Predicate Logics on Display

The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem's modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap's display logic by introduction rules for the existential and the universal quantifier. These rules for x and x are analogous to the display introduction rules for the modal operators and and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal modal predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules.     

A two-valued logic for reasoning about different types of consequence in Kleene's three-valued logic

A formal language of two-valued logic is developed, whose terms are formulas of the language of Kleene's three-valued logic. The atomic formulas of the former language are pairs of formulas of the latter language joined by consequence operators. These operators correspond to the three sensible types of consequence (strong-strong, strong-weak and weak-weak) in Kleene's logic in analogous way as the implication connective in the classical logic corresponds to the classical consequence relation. The composed formulas of the considered language are built from the atomic ones by means of the classical connectives and quantifiers.A deduction system for the developed language is given, consisting of a set of decomposition rules for sequences of formulas. It is shown that the deduction system is sound and complete.     

Determinism, Indeterminism And The Flow Of Time

A set of axioms implicitly defining the standard, though not instant-based but interval-based, time topology is used as a basis to build a temporal modal logic of events. The whole apparatus contains neither past, present, and future operators nor indexicals, but only B-series relations and modal operators interpreted in the standard way. Determinism and indeterminism are then introduced into the logic of events via corresponding axioms. It is shown that, if determinism and indeterminism are understood in accordance with their core meaning, the way in which they are formally introduced here represents the only right way to do this, given that we restrict ourselves to one real world and make no use of the many real worlds assumption. But then the result is that the very truth conditions for sentences about indeterministic events imply the existence of tensed truths, in spite of the fact that these conditions are formulated (in the indeterministic axiom) in terms of tenseless language. The tenseless theory of time implies determinism, while indeterminism requires the flow of time assumption.     

On the {↔, ∼} -reduct of the intuitionistic consequence operation

The intuitionistic consequence operation restricted to the language with (equivalence) and (negation) as the only connectives is axiomatized by means of a finite set of sequential rules of inference.To the memory of Professor Roman Suszko     

On Some Kripke Complete and Kripke Incomplete Intermediate Predicate Logics

The Kripke-completeness and incompleteness of some intermediate predicate logics is established. In particular, we obtain a Kripke-incomplete logic (H* +A+D+K) where H* is the intuitionistic predicate calculus, A is a disjunction-free propositional formula, D = x(P(x) V Q) xP(x) V Q, K = ¬¬x(P(x) V ¬P(x)) (the negative answer to a question of T. Shimura).     

Inferential Intensionality

The paper is a study of properties of quasi-consequence operation which is a key notion of the so-called inferential approach in the theory of sentential calculi established in [5]. The principal motivation behind the quasi-consequence, q-consequence for short, stems from the mathematical practice which treats some auxiliary assumptions as mere hypotheses rather than axioms and their further occurrence in place of conclusions may be justified or not. The main semantic feature of the q-consequence reflecting the idea is that its rules lead from the non-rejected assumptions to the accepted conclusions.First, we focus on the syntactic features of the framework and present the q-consequence as related to the notion of proof. Such a presentation uncovers the reasons for which the adjective inferential is used to characterize the approach and, possibly, the term inference operation replaces q-consequence. It also shows that the inferential approach is a generalisation of the Tarski setting and, therefore, it may potentially absorb several concepts from the theory of sentential calculi, cf. [10]. However, as some concrete applications show, see e.g.[4], the new approach opens perspectives for further exploration.The main part of the paper is devoted to some notions absent, in Tarski approach. We show that for a given q-consequence operation W instead of one W-equivalence established by the properties of W we may consider two congruence relations. For one of them the current name is kept preserved and for the other the term W-equality is adopted. While the two relations coincide for any W which is a consequence operation, for an arbitrary W the inferential equality and the inferential equivalence may differ. Further to this we introduce the concepts of inferential extensionality and intensionality for q-consequence operations and connectives. Some general results obtained in Section 2 sufficiently confirm the importance of these notions. To complete a view, in Section 4 we apply the new intensionality-extensionality distinction to inferential extensions of a version of the ukasiewicz four valued modal logic.     

First-order fuzzy logic

This paper is an attempt to develop the many-valued first-order fuzzy logic. The set of its truth, values is supposed to be either a finite chain or the interval 0, 1 of reals. These are special cases of a residuated lattice L, , , , , 1, 0. It has been previously proved that the fuzzy propositional logic based on the same sets of truth values is semantically complete. In this paper the syntax and semantics of the first-order fuzzy logic is developed. Except for the basic connectives and quantifiers, its language may contain also additional n-ary connectives and quantifiers. Many propositions analogous to those in the classical logic are proved. The notion of the fuzzy theory in the first-order fuzzy logic is introduced and its canonical model is constructed. Finally, the extensions of Gödel's completeness theorems are proved which confirm that the first-order fuzzy logic is also semantically complete.     

