Monomodal logic has exactly two maximally normal logics, which are also the only quasi-normal logics that are Post complete, and they are complete for validity in Kripke frames. Here we show that addition of a propositional constant to monomodal logic allows the construction of continuum many maximally normal logics that are not valid in any Kripke frame, or even in any complete modal algebra. We also construct continuum many quasi-normal Post complete logics that are not normal. The set of (...) extensions of S4.3 is radically altered by the addition of a constant: we use it to construct continuum many such normal extensions of S4.3, and continuum many non-normal ones, none of which have the finite model property. But for logics with weakly transitive frames there are only eight maximally normal ones, of which five extend K4 and three extend S4. (shrink)

We show that there are denumerably many Post-complete normal modal logics in the language which includes an additional propositional constant. This contrasts with the case when there is no such constant present, for which it is well known that there are only two such logics.

It is shown how to model propositional constants within the simplified Routley-Meyer semantics. Various axioms and rules allowingthe definition of modal operators, implicative negations, enthymematical conditionals, and propositions expressing various infinite conjunctions anddisjunctions are set forth and shown to correspond to specific frame conditions. Two propositional constants which are both often designated as “the Ackermann constant” are shown to capture two such “infinite” propositions: The conjunction of every logical law and the conjunction of every truth – (...) what Anderson and Belnap called the “world” constant. (shrink)

In this paper I give a characterization of the closed fragment of the provability logic of $I \triangle_0 + \mathrm{EXP}$ with a propositional constant for $\mathrm{EXP}$. In three appendices many details on arithmetization are provided.

This paper provides a semantic analysis of English rise-fall-rise (RFR) intonation as a focus quantifier over assertable alternative propositions. I locate RFR meaning in the conventional implicature dimension, and propose that its effect is calculated late within a dynamic model. With a minimum of machinery, this account captures disambiguation and scalar effects, as well as interactions with other focus operators like ‘only’ and clefts. Double focus data further support the analysis, and lead to a rejection of Ward and Hirschberg’s (Language (...) 61:747–776, 1985) claim that RFR never disambiguates. Finally, I draw out connections between RFR and contrastive topic (CT) intonation (Büring, Linguist Philos 26:511–545, 2003), and show that RFR cannot simply be reduced to a sub-case of CT. (shrink)

This article concerns the metatheory of a class of modal logics whose language includes propositional constants of various kinds. The main novelties are the use of general frames with specific restrictions and the definition of the strict range of a formula. Many examples from the literature are treated within the framework provided and some traditional model-theoretic issues such as preservation results concerning the validity of formulas and definability results concerning frame properties are addressed.

In this chapter, we start by spelling out three important features that distinguish expressives—utterances that express emotions and other affects—from descriptives, including those that describe emotions (Section 1). Drawing on recent insights from the philosophy of emotion and value (2), we show how these three features derive from the nature of affects, concentrating on emotions (3). We then spell out how theories of non-natural meaning and communication in the philosophy of language allow claims that expressives inherit their meaning from specificities (...) of emotions—namely, from being felt, evaluative attitudes toward propositional or non-propositional contents (4). (shrink)

Holton has drawn attention to a new semantic universal, according to which no natural language has contrafactive attitude verbs. Because factives are universal across natural languages, Holton’s universal is part of a major asymmetry between factive and contrafactive attitude verbs. We previously proposed that this asymmetry arises partly because the meaning of contrafactives is significantly harder to learn than that of factives. Here we extend our work by describing an additional computational experiment that further supports our hypothesis.

The present study examined the relationship between left–right discrimination (LRD) performance and handedness, sex and cognitive abilities. In total, 31 men and 35 women – with a balanced ratio of left-and right-handers – completed the Bergen Left–Right Discrimination Test. We found an advantage of left-handers in both identifying left hands and in verifying “left” propositions. A sex effect was also found, as women had an overall higher error rate than men, and increasing difficulty impacted their reaction time more than it (...) did for men. Moreover, sex interacted with handedness and manual preference strength. A negative correlation of LRD reaction time with visuo-spatial and verbal long-term memory was found independently of sex, providing new insights into the relationship between cognitive skills and performance on LRD. (shrink)

