In many actual arguments, the conclusion seems intuitively to follow from the premisses, even though we cannot show that it follows logically. The traditional approach to evaluating such arguments is to suppose that they have an unstated premiss whose explicit addition will produce an argument where the conclusion does follow logically. But there are good reasons for doubting that people so frequently leave the premisses of their arguments unstated. The inclination to suppose that they do stems from the belief that (...) the only way in which an argument's conclusion can follow definitely from its premisses is to follow logically. I argue that this belief is mistaken. I propose a revision of the current generic conception of logical consequence, and its variant specifications, to avoid the paradoxes of strict implication. The revised conception can then be naturally extended to include also what we might call 'enthymematic consequence'. This concept is a kind of consequence, whose properties merit investigation. (shrink)

P.F. Strawson contributed to many philosophical domains, including the philosophy of language, the history of philosophy, metaphysics, moral philosophy and philosophical methodology. Most of his contributions in these areas have influenced contemporary debates, either because his views are still defended or because they are still considered worthy of detailed responses. His views on the philosophy of logic have been only rarely discussed, however. My aim in this paper is threefold. First, I provide a systematic account of Strawson’s philosophy of logic. (...) I argue that Strawson is an important predecessor of logical expressivism, a contemporary position in the philosophy of logic most notably defended by Robert Brandom. My main focus is on Strawson’s largely-neglected 1982 paper ‘Logical Form and Logical Constants’, which contains his most mature ideas on the topic. Second, while Strawson’s position is of historical and independent philosophical interest, I argue that he leaves many points unclear. Finally, I propose several clarifications of Strawson’s position. (shrink)

Conventionalism about logic is the view that logical principles hold in virtue of some linguistic conventions. According to explicit conventionalism, these conventions have to be stipulated explicitly. Explicit conventionalism is subject to a famous criticism by Quine, who accused it of leading to an infinite regress. In response to the criticism, several authors have suggested reconstructing conventionalism as implicit in our linguistic behaviour. In this paper, drawing on a distinction from proof theory between derivable and admissible rules, I argue that (...) implicit conventionalism has not been stated in a sufficiently precise way, as it leaves open what one needs to say about admissible yet underivable rules. Moreover, it turns out that this challenge cannot be easily met, and that any attempt to meet the challenge makes conventionalism much less attractive a thesis. (shrink)

This article reviews David Hitchcock’s selected papers, On Reasoning and Argument.

We prove a generalization of Maehara’s lemma to show that the extensions of classical and intuitionistic first-order logic with a special type of geometric axioms, called singular geometric axioms, have Craig’s interpolation property. As a corollary, we obtain a direct proof of interpolation for (classical and intuitionistic) first-order logic with identity, as well as interpolation for several mathematical theories, including the theory of equivalence relations, (strict) partial and linear orders, and various intuitionistic order theories such as apartness and positive partial (...) and linear orders. (shrink)

The calculus of Natural Calculation is introduced as an extension of Natural Deduction by proper term rules. Such term rules provide the capacity of dealing directly with terms in the calculus instead of the usual reasoning based on equations, and therefore the capacity of a natural representation of informal mathematical calculations. Basic proof theoretic results are communicated, in particular completeness and soundness of the calculus; normalisation is briefly investigated. The philosophical impact on a proof theoretic account of the notion of (...) meaning is considered. (shrink)

In this paper investigates how natural deduction rules define connective meaning by presenting a new method for reading semantical conditions from rules called natural semantics. Natural semantics explains why the natural deduction rules are profoundly intuitionistic. Rules for conjunction, implication, disjunction and equivalence all express intuitionistic rather than classical truth conditions. Furthermore, standard rules for negation violate essential conservation requirements for having a natural semantics. The standard rules simply do not assign a meaning to the negation sign. Intuitionistic negation fares (...) much better. Not only do the intuitionistic rules have a natural semantics, that semantics amounts to familiar intuitionistic truth conditions. We will make use of these results to argue that intuitionistic connectives, rather than standard ones have a better claim to being the truly logical connectives. (shrink)

