The notion of local deduction theorem (which generalizes on the known instances of indeterminate deduction theorems, e.g. for the infinitely-valued ukasiewicz logic C ) is defined. It is then shown that a given finitary non-pathological logic C admits the local deduction theorem iff the class Matr(C) of all matrices validating C has the C-filter extension property (Theorem II.1).

Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.

We present an algebraic proof of the theorem stating that there are continuum many axiomatic extensions of global consequence associated with modal system E that do not admit the local deduction detachment theorem. We also prove that all these logics lack the finite frame property and have exactly three proper axiomatic extensions, each of which admits the local deduction detachment theorem.

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT —a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system (...) has a contextual DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case. (shrink)

This paper provides a finer analysis of the well-known form of the Local Deduction Theorem in contraction-free logics . An infinite hierarchy of its natural strengthenings is introduced and studied. The main results are the separation of its initial four members and the subsequent collapse of the hierarchy.

We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we (...) show that for the pure logic of one of these implicational connectives two – in general distinct – consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the ‘propositional operations’ associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives. (shrink)

We recapitulate (Section 1) some basic details of the system of implicative BCSK logic, which has two primitive binary implicational connectives, and which can be viewed as a certain fragment of the modal logic S5. From this modal perspective we review (Section 2) some results according to which the pure sublogic in either of these connectives (i.e., each considered without the other) is an exact replica of the material implication fragment of classical propositional logic. In Sections 3 and 5 we (...) show that for the pure logic of one of these implicational connectives two-in general distinct-consequence relations (global and local) definable in the Kripke semantics for modal logic turn out to coincide, though this is not so for the pure logic of the other connective, and that there is an intimate relation between formulas constructed by means of the former connective and the local consequence relation. (Corollary 5.8. This, as we show in an Appendix, is connected to the fact that the 'propositional operations' associated with both of our implicational connectives are close to being what R. Quackenbush has called pattern functions.) Between these discussions Section 4 examines some of the replacement-of-equivalents properties of the two connectives, relative to these consequence relations, and Section 6 closes with some observations about the metaphor of identical twins as applied to such pairs of connectives. (shrink)

In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACSs A of all algebraic closed-set systems C on A such that (A, C) ε FGModS. We relate some properties of these structures (...) with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACSs A is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to hold only for deductive systems in more restricted classes like the protoalgebraic or the weakly algebraizable ones. The paper ends with a section on examples and counterexamples. (shrink)

The paper discusses the origin of dark matter and dark energy from the concepts of time and the totality in the final analysis. Though both seem to be rather philosophical, nonetheless they are postulated axiomatically and interpreted physically, and the corresponding philosophical transcendentalism serves heuristically. The exposition of the article means to outline the “forest for the trees”, however, in an absolutely rigorous mathematical way, which to be explicated in detail in a future paper. The “two deductions” are two successive (...) stage of a single conclusion mentioned above. The concept of “transcendental invariance” meaning ontologically and physically interpreting the mathematical equivalence of the axiom of choice and the well-ordering “theorem” is utilized again. Then, time arrow is a corollary from that transcendental invariance, and in turn, it implies quantum information conservation as the Noether correlate of the linear “increase of time” after time arrow. Quantum information conservation implies a few fundamental corollaries such as the “conservation of energy conservation” in quantum mechanics from reasons quite different from those in classical mechanics and physics as well as the “absence of hidden variables” (versus Einstein’s conjecture) in it. However, the paper is concentrated only into the inference of another corollary from quantum information conservation, namely, dark matter and dark energy being due to entanglement, and thus and in the final analysis, to the conservation of quantum information, however observed experimentally only on the “cognitive screen” of “Mach’s principle” in Einstein’s general relativity. therefore excluding any other source of gravitational field than mass and gravity. Then, if quantum information by itself would generate a certain nonzero gravitational field, it will be depicted on the same screen as certain masses and energies distributed in space-time, and most presumably, observable as those dark energy and dark matter predominating in the universe as about 96% of its energy and matter quite unexpectedly for physics and the scientific worldview nowadays. Besides on the cognitive screen of general relativity, entanglement is available necessarily on still one “cognitive screen” (namely, that of quantum mechanics), being furthermore “flat”. Most probably, that projection is confinement, a mysterious and ad hoc added interaction along with the fundamental tree ones of the Standard model being even inconsistent to them conceptually, as far as it need differ the local space from the global space being definable only as a relation between them (similar to entanglement). So, entanglement is able to link the gravity of general relativity to the confinement of the Standard model as its projections of the “cognitive screens” of those two fundamental physical theories. (shrink)

