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).

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)

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)

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.

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)

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)

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)

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)

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 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)

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)

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.

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.

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)

According to the reading of Spinoza that Gilles Deleuze presents in Expressionism in Philosophy: Spinoza, Spinoza's philosophy should not be represented as a moment that can be simply subsumed and sublated within the dialectical progression of the history of philosophy, as it is figured by Hegel in the Science of Logic, but rather should be considered as providing an alternative point of view for the development of a philosophy that overcomes Hegelian idealism. Indeed, Deleuze demonstrates, by means of Spinoza, that (...) a more complex philosophy antedates Hegel's which cannot be supplanted by it. Spinoza therefore becomes a significant figure in Deleuze's project of tracing an alternative lineage in the history of philosophy, which, by distancing itself from Hegelian idealism, culminates in the construction of a philosophy of difference. Deleuze presents Spinoza's metaphysics as determined according to a 'logic of expression', which, insofar as it contributes to the determination of a philosophy of difference, functions as an alternative to the Hegelian dialectical logic. Deleuze's project in Expressionism in Philosophy is therefore to redeploy Spinoza in order to mobilize his philosophy of difference as an alternative to the dialectical philosophy determined by the Hegelian dialectic logic. (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)

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.

It is shown that the expressive power of second-order propositional modal logic whose modalities are S4.2 or weaker is the same as that of second-order predicate logic.

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.

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.

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)

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.

ABSTRACT Many temporal and modal logic languages can be regarded as subsets of first order logic, i.e. the semantics of a temporal logic formula is given as a first order condition on points of the underlying models (Kripke structures). Often the set of possible models is restricted to models which are trees. A temporal logic language is (first order) expressively complete, if for every first order condition for a node of a tree there exists an equivalent temporal formula which expresses (...) the same condition. In this paper expressive completeness of the temporal logic language with the set of operators U (until), S (since), and X k (k-next) is proved, and the result is extended to various other tree-like structures. (shrink)

We investigate the expressivity of many-valued modal logics based on an algebraic structure with a complete linearly ordered lattice reduct. Necessary and sufficient algebraic conditions for admitting a suitable Hennessy–Milner property are established for classes of image-finite and modally saturated models. Full characterizations are obtained for many-valued modal logics based on complete BL-chains that are finite or have the real unit interval [0, 1] as a lattice reduct, including Łukasiewicz, Gödel, and product modal logics.

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)

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)

The central question of Hans Urs von Balthasar’s Theo-Logic is how the infinite truth of God can be manifested in the finite structures of the created world. In the course of answering this question, Balthasar presents a philosophical understanding of expressive form and a theology of Christ as the expression of divine truth in the world. The philosophical discussion of truth provides support for the intelligibility of the theological claim that God’s truth has been manifested in the world in Christ. (...) The fullest expression of divine truth and the highest realization of worldly truth are found together in Christ, whom Balthasar calls the “truth of God.” Balthasar’s philosophical and theological understanding of expression, as set forth in the Theo-Logic, provides helpful insights for comparing his thought to that of Karl Rahner. (shrink)

We study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count. To this end an isomorphism preserving (`generic') model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. the arithmetical (...) side of the algorithms involved. In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics L r ∞ω (∃ ≥ m ) m∈ω and the corresponding pebble games. The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants (J r ) r≥ 1 with the following properties: $\bullet$ The invariants themselves can be evaluated in polynomial time. $\bullet$ A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants. $\bullet$ The invariant J r classifies all finite relational structures exactly up to equivalence with respect to the logic L r ∞ω (∃ ≥ m ) m∈ω . We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the L r ωω (∃ ≥ m ) m∈ω : It corresponds exactly to the polynomial time computable families (φ n ) n∈ω in these logics. Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindstrom quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting. (shrink)

The present text discusses whether there is a tension between aphorisms 6.1-6.13 of the Tractatus and the Church-Turing theorem about the decidability of predicate logic. We attempt to establish the following points: (i) Aphorisms 6.1-6.13 are not consistent with the Church-Turing theorem. (ii) The logical symbolism of the Tractatus, built from the N-operator, can (and should) be interpreted as expressively complete with respect to first-order formulas. (iii) Wittgenstein’s reasons for believing that Logic is decidable were purely philosophical and the (...) undecidability result shows that there are aspects of his criticisms of Frege and Russell that become unjustified in light of these results. (shrink)

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)

This paper compares the recently proposed Robust Full Computational Tree Logic to model robustness in concurrent systems with other computational tree logic -based logics. RoCTL* extends CTL* with the addition of the operators Obligatory and Robustly, which quantify over failure-free paths and paths with one more failure respectively. This paper focuses on examining the succinctness and expressiveness of RoCTL* by presenting translations to and from RoCTL*. The core result of this paper is to show that RoCTL* is expressively equivalent to (...) CTL* but is non-elementarily more succinct. That is, RoCTL* does not add any expressive power over CTL*, but can represent some properties using vastly reduced formulae. We present a translation from RoCTL* into CTL* that preserves truth but may result in non-elementary growth in the length of the translated formula, as each nested Robustly operator may result in an extra exponential blowup. However, we show that this translation is optimal in the sense th.. (shrink)

This paper presents a logical characterization of coalgebraic behavioral equivalence. The characterization is given in terms of coalgebraic modal logic, an abstract framework for reasoning about, and specifying properties of, coalgebras, for an endofunctor on the category of sets. Its main feature is the use of predicate liftings which give rise to the interpretation of modal operators on coalgebras. We show that coalgebraic modal logic is adequate for reasoning about coalgebras, that is, behaviorally equivalent states cannot be distinguished by (...) formulas of the logic. Subsequently, we isolate properties which also ensure expressiveness of the logic, that is, logical and behavioral equivalence coincide. (shrink)