This article presents a social learning perspective as a means to analyze and facilitate collective decision making and action in managed resource systems such as platforms. First, the social learning perspective is developed in terms of a normative and analytical framework. The normative framework entails three value principles, namely, systems thinking, experimentation, and communicative rationality. The analytical framework is built up around the following questions: who learns, what is learned, why it is learned, and how. Next, this perspective is used (...) to analyze two managed resource systems: Fishery management in Lake Aheme, Benin and water resources management in Gelderland, The Netherlands. To assess platform performance in resource use negotiation, emerging lessons from the case studies are combined with propositions concerning membership of platforms, accessibility of platform meetings, skills and relations of platform members, realization of platforms, and third party facilitation of platform activities. (shrink)

The main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility of the techniques is demonstrated by showing that (γ) is admissible for RQ* (R with constant domain quantifiers). The status (...) of the admissibility of (γ) for RQ* was, to the best of the author’s knowledge, an open problem. Further extensions of these results will be explored in the sequel(s). (shrink)

This paper studies the complexity of constant depth propositional proofs in the cedent and sequent calculus. We discuss the relationships between the size of tree-like proofs, the size of dag-like proofs, and the heights of proofs. The main result is to correct a proof construction in an earlier paper about transformations from proofs with polylogarithmic height and constantly many formulas per cedent.

LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives ¬ and $\bigwedge, \bigvee$. Then for every d ≥ 0 and n ≥ 2, there is a set Td n of depth d sequents of total size O which are refutable in LK by depth d + 1 proof of size exp) but such that every depth d refutation must have the size at least exp). The sets Td n express a weaker form of the (...) pigeonhole principle. (shrink)

Sentential constants have been part of the R environment since Church [1]. They have had diverse uses in explicating relevant ideas and in sim- plifying them technically. Of most interest have been the Ackermann pair of constants t; f, functioning conceptually as a least truth, and as a greatest , under the ordering of propositions under true impli- cation. Also interesting have been the Church constants F; T, functioning similarly as least greatest propositions.

This paper proves exponential separations between depth d-LK and depth -LK for every utilizing the order induction principle. As a consequence, we obtain an exponential separation between depth d-LK and depth -LK for . We investigate the relationship between the sequence-size, tree-size and height of depth d-LK-derivations for , and describe transformations between them. We define a general method to lift principles requiring exponential tree-size -LK-refutations for to principles requiring exponential sequence-size d-LK-refutations, which will be described for the Ramsey principle (...) and d=0. From this we also deduce width lower bounds for resolution refutations of the Ramsey principle. (shrink)

The Routley-Meyer relational semantics for relevant logics is extended to give a sound and complete model theory for many propositionally quantified relevant logics (and some non-relevant ones). This involves a restriction on which sets of worlds are admissible as propositions, and an interpretation of propositional quantification that makes ∀ pA true when there is some true admissible proposition that entails all p -instantiations of A . It is also shown that without the admissibility qualification many of the systems considered (...) are semantically incomplete, including all those that are sub-logics of the quantified version of Anderson and Belnap’s system E of entailment, extended by the mingle axiom and the Ackermann constant t . The incompleteness proof involves an algebraic semantics based on atomless complete Boolean algebras. (shrink)

Fully adequate answers to these questions are best provided in a comprehensive philosophy of logic. Within shorter compass, it is nevertheless possible to be guided by some conditions that are necessary to adequate answers. These will be results of the analysis of propositions and statements. They are necessary, since no answers to the questions about the constants will be acceptable if, for example, it follows from the answers that propositions or statements cannot be unities, or that propositions or statements (...) cannot be true or false. (shrink)

We present a new axiomatization of the deontic fragment of Anderson's relevant deontic logic, give an Andersonian reduction of a relevant version of Mally's deontic logic previously discussed in this journal, study the effect of adding propositional quantification to Anderson's system, and discuss the meaning of Anderson's propositional constant in a wide range of Andersonian deontic systems.

Relevant propositional dynamic logics have been sporadically discussed in the broader context of modal relevant logics, but have not come up for sustained investigation until recently. In this paper, we develop a philosophical motivation for these systems, and present some new results suggested by the proposed motivation. Among these, we’ll show how to adapt some recent work to show that the extensions of relevant logics by the extensional truth constants \ are complete with respect to a natural class (...) of ternary relation models to show a similar result for the constant-free versions of the logic. In addition, we prove that the logics in question satisfy the variable sharing property, vindicating the claim that they really are relevant logics. (shrink)