The pluralist sheds the more traditional ideas of truth and ontology. This is dangerous, because it threatens instability of the theory. To lend stability to his philosophy, the pluralist trades truth and ontology for rigour and other ‘fixtures’. Fixtures are the steady goal posts. They are the parts of a theory that stay fixed across a pair of theories, and allow us to make translations and comparisons. They can ultimately be moved, but we tend to keep them fixed temporarily. Apart (...) from considering rigour of proof as a fixture, I discuss fixed models, invariant notions and fixed information about objects across theories. There are other fixtures, but it is enough to start with these. (shrink)

We study structural rules in the context of multi-valued logics with finitely-many truth-values. We first extend Gentzen’s traditional structural rules to a multi-valued logic context; in addition, we propos some novel structural rules, fitting only multi-valued logics. Then, we propose a novel definition, namely, structural rules completeness of a collection of structural rules, requiring derivability of the restriction of consequence to atomic formulas by structural rules only. The restriction to atomic formulas relieves the need to concern logical rules in the (...) derivation. (shrink)

The paper highlights proof-theoretic semantics as providing natural resources for capturing semantic variation in natural language. The semantic variations include:Distinction between extensional predication and attribution to intensional transitive verbs a non-specific object.Omission of a verbal argument in a transitive verb.Obtaining sameness of meaning of sentences with transitive verbs with omitted object and existentially quantified object.Blocking unwarranted entailments in adjective–noun combinations.Capturing quantifier scope ambiguity.Obtaining context dependent quantifier domain restriction. The proof-theoretic resources employed to capture the above semantic variations include:The use of (...) different kinds of formal parameters and different ways of binding them to predicates.Omission of a premise from a proof-rule.Basing meanings on derivations in meaning-conferring proof-systems.Appealing to substructurality of I/E-rules.Controlling the order of rule applications in derivations.Controlling the open formulas participating in a context within a sequent. (shrink)

The paper presents a proof-theoretic semantics (PTS) for a fragment of natural language, providing an alternative to the traditional model-theoretic (Montagovian) semantics (MTS), whereby meanings are truth-condition (in arbitrary models). Instead, meanings are taken as derivability-conditions in a dedicated natural-deduction (ND) proof-system. This semantics is effective (algorithmically decidable), adhering to the meaning as use paradigm, not suffering from several of the criticisms formulated by philosophers of language against MTS as a theory of meaning. In particular, Dummett’s manifestation argument does not (...) obtain, and assertions are always warranted, having grounds of assertion. The proof system is shown to satisfy Dummett’s harmony property, justifying the ND rules as meaning conferring. The semantics is suitable for incorporation into computational linguistics grammars, formulated in type-logical grammar. (shrink)

In the proof-theoretic semantics approach to meaning, harmony , requiring a balance between introduction-rules (I-rules) and elimination rules (E-rules) within a meaning conferring natural-deduction proof-system, is a central notion. In this paper, we consider two notions of harmony that were proposed in the literature: 1. GE-harmony , requiring a certain form of the E-rules, given the form of the I-rules. 2. Local intrinsic harmony : imposes the existence of certain transformations of derivations, known as reduction and expansion . We propose (...) a construction of the E-rules (in GE-form) from given I-rules, and prove that the constructed rules satisfy also local intrinsic harmony. The construction is based on a classification of I-rules, and constitute an implementation to Gentzen’s (and Pawitz’) remark, that E-rules can be “read off” I-rules. (shrink)

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 . 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. (shrink)

Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way.To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce a family of logics called term-modal logics. A main feature of our logics is the use of modal (...) operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In term-modal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators. (shrink)

Since 1993, when Hudelmaier developed an O(n log n)-space decision procedure for propositional Intuitionistic Logic, a lot of work has been done to improve the efficiency of the related proof-search algorithms. In this paper a tableau calculus using the signs T, F and Fc with a new set of rules to treat signed formulas of the kind T((A → B) → C) is provided. The main feature of the calculus is the reduction of both the non-determinism in proof-search and the (...) width of proofs with respect to Hudelmaier's one. These improvements have a significant influence on the performances of the implementation. (shrink)

The hallmark of the deductive systems known as ‘conceptivist’ or ‘containment’ logics is that for all theorems of the form , all atomic formulae appearing in also appear in . Significantly, as a consequence, the principle of Addition fails. While often billed as a formalisation of Kantian analytic judgements, once semantics were discovered for these systems, the approach was largely discounted as merely the imposition of a syntactic filter on unrelated systems. In this paper, we examine a number of prima (...) facie unrelated deductive contexts in which Addition fails and attempt to harmonise them by developing a computational interpretation of conceptivist logics. (shrink)

