A non-monotonic logic, the Logic of Plausible Reasoning (LPR), capable of coping with the demands of what we call complex reasoning, is introduced. It is argued that creative complex reasoning is the way of reasoning required in many instances of scientific thought, professional practice and common life decision taking. For managing the simultaneous consideration of multiple scenarios inherent in these activities, two new modalities, weak and strong plausibility, are introduced as part of the Logic of Plausible Deduction (LPD), a deductive (...) logic specially designed to serve as the monotonic support for LPR. Axiomatics and semantics for LPD, together with a completeness proof, are provided. Once LPD has been given, LPR may be defined via a concept of extension over LPD. Although the construction of LPR extensions is first presented in standard style, for the sake of comparison with existing non-monotonic formalisms, alternative more elegant and intuitive ways for constructing non-monotonic LPR extensions are also given and proofs of their equivalence are presented. (shrink)

A first-order temporal non-commutative logic TN[l], which has no structural rules and has some l-bounded linear-time temporal operators, is introduced as a Gentzen-type sequent calculus. The logic TN[l] allows us to provide not only time-dependent, resource-sensitive, ordered, but also hierarchical reasoning. Decidability, cut-elimination and completeness (w.r.t. phase semantics) theorems are shown for TN[l]. An advantage of TN[l] is its decidability, because the standard first-order linear-time temporal logic is undecidable. A correspondence theorem between TN[l] and a resource indexed non-commutative logic RN[l] (...) is also shown. This theorem is intended to state that “time” is regarded as a “resource”. (shrink)

We argue that logical expressions in human language enable speakers to perform particular acts as well as stating propositions which may be true or false. We present a conversational action planning model of co-ordinated reasoning, which we use to predict choice of logical expressions in situations in which two people co-operate in the face of risk and uncertainty. We first show how this model predicts preferences for formulations of conditional directives where a principal instructs an agent on how (...) to behave in a hypothetical situation. Second, we show how this model accounts for choices of quantity and probability expressions that express risk, in situations where a professional advises a client on her options. We conclude that the pragmatic signals encoded in human logical vocabulary can facilitate the co-ordination of social interaction through aiding mutual recognition of intentions on joint projects where collaboration is likely to create value. (shrink)

In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics , we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern–Shoham’s interval logic HS.

The pretheoretical notions of logical consequence and of a logical expression are linked in vague and complex ways to modal and pragmatic intuitions. I offer an introduction to the difficulties that these intuitions create when one attempts to give precise characterizations of those notions. Special attention is given to Tarski’s theories of logical consequence and logical constancy. I note that the Tarskian theory of logical consequence has fared better in the face of the difficulties (...) than the Tarskian theory of logical constancy. Other theories of these notions are explained and criticized. (shrink)

The Editors express their gratitude and appreciation to the indi-viduals listed below who served as referees for Informal Logic for Volumes 31 (2011) and 32 (2012).

The Peanese convention for the use of dots as brackets has the disadvantage that it gives only an awkward method for representing chains of indefinite length, such as the compound implicationSuch chains occur frequently in logical investigations of a metatheoretic nature, and it is convenient to have a systematic method of abbreviating them. The most obvious method of doing this would be to leave the parentheses out entirely, and to understand that in such cases the implication sign or other (...) operation appearing on the extreme left is the most inclusive; but this method, which has been followed by Quine, is at variance with our ordinary algebraic usage, in which we write a − b + c − d for ((a − b) + c) − d, and consequently leads to confusion. It is desirable to have a modification of the Peanese convention which gives a simple method of representing such chain implications, and at the same time avoids this difficulty.Such a modification is easily obtained by generalizing somewhat a procedure of Church. The latter author uses only a single dot, which he writes on the right of an operator to signify a bracket extending from that point to the end of the formula (or parenthesized expression). If dots are used only on the right of operations this is all that can be desired. But the essential idea of this device can be extended to the case where dots are used also on the left, as follows: let us suppose that a group of dots on the right of an operation or prefix denotes the beginning of a bracket which extends to the right until it meets a group with an equal or larger number of dots on the left of an operation; and that the scope of a group of dots on the left of an operation shall extend to the left until it reaches a larger group of dots on the right of some operation. (shrink)