Frege famously argued that truth is not a property or relation. In the “Notes on Logic” Wittgenstein emphasised the bi-polarity of propositions which he called their sense. He argued that “propositions by virtue of sense cannot have predicates or relations.” This led to his fundamental thought that the logical constants do not represent predicates or relations. The idea, however, has wider ramifications than that. It is not just that propositions cannot have relations to other propositions but also that they (...) cannot have relations to anything at all. The paper explores the consequences of this insight for the way in which we should read the Tractatus. In the “Notes on Logic” the insight led to Wittgenstein's emphasis on “facts” in any attempt to understand the nature of symbolism. This emphasis is continued in the Tractatus. It is central to his view that propositions are facts which picture facts which prevent us from construing such picturing as a relation between what pictures and what is pictured. It illuminates the importance of context principle with regard to the distinction between showing and saying to which Wittgenstein attached so much importance and it underlies the non-relational view of psychological propositions which he advocates. Finally, if propositions by virtue of sense cannot have predicates or relations the paradox at the end of a work which consist largely of propositions about propositions becomes intelligible. (shrink)

According to the standard definition, a first-order theory is categorical if all its models are isomorphic. The idea behind this definition obviously is that of capturing semantic notions in axiomatic terms: to be categorical is to be, in this respect, successful. Thus, for example, we may want to axiomatically delimit the concept of natural number, as it is given by the pre-theoretic semantic intuitions and reconstructed by the standard model. The well-known results state that this cannot be done within first-order (...) logic, but it can be done within second-order one. Now let us consider the following question: can we axiomatically capture the semantic concept of conjunction? Such question, to be sure, does not make sense within the standard framework: we cannot construe it as asking whether we can form a first-order (or, for that matter, whatever-order) theory with an extralogical binary propositional operator so that its only model (up to isomorphism) maps the operator on the intended binary truth-function. The obvious reason is that the framework of standard logic does not allow for extralogical constants of this type. But of course there is also a deeper reason: an existence of a constant with this semantics is presupposed by the very definition of the framework1. Hence the question about the axiomatic capturability of concunction, if we can make sense of it at all, cannot be asked within the framework of standard logic, we would have to go to a more abstract level. To be able to make sense of the question we would have to think about a propositional ‘proto-language’, with uninterpreted logical constants, and to try to search out axioms which would fix the denotations of the constants as the intended truth-functions. Can we do this? It might seem that the answer to this question is yielded by the completeness theorem for the standard propositional calculus: this theorem states that the axiomatic delimitation of the calculus and the semantic delimitation converge to the same result.. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:76 Reviews PROPOSITIONAL ANALYSIS David Blitz Philosophy Dept. and Peace Studies / Central Connecticut State U. New Britain, ct 06050, usa [email protected] Graham Stevens. The Russellian Origins of Analytical Philosophy: Bertrand Russell and the Unity of the Proposition. London and New York: Routledge, 2005. Pp. xii, 185. isbn: 978-0-415-36044-9 (hb). £80.00. us$155.95. Graham Stevens has written a short book on a diUcult subject: the unity of the proposition. (...) While the title of the volume is The Russellian Origins of Analytic Philosophy, the underlying theme is the comparatively little discussed problem of how the elements of a proposition come together to form a unity, as indicated in the book’s subtitle: “Bertrand Russell and the Unity of the Prop­ osition”. Stevens traces this concept, as a unifying thread or “central concern”, throughout Russell’s many philosophies which constitute for Stevens a “consis­ tent and constantly evolving philosophy” (p. 2) stretching from the earliest log­ ical writings to the later works of the 1930s and 1940s. This is an important and well written volume, which Russellians should have in their personal and uni­ versity libraries. That the “unity of the proposition” is a problem at all requires the reader to return to the neo-Hegelian idealism which Russell encountered as a student at Cambridge.1 Russell, inXuenced by Peano and Frege, broke with the doctrine of internal relations, defended by Bradley and others, according to which external relations are unreal and false appearance. However, an important element of idealism remains even during the Principia period, due to Russell’s continued belief that a proposition is a non-linguistic, non-mental, abstract entity; these elements are all of the same ontological kind, which can be identiWed through analysis and recombined so that their synthesis reconstitutes the original “unity of the proposition”. This turns out to be a very tall order which will occupy Russell for nearly half a century. Consider the standard example Russell gives: “Desdemona loves Cassio”. According to Russell, this proposition (independently of its truth or falsity) actually contains its “simple constituents”z—zwith “Desdemona” and “Cassio” being things and “loves” a concept. Russell, as a founder of analytic philosophy, has to defend the legitimacy of this analysis from the claim that it has reduced 1 Chapter 1: “Russell, Frege and the Analysis of Unities”. December 2, 2009 (5:28 pm) E:\CPBR\RUSSJOUR\TYPE2901\russell 29,1 060 red.wpd E:\CPBR\RUSSJOUR\TYPE2901\russell 29,1 060 red.wpd Reviews 77 the structured unity of the proposition into an unstructured set of parts (or heap) where the original order of combination has been lost. The problem of is that merely listing “Desdemona”, “Cassio” and “loves” does not indicate the way in which the terms were originally combined or re­ lated. For reasons to be made clear later, this is referred to in the literature as the “narrow order problem”. In particular, the question arises as to how analysis enables us to assert that it is Desdemona who loves Cassio (supposing this to be the case), rather than the other way around (since love may not be requited). Stevens deals with logical form in a later chapter devoted to the problem of belief statements, but the notion applies here as well: analysis seems to have skip­ ped the logical form, the way in which the constituents are combined. But if the logical form is included, then there seems to be an extra constituent in the prop­ osition not given by direct inspection, and analysis would appear to have added something above and beyond what was there before the analysis. Adding linking relations to indicate how the components were originally combined does not help either. If there is some relation Lz which relates “Desdemona” and “loves” and another relation Rz which relates “loves” and “Cassio”, then there are Wve constituents, not three. Moreover, there is now room for inWnite regress, with a relation L2 linking “Desdemona” and L, R2 linking “loves” and “Cassio”, and so on. But without adding these additional items, there is an apparent failure of an­ alysis to fully provide the order of the elements present in the original... (shrink)