Simon Evnine examines various epistemic aspects of what it is to be a person. Persons are defined as finite beings that have beliefs, including second-order beliefs about their own and others' beliefs, and are agents, capable of making long-term plans. It is argued that for any being meeting these conditions, a number of epistemic consequences obtain. First, all such beings must have certain logical concepts and be able to use them in certain ways. Secondly, there are at least two principles (...) governing belief that it is rational for persons to satisfy and are such that nothing can be a person at all unless it satisfies them to a large extent. These principles are that one believe the conjunction of one's beliefs and that one treat one's future beliefs as, by and large, better than one's current beliefs. Thirdly, persons both occupy epistemic points of view on the world and show up within those views. This makes it impossible for them to be completely objective about their own beliefs. Ideals of rationality that require such objectivity, while not necessarily wrong, are intrinsically problematic for persons. This "aspectual dualism" is characteristic of treatments of persons in the Kantian tradition. In sum, these epistemic consequences support a traditional view of the nature of persons, one in opposition to much recent theorizing. (shrink)

I argue for the thesis (UL) that there are certain logical abilities that any rational creature must have. Opposition to UL comes from naturalized epistemologists who hold that it is a purely empirical question which logical abilities a rational creature has. I provide arguments that any creatures meeting certain conditions—plausible necessary conditions on rationality—must have certain specific logical concepts and be able to use them in certain specific ways. For example, I argue that any creature able to grasp theories must (...) have a concept of conjunction subject to the usual introduction and elimination rules. I also deal with disjunction, conditionality and negation. Finally, I put UL to work in showing how it could be used to define a notion of logical obviousness that would be well suited to certain contexts—e.g. radical translation and epistemic logic—in which a concept of obviousness is often invoked. (shrink)

Does there exist any equivalence between the notions of inconsistency and consequence in paraconsistent logics as is present in the classical two valued logic? This is the key issue of this paper. Starting with a language where negation ( ${\neg}$ ) is the only connective, two sets of axioms for consequence and inconsistency of paraconsistent logics are presented. During this study two points have come out. The first one is that the notion of inconsistency of paraconsistent logics turns out to (...) be a formula-dependent notion and the second one is that the characterization (i.e. equivalence) appears to be pertinent to a class of paraconsistent logics which have double negation property. (shrink)

Infinite regress arguments are a powerful tool in Aristotle, but this style of argument has received relatively little attention. Improving our understanding of infinite regress arguments has become pressing since recent scholars have pointed out that it is not clear whether Aristotle’s infinite regress arguments are, in general, effective or indeed what the logical structure of these arguments is. One obvious approach would be to hold that Aristotle takes infinite regress arguments to be per impossibile arguments, which derive an infinite (...) sequence. Due to his finitism, Aristotle then rejects such a sequence as impossible. This paper argues that this obvious approach does not work, even for its most amenable cases. The paper argues instead that infinite regress arguments involve domain-specific infinities, and so there is not a general finitism which underpins infinite regress arguments in Aristotle, but rather domain-specific reasons that there cannot be an infinite number of entities in each domain in which Aristotle invokes an infinite regress argument. (shrink)

On the Dummettian understanding, anti-realism regarding a particular discourse amounts to (or at the very least, involves) a refusal to accept the determinacy of the subject matter of that discourse and a corresponding refusal to assert at least some instances of excluded middle (which can be understood as expressing this determinacy of subject matter). In short: one is an anti-realist about a discourse if and only if one accepts intuitionistic logic as correct for that discourse. On careful examination, the strongest (...) Dummettian arguments for anti-realism of this sort fail to secure intuitionistic logic as the single, correct logic for anti-realist discourses. Instead, antirealists are placed in a situation where they fail to be justified in asserting monism (that intuitionistic logic is the unique correct logic). Thus, antirealists seem forced either to accept pluralism (i.e. one or more intermediate logic is at least as `correct’ as intuitionistic logic–an option I take to be unattractive from the anti-realist perspective), or they must be anti-realists about the realism/anti-realism debate (and, in particular, must refuse to assert the instance of excluded middle equivalent to logical monism or logical pluralism). (shrink)