BL-algebras rise as Lindenbaum algebras from many valued logic introduced by Hájek [2]. In this paper Boolean ds and implicative ds of BL-algebras are defined and studied. The following is proved to be equivalent: (i) a ds D is implicative, (ii) D is Boolean, (iii) L/D is a Boolean algebra. Moreover, a BL-algebra L contains a proper Boolean ds iff L is bipartite. Local BL-algebras, too, are characterized. These results generalize some theorems presented in [4], [5], [6] for MV-algebras (...) which are BL-algebras fulfiling an additional double negation law x = x **. (shrink)

The Local Computation Framework has been used to improve the efficiency of computation in various uncertainty formalisms. This paper shows how the framework can be used for the computation of logical deduction in two different ways; the first way involves embedding model structures in the framework; the second, and more direct, way involves embedding sets of formulae. This work can be applied to many of the logics developed for different kinds of reasoning, including predicate calculus, modal logics, possibilistic (...) logics, probabilistic logics and non-monotonic logics. (shrink)

Natural deduction (for short: nd-) calculi have not been used systematically as a basis for automated theorem proving in classical logic. To remove objective obstacles to their use we describe (1) a method that allows to give semantic proofs of normal form theorems for nd-calculi and (2) a framework that allows to search directly for normal nd-proofs. Thus, one can try to answer the question: How do we bridge the gap between claims and assumptions in heuristically motivated ways? (...) This informal question motivates the formulation of intercalation calculi. Ic-calculi are the technical underpinnings for (1) and (2), and our paper focuses on their detailed presentation and meta-mathematical investigation in the case of classical predicate logic. As a central theme emerges the connection between restricted forms of nd-proofs and (strategies for) proof search: normal forms are not obtained by removing local "detours", but rather by constructing proofs that directly reflect proof-strategic considerations. That theme warrants further investigation. (shrink)

Natural deduction (for short: nd-) calculi have not been used systematically as a basis for automated theorem proving in classical logic. To remove objective obstacles to their use we describe (1) a method that allows to give semantic proofs of normal form theorems for nd-calculi and (2) a framework that allows to search directly for normal nd-proofs. Thus, one can try to answer the question: How do we bridge the gap between claims and assumptions in heuristically motivated ways? (...) This informal question motivates the formulation of intercalation calculi. Ic-calculi are the technical underpinnings for (1) and (2), and our paper focuses on their detailed presentation and meta-mathematical investigation in the case of classical predicate logic. As a central theme emerges the connection between restricted forms of nd-proofs and (strategies for) proof search: normal forms are not obtained by removing local "detours", but rather by constructing proofs that directly reflect proof-strategic considerations. That theme warrants further investigation. (shrink)

Agents are often assumed to have degrees of belief (“credences”) and also binary beliefs (“beliefs simpliciter”). How are these related to each other? A much-discussed answer asserts that it is rational to believe a proposition if and only if one has a high enough degree of belief in it. But this answer runs into the “lottery paradox”: the set of believed propositions may violate the key rationality conditions of consistency and deductive closure. In earlier work, we showed that this problem (...) generalizes: there exists no local function from degrees of belief to binary beliefs that satisfies some minimal conditions of rationality and non-triviality. “Locality” means that the binary belief in each proposition depends only on the degree of belief in that proposition, not on the degrees of belief in others. One might think that the impossibility can be avoided by dropping the assumption that binary beliefs are a function of degrees of belief. We prove that, even if we drop the “functionality” restriction, there still exists no local relation between degrees of belief and binary beliefs that satisfies some minimal conditions. Thus functionality is not the source of the impossibility; its source is the condition of locality. If there is any non-trivial relation between degrees of belief and binary beliefs at all, it must be a “holistic” one. We explore several concrete forms this “holistic” relation could take. (shrink)

We prove that no logic (i.e. consequence operation) determined by any class of orthomodular lattices admits the deduction theorem (Theorem 2.7). We extend those results to some broader class of logics determined by ortholattices (Corollary 2.6).

We propose a new schema for the deduction theorem and prove that the deductive system S of a prepositional logic L fulfills the proposed schema if and only if there exists a finite set A(p, q) of propositional formulae involving only prepositional letters p and q such that A(p, p) L and p, A(p, q) s q.

In this paper logics defined by finite Sugihara matrices, as well as RM itself, are discussed both in their matrix (semantical) and in syntactical version. For each such a logic a deduction theorem is proved, and a few applications are given.