Public announcement logic is an extension of epistemic logic with dynamic operators that model the effects of all agents simultaneously and publicly acquiring the same piece of information. One of the extensions of PAL, group announcement logic, allows quantification over announcements made by agents. In GAL, it is possible to reason about what groups can achieve by making such announcements. It seems intuitive that this notion of coalitional ability should be closely related to the notion of distributed knowledge, the implicit (...) knowledge of a group. Thus, we study the extension of GAL with distributed knowledge, and in particular possible interaction properties between GAL operators and distributed knowledge. The perhaps surprising result is that, in fact, there are no interaction properties, contrary to intuition. We make this claim precise by providing a sound and complete axiomatisation of GAL with distributed knowledge. We also consider several natural variants of GAL with distributed knowledge, as well as some other related logic, and compare their expressive power. (shrink)

It is possible to understand the expressive power of a logic as issuing from its capacity to express properties of its models. There are some ways to formally capture whether a property of models is expressible, among them is one based on the notion of definability, and one based on the notion of discrimination. If the logics to be compared are defined within the same class of models, one can employ the notions of definability and discrimination directly to obtain formal (...) conditions for relative expressiveness. This paper studies generalizations of these formal conditions to cases where the compared logics are defined within different classes of models. There have been proposed in the literature formal conditions of two main kinds: with forward and with backward model-mappings. It is shown that none of them is adequate, despite their initial reasonableness. Moreover, we argue that general and reasonable formal conditions for relative expressiveness involving forward mappings are not likely to be found, given that they turn out to be highly dependent on specific features of the compared logics. On the other hand, it will be argued that there is a reasonable formal condition involving backward model-mappings. (shrink)

In a finitary closure space, irreducible sets behave like two-valued models, with membership playing the role of satisfaction. If f is a function on such a space and the membership of in an irreducible set is determined by the presence or absence of the inputs in that set, then f is a kind of truth function. The existence of some of these truth functions is enough to guarantee that every irreducible set is maximally consistent. The closure space is then said (...) to be expressive. This paper identifies the two-valued truth functional conditions that guarantee expressiveness. (shrink)

ABSTRACT We characterize those binary ramified frames for which propositional modal logic is as expressive as the corresponding first-order logic.

This paper is about the development of logic in the Aristotelian tradition, from Aristotle to the mid-fourteenth century. I will compare four systems of logic with regard to their expressive power. 1. Aristotle’s own logic, based mostly on chapters 1-2 and 4-7 of his Prior Analytics 2. An expanded version of Aristotle’s logic that one finds, e.g., in Sherwood’s Introduction to Logic and Peter of Spain’s Tractatus 3-5. Versions of the logic of later supposition theorists such as William Ockham, John (...) Buridan, and Paul of Venice. Version 4 is the logic without relatives ; version 5 adds relatives. I am ignoring modals, conditionals that are not ut nunc, infinitizing negation, exclusives and exceptives, all exponibles, all insolubles, and terms with simple or material supposition, ampliation and restriction, and many other things. (shrink)

Feature logics are the logical basis for so-called unification grammars studied in computational linguistics. We investigate the expressivity of feature terms with negation and the functional uncertainty construct needed for the description of long-distance dependencies and obtain the following results: satisfiability of feature terms is undecidable, sort equations can be internalized, consistency of sort equations is decidable if there is at least one atom, and consistency of sort equations is undecidable if there is no atom.

The general slogan in the title of this paper gives a general, but nevertheless accurate, expression of Strawson’s view concerning the nature of formal logic per se in relation to natural language. What is at stake here is the extent to which the formal methods and the formal semantics of contemporary symbolic logic can render the meanings of natural language expressions. Strawson sets up an agenda for logical theory which, although rather dated for a logic text, is what (...) one naturally expects from an introduction to elementary formal logic and a systematic analysis of certain crucial metalogical concepts, most notably entailment. However, my own reading is that Strawson’s approach to logic makes a lasting contribution to understanding the inner workings of natural language through his critical discussion of the limits of the formal renderings of the semantics of natural language expressions via canonical first-order notations. One can recognize in this critical attitude towards the limits of formal methods a consistent generalization of Strawson’s own strategy from his critical response to Russell’s theory of descriptions.In my paper I shall focus on certain examples which illustrate the general slogan that characterizes Strawson’s views on the nature of logic in connection with ordinary language and talk. (shrink)