The principle of tolerance characteristic of vague predicates is sometimes presented as a soft rule, namely as a default which we can use in ordinary reasoning, but which requires care in order to avoid paradoxes. We focus on two ways in which the tolerance principle can be modeled in that spirit, using special consequence relations. The first approach relates tolerant reasoning to nontransitive reasoning; the second relates tolerant reasoning to nonmonotonic reasoning. We compare the two approaches and examine three specific (...) consequence relations in relation to those, which we call: strict-to-tolerant entailment, pragmatic-to-tolerant entailment, and pragmatic-to-pragmatic entailment. The first two are nontransitive, whereas the latter two are nonmonotonic. (shrink)

In a recent paper, Barrio, Tajer and Rosenblatt establish a correspondence between metainferences holding in the strict-tolerant logic of transparent truth ST+ and inferences holding in the logic of paradox LP+. They argue that LP+ is ST+’s external logic and they question whether ST+’s solution to the semantic paradoxes is fundamentally different from LP+’s. Here we establish that by parity of reasoning, ST+ can be related to LP+’s dual logic K3+. We clarify the distinction between internal and external logic and (...) argue that while ST+’s nonclassicality can be granted, its self-dual character does not tie it to LP+ more closely than to K3+. (shrink)

This paper presents a semantical analysis of the Weak Kleene Logics Kw3 and PWK from the tradition of Bochvar and Halldén. These are three-valued logics in which a formula takes the third value if at least one of its components does. The paper establishes two main results: a characterisation result for the relation of logical con- sequence in PWK – that is, we individuate necessary and sufficient conditions for a set.

We present a construction of a truth class (an interpretation of a compositional truth predicate) in an arbitrary countable recursively saturated model of first-order arithmetic. The construction is fully classical in that it employs nothing more than the classical techniques of formal proof theory.

Suszko’s problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski’s structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. (...) In this article we give a more systematic perspective on Suszko’s problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those theorems to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos, and Wansing, for logics with various structural properties. We strengthen these results into exact rank results for nonpermeable logics. We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties. (shrink)

Given a consequence relation in many-valued logic, what connectives can be defined? For instance, does there always exist a conditional operator internalizing the consequence relation, and which form should it take? In this paper, we pose this question in a multi-premise multi-conclusion setting for the class of so-called intersective mixed consequence relations, which extends the class of Tarskian relations. Using computer-aided methods, we answer extensively for 3-valued and 4-valued logics, focusing not only on conditional operators, but also on what we (...) call Gentzen-regular connectives. For arbitrary N-valued logics, we state necessary and sufficient conditions for the existence of such connectives in a multi-premise multi-conclusion setting. The results show that mixed consequence relations admit all classical connectives, and among them pure consequence relations are those that admit no other Gentzen-regular connectives. Conditionals can also be found for a broader class of intersective mixed consequence relations, but with the exclusion of order-theoretic consequence relations. (shrink)

Proof-theoretic semantics is an inferentialist theory of meaning, usually developed in a multiple-assumption and single-conclusion framework. In that framework, this theory seems unable to justify classical logic, so some authors have proposed a multiple-conclusion reformulation to accomplish this goal. In the first part of this paper, the debate originated by this proposal is briefly exposed and used to defend the diverging opinion that proof-theoretic semantics should always endorse a single-assumption and single-conclusion framework. In order to adopt this approach some of (...) its _criteria_ of validity, especially separability, need to be weakened. This choice is evaluated and defended. The main argument in this direction is based on the circular dependences of meaning between multiple assumptions and conjunctions, and between multiple conclusions and disjunctions. In the second part of this paper, some systems that suit the new requirements are proposed for both intuitionistic and classical logic. A proof that they are valid, according to the weakened _criteria_, is sketched. (shrink)

ellucci, C., Existential instantiation and normalization in sequent natural deduction, Annals of Pure and Applied Logic 58 111–148. A sequent conclusion natural deduction system is introduced in which classical logic is treated per se, not as a special case of intuitionistic logic. The system includes an existential instantiation rule and involves restrictions on the discharge rules. Contrary to the standard formula conclusion natural deduction systems for classical logic, its normal derivations satisfy both the subformula property and the separation property and (...) allow to establish a version of the midsequent theorem and Herbrand's theorem. (shrink)