Modal Logics of Succession for 2-Dimensional Integral Spacetime

We consider the problem of axiomatizing various natural successor logics for 2-dimensional integral spacetime. We provide axiomatizations in monomodal and multimodal languages, and prove completeness theorems. We also establish that the irreflexive successor logic in the standard modal language (i.e. the language containing and ) is not finitely axiomatizable.     

Lexical insertion,inflection, and derivation: Creative processes in word production

This study examines the distinction between creative and noncreative behavior as applied to the production of words and sentences. Under one hypothesis words such as governor and government are stored as independent units and produced by rote, but under another these nouns are not produced as fully integrated units but are generated by rules for combining stems (govern) and affixes (-ment and -or). Analyses of German and English speech errors supported the generation-by-rule thesis and indicated that word stems, prefixes, and suffixes must be separately stored in the internal lexicon and marked as to syntactic function in combining with other word components. The data also suggested a three-stage model of lexical processes in the production of speech: a lexical insertion stage whereby abstract lexical formatives are called on or introduced into sentences by means of abstract syntactic and semantic features; a rule application stage whereby feature agreement rules are applied to the formative and then inflectional, derivational, and phonological rules are applied to derive the phonetic string; finally a serial output stage whereby the phonetic string is translated into serially ordered motor commands.This research was supported by NIMH Grant RO 19964 to Donald Mackay.     

Logic of Knowledge and Utterance and the Liar

We extend the ordinary logic of knowledge based on the operator K and the system of axioms S₅ by adding a new operator U, standing for the agent utters , and certain axioms and a rule for U, forming thus a new system KU. The main advantage of KU is that we can express in it intentions of the speaker concerning the truth or falsehood of the claims he utters and analyze them logically. Specifically we can express in the new language various notions of lying, as well as of telling the truth. Consequently, as long as lying or telling the truth about a fact is an intentional mode of the speaker, we can resolve the Liar paradox, or at least some of its variants, turning it into an ordinary (false or true) sentence. Also, using Kripke structures analogous to those employed by S. Kraus and D. Lehmann in [3] for modelling the logic of knowledge and belief, we offer a sound and complete semantics for KU.     

Is “some-other-time” sometimes better than “sometime” for proving partial correctness of programs?

The main result of this paper belongs to the field of the comparative study of program verification methods as well as to the field called nonstandard logics of programs. We compare the program verifying powers of various well-known temporal logics of programs, one of which is the Intermittent Assertions Method, denoted as Bur. Bur is based on one of the simplest modal logics called S5 or sometime-logic. We will see that the minor change in this background modal logic increases the program verifying power of Bur. The change can be described either technically as replacing the reflexive version of S5 with an irreflexive version, or intuitively as using the modality some-other-time instead of sometime. Some insights into the nature of computational induction and its variants are also obtained.This project was supported by the Hungarian National Foundation for Scientific Research, Grant No. 1810.     

A New Representation Theorem for Contranegative Deontic Logic

The logic of an ought operator O is contranegative with respect to an underlying preference relation if it satisfies the property Op & (¬p)(¬q) Oq. Here the condition that is interpolative ((p (pq) q) (q (pq) p)) is shown to be necessary and sufficient for all -contranegative preference relations to satisfy the plausible deontic postulates agglomeration (Op & OqO(p&q)) and disjunctive division (O(p&q) Op Oq).     

What is the upper part of the lattice of bimodal logics?

We define an embedding from the lattice of extensions ofT into the lattice of extensions of the bimodal logic with two monomodal operators ₁ and ₂, whose ₂-fragment isS5 and ₁-fragment is the logic of a two-element chain. This embedding reflects the fmp, decidability, completenes and compactness. It follows that the lattice of extension of a bimodal logic can be rather complicated even if the monomodal fragments of the logic belong to the upper part of the lattice of monomodal logics.Presented byWolfgang Rautenberg     

The endorsement of irrational beliefs in a general clinical population

Burgess' Attitude and Belief Scale, a measure of Ellis' irrational beliefs, was administered to a large sample of outpatients. All 13 subscales demonstrate very adequate internal consistency. A factor analysis yielded one factor accounting for 83% of the variance, which was labeled irrationality. Clients endorsed focused items more than overgeneralized items, self-referential items more than non-self-referential items, and preferential items more than irrational items. Clients also received higher scores on the irrational process of demandingness than they did on the irrational process of awfulizing, self-worth and low frustration tolerance. The results were consistent with new formulations in Rational-Emotive theory. Suggestions were made for the construction of measures of irrational beliefs.     