We use a deontic logic of collective agency to study reducibility questions about collective agency and collective obligations. The logic that is at the basis of our study is a multi-modal logic in the tradition of *stit* logics of agency. Our full formal language has constants for collective and individual deontic admissibility, modalities for collective and individual agency, and modalities for collective and individual obligations. We classify its twenty-seven sublanguages in terms of their expressive power. This classification enables us to (...) investigate reducibility relations between collective deontic admissibility, collective agency, and collective obligations, on the one hand, and individual deontic admissibility, individual agency, and individual obligations, on the other. (shrink)

Some expressions of English, like the demonstratives ‘this’ and ‘that’, are referentially promiscuous: distinct free occurrences of them in the same sentence can differ in content relative to the same context. One lesson of referentially promiscuous expressions is that basic logical properties like validity and logical truth obtain or fail to obtain only relative to a context. This approach to logic can be developed in just as rigorous a manner as David Kaplan’s classic logic of demonstratives. The result (...) is a logic that applies to arguments in English containing multiple occurrences of referentially promiscuous expressions. (shrink)

This paper investigates the question “when is a logic more expressive than another?” In order to approach it, “logic” is understood in the model-theoretic sense and, contrary to other proposals in the literature, it is argued that relative expressiveness between logics is best framed with respect to the notion of expressing properties of models, a notion that can be captured precisely in various ways. It is shown that each precise rendering can give rise to a formal condition for relative expressiveness (...) that has appeared in the literature. Five such conditions are exposed, tested for some properties and compared to each other. As the formal conditions for relative expressiveness have various levels of stringency, some results on lifting some conditions to stricter ones are explored. Finally, a discussion on the properties of these formal conditions is presented. Special attention is given to notion of meaning equivalence, and how one may consider that it holds or not, depending on the weight attributed to logical and non-logical constants in expressiveness comparisons. (shrink)

We propose a probabilistic interpretation of Propositional Dynamic Logic (PDL). We show that logical and behavioral equivalence are equivalent over general measurable spaces. This is done first for the fragment of straight line programs and then extended to cater for the nondeterministic nature of choice and iteration, expanded to PDL as a whole. Bisimilarity is also discussed and shown to be equivalent to logical and behavioral equivalence, provided the base spaces are Polish spaces. We adapt techniques from coalgebraic (...) stochastic logic and point out some connections to Souslin's operation A from descriptive set theory. This leads to a discussion of complete stochastic Kripke models and model completion, which permits an adequate treatment of the test operator. (shrink)

We consider second-order propositional modal logic (SOPML), an extension of the basic modal language with propositional quantifiers introduced by Kit Fine in 1970. We determine the precise expressive power of SOPML by giving analogues of the Van Benthem–Rosen theorem and the Goldblatt Thomason theorem. Furthermore, we show that the basic modal language is the bisimulation invariant fragment of SOPML, and we characterize the bounded fragment of first-order logic as being the intersection of first-order logic and SOPML.

Against the backdrop of the frequent comparison of theories of truth in the literature on semantic paradoxes with regard to which inferences and metainferences are deemed valid, this paper develops a novel approach to defining a binary predicate for representing the valid inferences and metainferences of a theory within the theory itself under the assumption that the theory is defined with a classical meta-theory. The aim with the approach is to obtain a tool which facilitates the comparison between a theory (...) and its competitors within the theory itself, thereby expressing the disagreement between the theories within the theories. After discussing what we can and should require of an object-linguistic representation of a theory for that purpose, this paper proposes to restrict the representation of valid metainferences to locally valid metainferences, a requirement which turns out to be \-consistent and conservative over classical first-order arithmetic. This approach is then applied to four theories definable on strong Kleene models using a labelled nested sequent calculus. (shrink)