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express metalogical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the logics of formal inconsistency and by the logics of formal undeterminedness. LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of (...) the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core—in the case studied here, this common core is classical positive propositional logic. The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of LFI and undeterminedness, namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion one at a time. The last sections offer an algebraic account for such logics by adapting the swap structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite nondeterministic matrices. (shrink)

From the technical point of view, philosophically neutral, the duality between a paraconsistent and a paracomplete logic lies in the fact that explosion does not hold in the former and excluded middle does not hold in the latter. From the point of view of the motivations for rejecting explosion and excluded middle, this duality can be interpreted either ontologically or epistemically. An ontological interpretation of intuitionistic logic is Brouwer’s idealism; of paraconsistency is dialetheism. The epistemic interpretation of intuitionistic logic is (...) in terms of preservation of constructive proof; of paraconsistency is in terms of preservation of evidence. In this paper, we explain and defend the epistemic approach to paraconsistency. We argue that it is more plausible than dialetheism and allows a peaceful and fruitful coexistence with classical logic. (shrink)

The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true. The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence, and extends it to the Logic of Evidence and Truth. The latter is a logic of (...) formal inconsistency and undeterminedness that is able to express not only preservation of evidence but also preservation of truth. LETj is anti-dialetheist in the sense that, according to the intuitive interpretation proposed here, its consequence relation is trivial in the presence of any true contradiction. Adequate semantics and a decision method are presented for both BLE and LETj, as well as some technical results that fit the intended interpretation. (shrink)

It is argued that truth value of a sentence containing free variables in a context of use, just as the reference of the free variables concerned, depends on the assumptions and posits given by the context. However, context may under-determine the reference of a free variable and the truth value of sentences in which it occurs. It is argued that in such cases a free variable has indeterminate reference and a sentence in which it occurs may have indeterminate truth value. (...) On letting, say, x be such that \, the sentence ‘Either \ or \’ is true but the sentence ‘\’ has an indeterminate truth value: it is determinate that the variable x refers to either 2 or \, but it is indeterminate which of the two it refers to, as a result ‘\’ has a truth value but its truth value is indeterminate. The semantic indeterminacy is analysed in a ‘radically’ supervaluational semantic framework closely analogous to the treatment of vagueness in McGee and McLaughlin and Smith, which saves bivalence, the T-schema and the truth-functional analysis of the boolean connectives. It is shown that on such an analysis the modality ‘determinately’ is quite clearly not an epistemic modality, avoiding a potential objection raised by Williamson against such ‘radically’ supervaluational treatments of vagueness, and that determinate truth is the semantic value preserved in classically valid arguments. The analysis is contrasted with the epistemicist proposal of Breckenridge and Magidor which implies that ‘\’ has a determinate but unknowable truth value. (shrink)

This paper provides finitary jointly necessary and sufficient acceptance and rejection conditions for the logical constants of a first order quantificational language. By introducing the notion of making an assignment as a distinct object level practice—something you do with a sentence—(as opposed to a meta-level semantic notion) and combining this with the practice of (hypothetical and categorical) acceptance and rejection and the practice of making suppositions one gains a structure that is sufficiently rich to fully characterize the class of classical (...) first order theories. The analysis thus provides a way of characterizing classical first order quantification by expressivist means. (shrink)

The connectives of classical propositional logic are given an analysis in terms of necessary and sufficient conditions of acceptance and rejection, i.e. the connectives are analyzed within an expressivist bilateral meaning-is-use framework. It is explained how such a framework differs from standard inferentialist frameworks and it is argued that it is better suited to address the particular issues raised by the expressivist thesis that the meaning of a sentence is determined by the mental state that it is conventionally used to (...) express. Furthermore, it is shown that the classical requirements governing the connectives completely characterize classical logic, are conservative and separable, are in bilateral harmony, are structurally preservative with respect to the classical coordination requirements and resolve the categoricity problem. These results are taken to show that one can give an expressivist bilateral meaning-is-use analysis of the connectives that confer on them a determinate coherent classical interpretation. (shrink)

