Algebraic aspects of deduction theorems

The first known statements of the deduction theorems for the first-order predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas and , C(X{{a}}) iff P(, ) AC(X). [P(, ) denotes the set of formulas which result by the simultaneous substitution of for p and for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian.A part of this paper was presented in abstracted form in Bulletin of the Section of Logic, Vol. 12, No. 3 (1983), pp. 111–116, and in The Journal of Symbolic 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.     

A topological logic of action

We consider a quantifier-free language in which there are terms as well as formulas. The proposition-forming propositional operators are the usual ones, and the term-making term operators are the usual lattice theoretical ones. In addition there is a formula-making term operator, does. We study a new logic in which does is claimed to approximate some features of the informal concept the agent performs the action .     

A deontic logic of action

The formal language studied in this paper contains two categories of expressions, terms and formulas. Terms express events, formulas propositions. There are infinitely many atomic terms and complex terms are made up by Boolean operations. Where and are terms the atomic formulas have the form = ( is the same as ), Forb ( is forbidden) and Perm ( is permitted). The formulae are truth functional combinations of these. An algebraic and a model theoretic account of validity are given and an axiomatic system is provided for which they are characteristic.The closure principle, that what is not forbidden is permitted is shown to hold at the level of outcomes but not at the level of events. In the two final sections some other operators are considered and a semantics in terms of action games.     

The elimination of de re formulas

It is shown that de re formulas are eliminable in the modal logic S5 extended with the axiom scheme x x.     

Paraconsistent logic and model theory

The object of this paper is to show how one is able to construct a paraconsistent theory of models that reflects much of the classical one. In other words the aim is to demonstrate that there is a very smooth and natural transition from the model theory of classical logic to that of certain categories of paraconsistent logic. To this end we take an extension of da Costa'sC ₁ ⁼ (obtained by adding the axiom A A) and prove for it results which correspond to many major classical model theories, taken from Shoenfield [5]. In particular we prove counterparts of the theorems of o-Tarski and Chang-o-Suszko, Craig-Robinson and the Beth definability theorem.     

A Revenge-Immune Solution to the Semantic Paradoxes

The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema True(A)A, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in ordinary contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(A) with A within the language.The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are defective. We can in fact define a hierarchy of defectiveness predicates within the language. Contrary to claims that any solution to the paradoxes just breeds further paradoxes (revenge problems) involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various levels of defectiveness can all be made coherent together within a single object language.     

A note on the logic of signed equations

A signed -equation is an expression of the form t t or t t, where t and t are -terms (for some ranked set ). We characterize those classes of -algebras which are models of a set of signed -equations. Further we consider the problem of finding a complete deductive system analogous to equational logic for the logical consequence operation restricted to signed equations.     

On the logic of conscious belief

In this paper we are discussing a version of propositional belief logic, denoted by LB, in which so-called axioms of introspection (B BB and B B B) are added to the usual ones. LB is proved to be sound and complete with respect to Boolean algebras equipped with proper filters (Theorem 5). Interpretations in classical theories (Theorem 4) are also considered. A few modifications of LB are further dealt with, one of which turns out to be S5.     

Modal logic with names

We investigate an enrichment of the propositional modal language with a universal modality having semanticsx iff y(y ), and a countable set of names — a special kind of propositional variables ranging over singleton sets of worlds. The obtained language _c proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment () of, where is an additional modality with the semanticsx iff y(y x y ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in _c. Strong completeness of the normal _c-logics is proved with respect to models in which all worlds are named. Every _c-logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from to _c are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched.     

Axioms for Deliberative Stit

Based on a notion of companions to stit formulas applied in other papers dealing with astit logics, we introduce choice formulas and nested choice formulas to prove the completeness theorems for dstit logics in a language with the dstit operator as the only non-truth-functional operator. The main logic discussed in this paper is the basic logic of dstit with multiple agents, other logics discussed include the basic logic of dstit with a single agent and some logics of dstit with multiple agents each of which corresponds to a semantic condition concerning the number of possible choices for agents.     

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.     

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     

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.     

The logic of linear tolerance

A nonempty sequence T₁,...,T_n of theories is tolerant, if there are consistent theories T ₁ ⁺ ,..., T _n ⁺ such that for each 1 i n, T _i ⁺ is an extension of T_i in the same language and, if i n, T _i ⁺ interprets T _i+1 ⁺ . We consider a propositional language with the modality , the arity of which is not fixed, and axiomatically define in this language the decidable logics TOL and TOL. It is shown that TOL (resp. TOL) yields exactly the schemata of PA-provable (resp. true) arithmetical sentences, if (A₁,..., A_n) is understood as (a formalization of) PA+A₁, ..., PA+A_n is tolerant.     

A note on the normal form of closed formulas of interpretability logic

Each closed (i.e. variable free) formula of interpretability logic is equivalent in ILF to a closed formula of the provability logic G, thus to a Boolean combination of formulas of the form ⁿ.     

Parallel action: Concurrent dynamic logic with independent modalities

Regular dynamic logic is extended by the program construct, meaning and executed in parallel. In a semantics due to Peleg, each command is interpreted as a set of pairs (s,T), withT being the set of states reachable froms by a single execution of, possibly involving several processes acting in parallel. The modalities << and [] are given the interpretations<>A is true ats iff there existsT withsRT andA true throughoutT, and[]A is true ats iff for allT, ifsRT thenA is true throughoutT, which make <> and [] no longer interdefinable via negation, as they are in the regular case.We prove that the logic defined by this modelling is finitely axiomatisable and has the finite model property, hence is decidable. This requires the development a new theory of canonical models and filtrations for reachability relations.     

Reasoning and logic

Gilbert Harman, in Logic and Reasoning (Synthese 60 (1984), 107–127) describes an unsuccessful attempt ... to develop a theory which would give logic a special role in reasoning. Here reasoning is psychological, a procedure for revising one's beliefs. In the present paper, I construe reasoning sociologically, as a process of linguistic interaction; and show how both reasoning in the psychologistic sense and logic are related to that process.     

Equivalents of Mingle and Positive Paradox

Relevant logic is a proper subset of classical logic. It does not include among its theorems any ofpositive paradox A (B A)mingle A (A A)linear order (A B) (B A)unrelated extremes (A ) (B B¯)This article shows that those four formulas have different effects when added to relevant logic, and then lists many formulas that have the same effect as positive paradox or mingle.     

Identity in modal logic theorem proving

THINKER is an automated natural deduction first-order theorem proving program. This paper reports on how it was adapted so as to prove theorems in modal logic. The method employed is an indirect semantic method, obtained by considering the semantic conditions involved in being a valid argument in these modal logics. The method is extended from propositional modal logic to predicate modal logic, and issues concerning the domain of quantification and existence in a world's domain are discussed. Finally, we look at the very interesting issues involved with adding identity to the theorem prover in the realm of modal predicate logic. Various alternatives are discussed.