La réaffirmation des constantes de la responsabilité dans les différents projets de réforme de cette matière marque la fonction réparatrice de cette institution, bien que les auteurs des propositions se soient, pour la plupart, contentés d’une codification à droit constant, loin de grandes ruptures conceptuelles, marquant la stabilité des conditions relatives au préjudice et au lien de causalité.

I build a canonical model for constant domain basic first-order logic (BQLCD), the constant domain first-order extension of Visser’s basic propositional logic, and use the canonical model to verify that BQLCD satisfies the disjunction and existence properties.

In this thesis, I deal with the notions of a condition holding for some proposition and a proposition being true in a certain number of possible worlds. These notions are called propositional quantifiers and numerical modalizers respectively. In each chapter, I attempt to dispose of a system. A system consists of: a language; axioms and rules of inference; and an interpretation. To dispose of a system is to prove its decidability and its consistency and completeness for the given interpretation. (...) I shall, in passing, make applications to de definability, translatability and other topics. In Chapter 1, I consider the system 5SQ. Its language is that of 55 with Q as a fresh unary operator. Its axioms and rules of inference are those for 85 plus the following special axiom-schemes for Q: 1) QA::>MA 2) Q A ::> L V L 3) L ::> 4) Q A ::> L Q A. 'QA' is interpreted as 'A is true in exactly one possible world.' I dispose of the system by showing that every formula in it is equivalent to one in normal form. In Chapter 2 I consider the system S5n. Its language is that of S5 but with the unary operators Qk for each non-negative integer k. Its axioms and rules are those of S5 plus the following special axiom-schemes for Qk: 1) Qk A>~Q1 A, 1˂K 2) Qk A ≡ Vki≡oQi ˄ Qk-I 3) L ˂ 4) Qk A ˂ L Qk A 5) Qo A ≡ L ~ A, 1 ≥ 0, k ≥ 1 ‘Qk A’ is interpreted as 'A is true in exactly k possible worlds.' I dispose of this system by generalising the normal forms of S5Q. In the chapters 3-5, I consider three systems which result from adding propositional quantifiers to S5. The first two systems, S5╥+ and S5╥, contain the usual axioms and rules for quantifiers. The first contains, in addition, the axiom-scheme G = )). The last, S5╥-, results from S5╥ by restricting the Scheme of Specification, viz., A > A, B free for P in A, to formulas B of the propositional calculus. To interpret these systems we must specify which propositions the variable P ranges over. For St╥-, we merely require that if p and q are propositions, then and are also propositions. For S5╥+, we also require that each possible world be describable i.e. that there be a proposition which is true in that world alone. And for S5╥, we require not that each possible world be describable but that there be a proposition which is true in just those possible worlds which are describable. Again, we dispose of the systems by normal forms. This requires that we eliminate quantifiers and nested occurrences of L by adding new symbols to the language. For S5╥+, the operators Qk suffice. For S5╥, the operators Qk suffice. For S5╥, we also require the constant g and a fresh unary operator N. For S5╥-, even greater additions are required. In the last two chapters, 6 and 7, I turn to systems which have essentially the same language as S5╥. However, ‘Qk A' is now interpreted as 'A is true in exactly k possible worlds accessible from the given world.' Different conditions on R, the relation of accessibility, lead to different axioms. In chapter 6 I consider the conditions of reflexivity, symmetry and transitivity, and in Chapter 7 the conditions of being a partial, convergent, total or dense order. I prove consistency and completeness by the method of maximally consistent systems. The method can yield decidability results, but I do not go into the matter. I have, as a rule, not given acknowledgements for well-established results or terminology. The main references are at the end of each chapter. Fuller references are in the bibliography. (shrink)