Starting from certain metalogical results, I argue that first-order logical truths of classical logic are a priori and necessary. Afterwards, I formulate two arguments for the idea that first-order logical truths are also analytic, namely, I first argue that there is a conceptual connection between aprioricity, necessity, and analyticity, such that aprioricity together with necessity entails analyticity; then, I argue that the structure of natural deduction systems for FOL displays the analyticity of its truths. Consequently, each philosophical approach to these (...) truths should account for this evidence, i.e., that first-order logical truths are a priori, necessary, and analytic, and it is my contention that the semantic account is a better candidate. (shrink)

A hybrid logic is obtained by adding to an ordinary modal logic further expressive power in the form of a second sort of propositional symbols called nominals and by adding so-called satisfaction operators. In this paper we consider hybridized versions of S5 (“the logic of everywhere”) and the modal logic of inequality (“the logic of elsewhere”). We give natural deduction systems for the logics and we prove functional completeness results.

This is a companion paper to Braüner where a natural deduction system for propositional hybrid logic is given. In the present paper we generalize the system to the first-order case. Our natural deduction system for first-order hybrid logic can be extended with additional inference rules corresponding to conditions on the accessibility relations and the quantifier domains expressed by so-called geometric theories. We prove soundness and completeness and we prove a normalisation theorem. Moreover, we give an axiom system first-order hybrid logic.

The main aim of the present paper is to use a proof system for hybrid modal logic to formalize what are called false-belief tasks in cognitive psychology, thereby investigating the interplay between cognition and logical reasoning about belief. We consider two different versions of the Smarties task, involving respectively a shift of perspective to another person and to another time. Our formalizations disclose that despite this difference, the two versions of the Smarties task have exactly the same underlying logical structure. (...) We also consider the Sally-Anne task, having a more complicated logical structure, presupposing a “principle of inertia” saying that a belief is preserved over time, unless there is belief to the contrary. (shrink)

The fixed point combinator (Y) is an important non-proper combinator, which is defhable from a combinatorially complete base. This combinator guarantees that recursive equations have a solution. Structurally free logics (LC) turn combinators into formulas and replace structural rules by combinatory ones. This paper introduces the fixed point and the dual fixed point combinator into structurally free logics. The admissibility of (multiple) cut in the resulting calculus is not provable by a simple adaptation of the similar proof for LC with (...) proper combinators. The novelty of our proof—beyond proving the cut for a newly extended calculus–is that we add a fourth induction to the by-and-large Gentzen-style proof. (shrink)

This paper briefly overviews some of the results and research directions. In the area of substructural logics from the last couple of decades. Substructural logics are understood here to include relevance logics, linear logic, variants of Lambek calculi and some other logics that are motivated by the idea of omitting some structural rules or making other structural changes in LK, the original sequent calculus for classical logic.

I provide an interpretation of Wittgenstein's much criticized remarks on Gödel's First Incompleteness Theorem in the light of paraconsistent arithmetics: in taking Gödel's proof as a paradoxical derivation, Wittgenstein was right, given his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. I show that the models of paraconsistent arithmetics (obtained via the Meyer-Mortensen (...) collapsing filter on the standard model) match with many intuitions underlying Wittgensteins philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any meaningful mathematical question. (shrink)

We present a framework for machine implementation of families of non-classical logics with Kripke-style semantics. We decompose a logic into two interacting parts, each a natural deduction system: a base logic of labelled formulae, and a theory of labels characterizing the properties of the Kripke models. By appropriate combinations we capture both partial and complete fragments of large families of non-classical logics such as modal, relevance, and intuitionistic logics. Our approach is modular and supports uniform proofs of soundness, completeness and (...) proof normalization. We have implemented our work in the Isabelle Logical Framework. (shrink)

Using the programming-language concept of continuations, we propose a new, multimodal analysis of quantification in Type Logical Grammar. Our approach provides a geometric view of in-situ quantification in terms of graphs, and motivates the limited use of empty antecedents in derivations. Just as continuations are the tool of choice for reasoning about evaluation order and side effects in programming languages, our system provides a principled, type-logical way to model evaluation order and side effects in natural language. We illustrate with an (...) improved account of quantificational binding, weak crossover, wh-questions, superiority, and polarity licensing. (shrink)