Various sources in the literature claim that the deduction theorem does not hold for normal modal or epistemic logic, whereas others present versions of the deduction theorem for several normal modal systems. It is shown here that the apparent problem arises from an objectionable notion of derivability from assumptions in an axiomatic system. When a traditional Hilbert-type system of axiomatic logic is generalized into a system for derivations from assumptions, the necessitation rule has to be modified (...) in a way that restricts its use to cases in which the premiss does not depend on assumptions. This restriction is entirely analogous to the restriction of the rule of universal generalization of first-order logic. A necessitation rule with this restriction permits a proof of the deduction theorem in its usual formulation. Other suggestions presented in the literature to deal with the problem are reviewed, and the present solution is argued to be preferable to the other alternatives. A contraction-and cut-free sequent calculus equivalent to the Hubert system for basic modal logic shows the standard failure argument untenable by proving the underivability of DA from A. (shrink)

Attempts to articulate the real meaning or ultimate significance of a famous theorem comprise a major vein of philosophical writing about mathematics. The subfield of mathematical logic has supplie...

By indirect-deduction theorems introduced in the present paper we mean the theorems that allow to formalize indirect reasonings occurring in deductive practice in general and in mathematics in particular. We discuss the relationship between the introduced theorems and some logical calculi being virtually confined to propositional calculi with implication and negation. It is worth to notice that the above theorems are very handy and effective in proving logical theses.

The standard deduction theorem or introduction rule for implication, for classical logic is also valid for intuitionistic logic, but just as with predicate logic, other rules of inference have to be restricted if the theorem is to hold for weaker implicational logics.In this paper we look in detail at special cases of the Gentzen rule for and show that various subsets of these in effect constitute deduction theorems determining all the theorems of many well known as (...) well as not well known implicational logics. In particular systems of rules are given which are equivalent to the relevance logics E,R, T, P-W and P-W-I. (shrink)

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. (shrink)

A running commentary is offered on the first half of Frege’s Grundlagen der Arithmetik, §17, and suggests that Frege anticipated the method of demonstration used by Paul Bernays for the Deduction Theorem.

In [13], M. Tokarz specified some infinite family of consequence operations among all ones associated with the relevant logic RM or with the extensions of RM and proved that each of them admits a deduction theorem scheme. In this paper, we show that the family is complete in a sense that if C is a consequence operation with $C_{RM} \leq C$ and C admits a deduction theorem scheme, then C is equal to a consequence operation specified (...) in [13]. In algebraic terms, this means that the only quasivarieties of Sugihara algebras with the relative congruence extension property are the quasivarieties corresponding, via the algebraization process, to the consequence operations specified in [13]. (shrink)

This paper studies, with techniques ofAlgebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential logics have been isolated. These logics are weaker than the positive implicative logic. We have determined their (...) algebraic models and the relationships between them, and have classified them according to several standard criteria of Abstract Algebraic Logic. One of the logics is protoalgebraic but neither equivalential nor weakly algebraizable, a rare situation where very few natural examples were hitherto known. In passing we have found new, alternative presentations of positive implicative logic, both in Hilbert style and in Gentzen style, and have characterized it in terms of the restricted Deduction Theorem: it is the weakest logic satisfying Modus Ponens and the Deduction Theorem restricted to at most 2 side formulas. The algebraic part of the work has lead to the class of quasi-Hilbert algebras, a quasi-variety of implicative algebras introduced by Pla and Verdú in 1980, which is larger than the variety of Hilbert algebras. Its algebraic properties reflect those of the corresponding logics and Gentzen systems. (shrink)

This paper studies, with techniques ofAlgebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential logics have been isolated. These logics are weaker than the positive implicative logic. We have determined their (...) algebraic models and the relationships between them, and have classified them according to several standard criteria of Abstract Algebraic Logic. One of the logics is protoalgebraic but neither equivalential nor weakly algebraizable, a rare situation where very few natural examples were hitherto known. In passing we have found new, alternative presentations of positive implicative logic, both in Hilbert style and in Gentzen style, and have characterized it in terms of the restricted Deduction Theorem: it is the weakest logic satisfying Modus Ponens and the Deduction Theorem restricted to at most 2 side formulas. The algebraic part of the work has lead to the class of quasi-Hilbert algebras, a quasi-variety of implicative algebras introduced by Pla and Verdú in 1980, which is larger than the variety of Hilbert algebras. Its algebraic properties reflect those of the corresponding logics and Gentzen systems. (shrink)

By a sentential logic we understand a pair, where S is a sentential language, i.e. an absolutely free algebra freely generated by an infinite set p, q, r,... of sentential variables and endowed with countably many finitary connectives §1, §2,... and C is a consequence operation on S, the underlying set of S, satisfying the condition of structurality: eC ⊆ C, for every endomorphism e of S and for every X ⊆ S. If no confusion is likely we shall identify (...) a logic with its consequence operation C. A logic C is standard if C = [ {C : Y ⊆ X & Y is finite}, for all X ⊆ S. (shrink)