrink)

Logic is a celebrated representation language because of its formal generality. But there are two senses in which a logic may be considered general, one that concerns a technical ability to discriminate between different types of individuals, and another that concerns constitutive norms for reasoning as such. This essay embraces the former, permutation-invariance conception of logic and rejects the latter, Fregean conception of logic. The question of how to apply logic under this pure invariantist (...) view is addressed, and a methodology is given. The pure invariantist view is contrasted with logical pluralism, and a methodology for applied logic is demonstrated in remarks on a variety of issues concerning non-monotonic logic and non-monotonic inference, including Charles Morgan’s impossibility results for non-monotonic logic, David Makinson’s normative constraints for non-monotonic inference, and Igor Douven and Timothy Williamson’s proposed formal constraints on rational acceptance. (shrink)

The paper begins with a more carefully stated version of ontologically neutral (ON) logic, originally introduced in (Hailperin, 1997). A non-infinitistic semantics which includes a definition of potential infinite validity follows. It is shown, without appeal to the actual infinite, that this notion provides a necessary and sufficient condition for provability in ON logic.

It is claimed hereby that, against a current view of logic as a theory of consequence, opposition is a basic logical concept that can be used to define consequence itself. This requires some substantial changes in the underlying framework, including: a non-Fregean semantics of questions and answers, instead of the usual truth-conditional semantics; an extension of opposition as a relation between any structured objects; a definition of oppositions in terms of basic negation. Objections to this claim will be (...) reviewed. (shrink)

In this paper, we introduce a variant of second-order propositional modal logic interpreted on general (or Henkin) frames, \(SOPML^{\mathcal {H}}\), and present a decidable fragment of this logic, \(SOPML^{\mathcal {H}}_{dec}\), that preserves important expressive capabilities of \(SOPML^{\mathcal {H}}\). \(SOPML^{\mathcal {H}}_{dec}\) is defined as a _modal loosely guarded fragment_ of \(SOPML^{\mathcal {H}}\). We demonstrate the expressive power of \(SOPML^{\mathcal {H}}_{dec}\) using examples in which modal operators obtain (a) the epistemic interpretation, (b) the dynamic interpretation. \(SOPML^{\mathcal {H}}_{dec}\) partially satisfies the (...) principle of non-Fregean logic: two different _atomic_ propositions with the same truth value can have different contents. In \(SOPML^{\mathcal {H}}_{dec}\), we also define _relating connectives_ and show that the _weak Boethius’ Thesis_ built using these connectives is a valid formula of \(SOPML^{\mathcal {H}}_{dec}\). (shrink)

In Frege’s analysis of the relationship between science and fiction there are two important aspects, which the paper will discuss. It shows that Frege makes a strict distinction between Dichtung und Wissenschaft on the level of object language but not on the level of metalanguage. In his “On Sense and Reference” and in scattered remarks elsewhere Frege explains the semantics of scientific and everyday discourse. As a kind of side product he presents an explication of the concept of fictional discourse (...) concerning questions of illocutionary force and reference. Here Frege anticipates J. R. Searle’s speech-act-theory of fictional discourse, which allows to understand works of fiction as consisting of fictional discourse. On the basis of Frege’s distinctions this approach is defended against ontological arguments, which make use of terms like ‘fictive entities’ or ‘non-existent objects’ in the Meinongian tradition. Frege excludes the connotative or figurative elements of language, called “colourings and shadings” of sense or thought, from the scientific use of language and assigns such elements to “the art of poetry” or “eloquence”. The fact that the expression ‘colouring’ is itself a figurative term, raises a paradoxical question: To what extent does understanding Frege’s own explanation of the difference between sense and colouring depend on the poetic or at least rhetoric use of language? There are reasons to believe that Frege was—even if only reluctantly—aware of this paradox. Otherwise he would not have repeatedly emphasized that explanations of categorial logical distinctions cannot dispense with “figurative expressions”. Insofar as such distinctions are “reliant upon the accommodating understanding of the reader”, they pay tribute to a rhetoric of cognition. While Frege denies that colourings contribute to cognitive content, he must still admit that they make an indispensable protreptic contribution to conveying cognition. (shrink)

The volume deals with the history of logic, the question of the nature of logic, the relation of logic and mathematics, modal or alternative logics (many-valued, relevant, paraconsistent logics) and their relations, including translatability, to classical logic in the Fregean and Russellian sense, and, more generally, the aim or aims of philosophy of logic and mathematics. Also explored are several problems concerning the concept of definition, non-designating terms, the interdependence of quantifiers, and the idea (...) of an assertion sign. The contributions concerned with Wittgenstein's investigations into the philosophy of logic and mathematics pursue issues relating to logical necessity, the undeniability of the law of the excluded middle, and the source of self-evidence, often characterized in the literature as the "rule-following considerations". Additionally, they examine Wittgenstein's attitudes towards the very idea of set-theory as a possible foundation for arithmetic. The volume also includes a number of contributions on specific issues concerning Wittgenstein's views on moral and religious judgements. (shrink)

A discriminator logic is the 1 -assertional logic of a discriminator variety V having two constant terms 0 and 1 such that V ⊨ 0 1 iff every member of V is trivial. Examples of such logics abound in the literature. The main result of this research announcement asserts that a certain non-Fregean deductive system SBPC, which closely resembles the classical propositional calculus, is canonical for the class of discriminator logics in the sense that any discriminator (...) class='Hi'>logic S can be presented as an axiomatic extension of SBPC by a set of extensional logical connectives taken from the language of S. The results outlined in this research announcement are extended to several generalisations of the class of discriminator logics in the main work. (shrink)

The paper shows that it is possible to obtain a "slingshot" result in Gödel's theory of positiveness in the presence of the theorem of the necessary existence of God. In the context of the reconstruction of Gödel's original "slingshot" argument on the suppositions of non-Fregean logic, this is a natural result. The "slingshot" result occurs in sufficiently strong non-Fregean theories accepting the necessary existence of some entities. However, this feature of a Gödelian theory may be considered not (...) as a trivialisation, but as an intended consequence of Gödel's ontotheological views. (shrink)

The well-known algebraic semantics and topological semantics for intuitionistic logic (Int) is here extended to Wansing's bi-intuitionistic logic (2Int). The logic 2Int is also characterised by a quasi-twist structure semantics, which leads to an alternative topological characterisation of 2Int. Later, notions of Fregean negation and of unilateralisation are proposed. The logic 2Int is extended with a ‘Fregean negation’ connective ∼, obtaining 2Int∼, and it is showed that the logic N4⋆ (an extension of Nelson's (...) paraconsistent logic) results to be the unilateralisation of 2Int∼ via ∼. The logic 2Int∼ is also characterised by a Kripke-style semantics, a twist structure semantics and a topological semantics. (shrink)