Terms of logic like 'all' and 'some' can be understood in relation to one another purely in thought. Individual objects of which one becomes aware by sense perception are not objects of pure thought. Hence they cannot be uniquely named. Since Aristotelian and mathematical logic deals with pure thought alone there are no singular terms in either of them. The individual constants of mathematical logic are not objects of sense perception. In Nyaya and Buddhist logic singular terms are indispensable, (...) because inference does not deal with pure thought, but with knowledge which includes both thought and sense perception. (shrink)

Many philosophers claim that understanding a logical constant (e.g. ‘if, then’) fundamentally consists in having dispositions to infer according to the logical rules (e.g. Modus Ponens) that fix its meaning. This paper argues that such dispositionalist accounts give us the wrong picture of what understanding a logical constant consists in. The objection here is that they give an account of understanding a logical constant which is inconsistent with what seem to be adequate manifestations of such understanding. I then outline an (...) alternative account according to which understanding a logical constant is not to be understood dispositionally, but propositionally. I argue that this account is not inconsistent with intuitively correct manifestations of understanding the logical constants. (shrink)

It is shown how to model propositional constants within the simplified Routley-Meyer semantics. Various axioms and rules allowing the definition of modal operators, implicative negations, enthymematical conditionals, and propositions expressing various infinite conjunctions and disjunctions are set forth and shown to correspond to specific frame conditions. Two propositional constants which are both often designated as “the Ackermann constant” are shown to capture two such “infinite” propositions: The conjunction of every logical law and the conjunction of every (...) truth –what Anderson and Belnap called the “world” constant. (shrink)

In a recent paper, Godblatt and Kowalski show that if we add to monomodal logic just a single propositional constant then instead of two coatoms, we suddenly have continuum many. In this note we shall provide an alternative proof of that fact by showing that the simulation results of Kracht and Wolter can be sharpened.

Extending the language of the intuitionistic propositional logic Int with additional logical constants, we construct a wide family of extensions of Int with the following properties: (a) every member of this family is a maximal conservative extension of Int; (b) additional constants are independent in each of them.

This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then treated, including polynomial (...) simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and complexity theory within bounded arithmetic, and relations to complexity issues of predicate calculus. Students and researchers in mathematical logic and complexity theory will find this comprehensive treatment an excellent guide to this expanding interdisciplinary area. (shrink)

This paper considers the question of what knowing a logical rule consists in. I defend the view that knowing a logical rule is having propositional knowledge. Many philosophers reject this view and argue for the alternative view that knowing a logical rule is, at least at the fundamental level, having a disposition to infer according to it. To motivate this dispositionalist view, its defenders often appeal to Carroll’s regress argument in ‘What the Tortoise Said to Achilles’. I show that (...) this dispositionalist view, and the regress that supposedly motivates it, operate with the wrong picture of what is involved in knowing a logical rule. In particular I show that it gives us the wrong picture of the relation between knowing a logical rule and actions of inferring according to it, as well as of the way in which knowing a logical rule might be a priori. (shrink)

In the Tractatus, Wittgenstein maintains that “The world is all that is the case.” Some philosophers have seen an advantage in introducing into a formal language either a constant which will represent the world, or an operator, e.g., ‘Max’, such that indicates that p gives a complete description of the actual world, of the world at some instant of time, or of a possible world. Such propositions are called world propositions, possible world propositions, or maximal propositions. For us, a maximal (...) proposition is a possible world proposition; it gives a complete description of one way the world might be. A maximal proposition p is not logically false, and for any proposition q, either p entails q or p entails not-q. (shrink)