Inverses for Normal Modal Operators

Given a 1-ary sentence operator , we describe L - another 1-ary operator - as as a left inverse of in a given logic if in that logic every formula is provably equivalent to L. Similarly R is a right inverse of if is always provably equivalent to R. We investigate the behaviour of left and right inverses for taken as the operator of various normal modal logics, paying particular attention to the conditions under which these logics are conservatively extended by the addition of such inverses, as well as to the question of when, in such extensions, the inverses behave as normal modal operators in their own right.     

Remarks on Gregory's “Actually” Operator

In this note we show that the classical modal technology of Sahlqvist formulas gives quick proofs of the completeness theorems in [8] (D. Gregory, Completeness and decidability results for some propositional modal logics containing actually operators, Journal of Philosophical Logic 30(1): 57–78, 2001) and vastly generalizes them. Moreover, as a corollary, interpolation theorems for the logics considered in [8] are obtained. We then compare Gregory's modal language enriched with an actually operator with the work of Arthur Prior now known under the name of hybrid logic. This analysis relates the actually axioms to standard hybrid axioms, yields the decidability results in [8], and provides a number of complexity results. Finally, we use a bisimulation argument to show that the hybrid language is strictly more expressive than Gregory's language.     

The American plan completed: Alternative classical-style semantics,without stars,for relevant and paraconsistent logics

American-plan semantics with 4 values 1, 0, { {1, 0}} {{}}, interpretable as True, False, Both and Neither, are furnished for a range of logics, including relevant affixing systems. The evaluation rules for extensional connectives take a classical form: in particular, those for negation assume the form 1 (A, a) iff 0 (A, a) and 0 (A, a) iff 1 (A, a), so eliminating the star function *, on which much criticism of relevant logic semantics has focussed. The cost of these classical features is a further relation (or operation), required in evaluating falsity assignments of implication formulae.Two styles of 4 valued relational semantics are developed; firstly a semantics using notions of double truth and double validity for basic relevant systemB and some extensions of it; and secondly, since the first semantics makes heavy weather of validating negation principles such as Contraposition, a reduced semantics using more complex implicational rules for relevant systemC and various of its extensions. To deal satisfactorily with elite systemsR,E andT, however, further complication is inevitable; and a relation of mateship (suggested by the Australian plan) is introduced to permit cross-over from 1 to 0 values and vice versa.     

On superintuitionistic logics as fragments of proof logic extensions

Coming fromI andCl, i.e. from intuitionistic and classical propositional calculi with the substitution rule postulated, and using the sign to add a new connective there have been considered here: Grzegorozyk's logicGrz, the proof logicG and the proof-intuitionistic logicI set up correspondingly by the calculiFor any calculus we denote by the set of all formulae of the calculus and by the lattice of all logics that are the extensions of the logic of the calculus, i.e. sets of formulae containing the axioms of and closed with respect to its rules of inference. In the logiclG the sign is decoded as follows: A = (A & A). The result of placing in the formulaA before each of its subformula is denoted byTrA. The maps are defined (in the definitions of x and the decoding of is meant), by virtue of which the diagram is constructedIn this diagram the maps, x and are isomorphisms, thereforex ^–1 = ; and the maps and are the semilattice epimorphisms that are not commutative with lattice operation +. Besides, the given diagram is commutative, and the next equalities take place: ^–1 = ^–1 and = ^–1 x. The latter implies in particular that any superintuitionistic logic is a superintuitionistic fragment of some proof logic extension.     

Context and strategy in acquiring temporal connectives

There has been considerable debate over the role of comprehension strategies in the acquisition of temporal connectives. This study examined the role of caluse logic and interpretational strategies in the acquisition of temporal words; age-related changes were also considered. Thirty-two children between 3 and 5 years of age served as subjects. Sentences with a variety of temporal words were used to tap children's comprehension of before, after, when, while, just before that, and and after that. Clause logic was found to significantly improve the understanding of these sentences. However, the order of mention and main clause first strategies were used infrequently. Apparently, when children respond to sentences that describe information in a way that is consistent with that they normally hear, these strategies are seldom applied.I wish to express my thanks to Janice Bogen, who assisted in data collection, and to Erika Hoff-Ginsberg, who assisted in some of the data analysis.     

On Ultrafilter Logic and Special Functions

Logics for generally were introduced for handling assertions with vague notions,such as generally, most, several, etc., by generalized quantifiers, ultrafilter logic being an interesting case. Here, we show that ultrafilter logic can be faithfully embedded into a first-order theory of certain functions, called coherent. We also use generic functions (akin to Skolem functions) to enable elimination of the generalized quantifier. These devices permit using methods for classical first-order logic to reason about consequence in ultrafilter logic.Presented by André Fuhrmann