This paper considers various objections to Carnap’s logical syntax definition of ’logical expression’, including those by Saunders Mac Lane and W. V. O. Quine. While the specific objections of these two authors can be answered, if necessary by a slight modification of Carnap’s definition, there are other objections that I do not see how to meet. I also consider the proposal by Denis Bonnay for avoiding the objections to Carnap’s definition. In light of the unresolved problems with (...) Carnap’s definition, I go on to consider what Carnap’s assumptions must have been in framing his definition and to assess how much damage is caused by this failure of Carnap’s definition. This damage is not as much as might be assumed. (shrink)

Natural deduction systems were motivated by the desire to define the meaning of each connective by specifying how it is introduced and eliminated from inference. In one sense, this attempt fails, for it is well known that propositional logic rules underdetermine the classical truth tables. Natural deduction rules are too weak to enforce the intended readings of the connectives; they allow non-standard models. Two reactions to this phenomenon appear in the literature. One is to try to restore the standard readings, (...) for example by adopting sequent rules with multiple conclusions. Another is to explore what readings the natural deduction rules do enforce. When the notion of a model of a rule is generalized, it is found that natural deduction rules express “intuitionistic” readings of their connectives. A third approach is presented here. The intuitionistic readings emerge when models of rules are defined globally, but the notion of a local model of a rule is also natural. Using this benchmark, natural deduction rules enforce exactly the classical readings of the connectives, while this is not true of axiomatic systems. This vindicates the historical motivation for natural deduction rules. One odd consequence of using the local model benchmark is that some systems of propositional logic are not complete for the semantics that their rules express. Parallels are drawn with incompleteness results in modal logic to help make sense of this. (shrink)

The expressive truth functions of two-valued logic have all been identified. This paper begins the task of identifying the expressive truth functions of n-valued logic by characterizing the unary ones. These functions have distinctive algebraic, semantic, and closure-theoretic properties.

It seems to make perfectly good sense to distinguish between what is expressed and the way in which it is expressed. There is little doubt that there are many different ways of saying the same thing open to us. If I denied this, I would certainly be wrong. And yet a word of caution may not be amiss. Among logicians a tendency has grown up to concentrate their attention on those properties of a statement which make it true or false, (...) what they call a “proposition”, and to neglect the form in which it is expressed. I think it is a dangerous tendency as it may lead to overlooking all sorts of differences which are due to the form. Let me give an example taken from F. P. Ramsey. (shrink)

The optimal balance between decidability and expressiveness is a big problem of logical systems, in particular, of quantified epistemic logics (QELs). On the one hand, decidability is a very significant characteristic of logics that allows us to use such logics in the framework of artificial intelligence. On the other hand, QELs have important expressive capabilities that should not be lost when we construct decidable fragments of these logics. QELs are known to be much more expressive than first-order logics. One (...) important example of their extra expressive power is that they allow us to distinguish between de dicto and de re readings of epistemic sentences. It is clear that such capabilities should be preserved as much as possible in decidable fragments. In this paper, we consider extensions of QELs that include quantification over modalities. Denote this extensions by $$\hbox {Q}_{\Box }$$ Ls. $$\hbox {Q}_{\Box }$$ Ls allows us to make more subtle distinctions between de dicto and de re readings of epistemic sentences, and we also should keep these new features as much as possible in decidable fragments. It is known that there are not much interesting decidable QELs. The situation with $$\hbox {Q}_{\Box }$$ Ls is the same. But in recent years (after 2018), we have obtained a variety of decidable $$\hbox {Q}_{\Box }$$ Ls constructed in different ways. We distinguish between (1) the approach in which for every undecidable $$\hbox {Q}_{\Box }$$ L and for every variant of its decidable fragment, a specific proof is constructed, and (2) the approach in which a class of decidable $$\hbox {Q}_{\Box }$$ Ls is obtained using general tools and a uniform method for all $$\hbox {Q}_{\Box }$$ Ls of this class. In this paper, we compare the results of these approaches. (shrink)