The thesis that, in a system of natural deduction, the meaning of a logical constant is given by some or all of its introduction and elimination rules has been developed recently in the work of Dummett, Prawitz, Tennant, and others, by the addition of harmony constraints. Introduction and elimination rules for a logical constant must be in harmony. By deploying harmony constraints, these authors have arrived at logics no stronger than intuitionist propositional logic. Classical logic, they maintain, cannot be (...) justified from this proof-theoretic perspective. This paper argues that, while classical logic can be formulated so as to satisfy a number of harmony constraints, the meanings of the standard logical constants cannot all be given by their introduction and/or elimination rules; negation, in particular, comes under close scrutiny. (shrink)

Can we find propositions that cannot rationally be denied in any possible world without assuming the existence of that same proposition, and so involving ourselves in a contradiction? In other words, can we find transworld propositions needing no further foundation or justification? Basically, three differing positions can be imagined: firstly, a relativist position, according to which ultimately founded propositions are impossible; secondly, a meta-relativist position, according to which ultimately founded propositions are possible but unnecessary; and thirdly, an absolute position, according (...) to which such propositions are necessary. In this short essay I show that under the premise of modal logic S5 with constant domain there are ultimately founded propositions and that their existence is even necessary, and I will give some reasons for the superiority of S5 over other logics. (shrink)

Automated reasoning issues are addressed for a finite lattice-valued propositional logic LnP(X) with truth-values in a finite lattice-valued logical algebraic structure—lattice implication algebra. We investigate extended strategies and rules from classical logic to LnP(X) to simplify the procedure in the semantic level for testing the satisfiability of formulas in LnP(X) at a certain truth-value level α (α-satisfiability) while keeping the role of truth constant formula played in LnP(X). We propose a lock resolution method at a certain truth-value level α (...) (α-lock resolution) in LnP(X) and have proved its theorems of soundness and weak completeness, respectively. We provide more efficient resolution based automated reasoning in LnP(X) and key supports for α-resolution-based automated reasoning approaches and algorithms in lattice based linguistic truth-valued logic. (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both (...) translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). (shrink)

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, (...) in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. (shrink)

In the first part of the paper it is proved that there exists a one–one mapping between a minimal contingential logic extended with a suitable axiom for a propositional constant τ, named KΔτw, and a logic of necessity ${K\square \tau{w}}$ whose language contains ${\square}$ and τ. The form of the proposed translation aims at giving a solution to a problem which was left open in a preceding paper. It is then shown that the presence of τ in the language (...) of KΔτw allows for the definition, in terms of the non-contingency operator Δ, not only of ${\square}$ but of a second necessity operator O. It is observed that this fact opens the road to an investigation of the τ-free multimodal fragments of KΔτw. (shrink)

We investigate the variety corresponding to a logic, which is the combination of ukasiewicz Logic and Product Logic, and in which Gödel Logic is interpretable. We present an alternative axiomatization of such variety. We also investigate the variety, called the variety of algebras, corresponding to the logic obtained from by the adding of a constant and of a defining axiom for one half. We also connect algebras with structures, called f-semifields, arising from the theory of lattice-ordered rings, and prove that (...) every algebra can be regarded as a structure whose domain is the interval [0, 1] of an f-semifield, and whose operations are the truncations of the operations of to [0, 1]. We prove that such a structure is uniquely determined by up to isomorphism, and we establish an equivalence between the category of algebras and that of f-semifields. (shrink)

This monograph is a defence of the Fregean take on logic. The author argues that Frege ́s projects, in logic and philosophy of language, are essentially connected and that the formalist shift produced by the work of Peano, Boole and Schroeder and continued by Hilbert and Tarski is completely alien to Frege's approach in the Begriffsschrift. A central thesis of the book is that judgeable contents, i.e. propositions, are the primary bearers of logical properties, which makes logic embedded in our (...) conceptual system. This approach allows coherent and correct definitions of logical constants, logical consequence, and truth and connects their use to the practices of rational agents in science and everyday life. (shrink)

This work concerns the use of operators for past and future contingency in Priorean temporal logic. We will develop a system named C_t, whose language includes a propositional constant and prove that (I) C_t is complete with respect to a certain class of general frames and (II) the usual operators for past and future necessity are definable in such system. Furthermore, we will introduce the extension C_t(lin) that can be interpreted on linear and transitive general frames. The theoretical result (...) of the current work is that contingency can be treated as a primitive notion in reasoning about temporal modalities. (shrink)

A Ramsey statement denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \longrightarrow (k)^2_2}$$\end{document} says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formula RAM(n, k) of size O(nk) and with terms of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}$$\end{document}. Let rk be the minimal n for which the statement holds. We prove that (...) RAM(rk, k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlák’s work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4k, k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R. (shrink)

Work on the nature and scope of formal logic has focused unduly on the distinction between logical and extra-logical vocabulary; which argument forms a logical theory countenances depends not only on its stock of logical terms, but also on its range of grammatical categories and modes of composition. Furthermore, there is a sense in which logical terms are unnecessary. Alexandra Zinke has recently pointed out that propositional logic can be done without logical terms. By defining a logical-term-free language with (...) the full expressive power of first-order logic with identity, I show that this is true of logic more generally. Furthermore, having, in a logical theory, non-trivial valid forms that do not involve logical terms is not merely a technical possibility. As the case of adverbs shows, issues about the range of argument forms logic should countenance can quite naturally arise in such a way that they do not turn on whether we countenance certain terms as logical. (shrink)

Inferentialism claims that expressions are meaningful by virtue of rules governing their use. In particular, logical expressions are autonomous if given meaning by their introduction-rules, rules specifying the grounds for assertion of propositions containing them. If the elimination-rules do no more, and no less, than is justified by the introduction-rules, the rules satisfy what Prawitz, following Lorenzen, called an inversion principle. This connection between rules leads to a general form of elimination-rule, and when the rules have this form, they may (...) be said to exhibit “general-elimination” harmony. Ge-harmony ensures that the meaning of a logical expression is clearly visible in its I-rule, and that the I- and E-rules are coherent, in encapsulating the same meaning. However, it does not ensure that the resulting logical system is normalizable, nor that it satisfies the conservative extension property, nor that it is consistent. Thus harmony should not be identified with any of these notions. (shrink)

The Moral Law is fulfilled iff everything that ought to be the case is the case, and The Good is realised in a possible world w at a time t iff w is deontically accessible from w at t. In this paper, I will introduce a set of temporal modal deontic systems with propositional quantifiers that can be used to prove some interesting theorems about The Moral Law and The Good. First, I will describe a set of systems without (...) any propositional quantifiers. Then, I will show how these systems can be extended by a couple of propositional quantifiers. I will use a kind of TxW semantics to describe the systems semantically and semantic tableaux to describe them syntactically. Every system will include a constant · that stands for The Good. ‘·’ is read as ‘The Good is realised’. All systems that contain the propositional quantifiers will also include a constant '*' that stands for The Moral Law. '*' is read as ‘The Moral Law is fulfilled’. I will prove that all systems (without the propositional quantifiers) are sound and complete with respect to their semantics and that all systems (including the extended systems) are sound with respect to their semantics. It is left as an open question whether or not the extended systems are complete. (shrink)

Let L be a first-order language and Φ and ψ two $\Sigma _{1}^{1}$ L-sentences that cannot be satisfied simultaneously in any finite L-structure. Then obviously the following principle Chain L,Φ,ψ (n,m) holds: For any chain of finite L-structures C 1 ,...,C m with the universe [n] one of the following conditions must fail: 1. $C_{1}\vDash \Phi $ , 2. C i ≅ C i+1 , for i = 1,...,m — 1, 3. $C_{m}\vDash \Psi $ . For each fixed L and (...) parameters n,m the principle Chain L,Φ,ψ (n,m) can be encoded into a propositional DNF formula of size polynomial in n,m. For any language L containing only constants and unary predicates we show that there is a constant c L such that the following holds: If a constant depth Frege system in DeMorgan language proves Chain L,Φ,ψ (n, c L · n) by a size s proof then the class of finite L-structures with universe [n] satisfying Φ can be separated from the class of those L-structures on [n] satisfying ψ by a depth 3 formula of size $2^{{\rm log}(s)^{O(1)}}$ and with bottom fan-in ${\rm log}(s)^{O(1)}$. (shrink)

The provability problem in intuitionistic propositional second-order logic with existential quantifier and implication (∃,→) is proved to be undecidable in presence of free type variables (constants). This contrasts with the result that inutitionistic propositional second-order logic with existential quantifier, conjunction and negation is decidable.