Lewis (Noûs 13:455–476, 1979) claimed that branching-time(-like) models can be derived from his sphere models. However, he did not present any specific construction of branching-time(-like) models from his sphere models formally. Meanwhile, Hosokawa (in: Modern logic of modality and its philosophical range: counterfactuals, Gettier problem, and information flow, Tokyo Metropolitan University, 2018) presented a logico-mathematically strict manner in which sphere models can be reconstructed from branching-time models. Subsequently, Hosokawa (J Logic Lang Inf 32:677–706, 2023) presented a proof-theoretically refined version of (...) hybrid tense logic for a certain type of conditionals, which is referred to as hybrid tense logic for temporal conditionals ( $$\textbf{HTL}_{TC}$$ ). In this paper, we interpret a hybrid version $${\textbf{V}}_{\mathcal {HC(@, \downarrow )}}$$ of Lewis’s counterfactual logic $${\textbf{V}}$$ into $$\textbf{HTL}_{TC}$$, and then prove that $$\textbf{HTL}_{TC}$$ is more expressive than $${\textbf{V}}_{\mathcal {HC(@, \downarrow )}}$$ on the class of temporal sphere models, i.e., sphere models derived from a type of branching-time models. (shrink)

In this paper, we present several extensions of epistemic logic with update operators modelling public information change. Next to the well-known public announcement operators, we also study public substitution operators. We prove many of the results regarding expressivity and completeness using so-called reduction axioms. We develop a general method for using reduction axioms and apply it to the logics at hand.

In this article we investigate the family of independence-friendly (IF) logics in the equality-free setting, concentrating on questions related to expressive power. Various natural equality-free fragments of logics in this family translate into existential second-order logic with prenex quantification of function symbols only and with the first-order parts of formulae equality-free. We study this fragment of existential second-order logic. Our principal technical result is that over finite models with a vocabulary consisting of unary relation symbols only, this fragment of second-order (...) logic is weaker in expressive power than first-order logic (with equality). Results about the fragment could turn out useful for example in the study of independence-friendly modal logics. In addition to proving results of a technical nature, we address issues related to a perspective from which IF logic is regarded as a specification framework for games, and also discuss the general significance of understanding fragments of second-order logic in investigations related to non-classical logics. (shrink)

Truth-theoretic deflationism holds that truth is simple, and yet that it can fulfil many useful logico-linguistic roles. Deflationism focuses on axioms for truth: there is no reduction of the notion of truth to more fundamental ones such as sets or higher-order quantifiers. In this paper I argue that the fundamental properties of reasonable, primitive truth predicates are at odds with the core tenets of classical truth-theoretic deflationism that I call fix, express, and quantify. Truth may be regarded as a broadly (...) logical, sui generis notion. However, this has little to do with the original aims of the deflationary theory of truth. (shrink)

We investigate the extension of monadic second-order logic of order with cardinality quantifiers "there exists uncountably many sets such that... " and "there exists continuum many sets such that... ". We prove that over the class of countable linear orders the two quantifiers are equivalent and can be effectively and uniformly eliminated. Weaker or partial elimination results are obtained for certain wider classes of chains. In particular, we show that over the class of ordinals the uncountability quantifier can be effectively (...) and uniformly eliminated. Our argument makes use of Shelah's composition method and Ramsey-like theorem for dense linear orders. (shrink)

Engaging with the challenging and controversial reading of Spinoza presented by Gilles Deleuze in Expressionism in Philosophy (1968), this book focuses on Deleuze's redeployment of Spinozist concepts within the context of his own philosophical project of constructing a philosophy of difference as an alternative to the Hegelian dialectical philosophy. Duffy demonstrates that a thorough understanding of Deleuze's Spinozism is necessary in order to fully engage with Deleuze's philosophy of difference.

Intuitionistic dependence logic was introduced by Abramsky and Väänänen [1] as a variant of dependence logic under a general construction of Hodges’ (trump) team semantics. It was proven that there is a translation from intuitionistic dependence logic sentences into second order logic sentences. In this paper, we prove that the other direction is also true, therefore intuitionistic dependence logic is equivalent to second order logic on the level of sentences.

In this article we study linear temporal logics with team semantics that are novel logics for defining hyperproperties. We define Kamp-type translations of these logics into fragments of first-order team logic and second-order logic. We also characterize the expressive power and the complexity of model-checking and satisfiability of team logic and second-order logic by relating them to second- and third-order arithmetic. Our results set in a larger context the recent results of Lück showing that the extension of TeamLTL by the (...) Boolean negation is highly undecidable under the so-called synchronous semantics. We also study stutter-invariant fragments of extensions of TeamLTL. (shrink)