Algebraic proof of the separation theorem on classical propositional calculus

(1978). Orthomodular Lattices and Quantum Physics. Mathematics Magazine: Vol. 51, No. 5, pp. 299-303.

A κ-denotational semantics for map theory in ZFC + SI

The first part of the paper_ outlines the development of causality in Greek philosophy. Some remarks are made on how some medieval philosophers approached to the problem. The paper shows also how modern philosophy understood causation. The paper inquires into the characteristics of causal relation as it is accepted in the domain of modern and recent physics.In the second part of the paper one finds some remarks concerning the programme of construing a new system CI of logic of causal propositions. This system is adequate to the manner in which causality is presented in physics. The system CI is being construed to characterize the connective of relativistic conditionals ‘$\rightsquigarrow$’ by means of the methods of recent logic. This connective should be read as follows: ‘if …, then for that reason …’. The arguments of the connective ‘$\rightsquigarrow$’ may be propositional formulas describing a particular event. The new system of non-classical logic is based on the classical propositional calculus, on the one of systems of temporal logic, and on a system ZI of logic of change. The findings concerning causation, as outlined in the first part of the paper, constitute descriptive semantics of the system CI presented in the second part of the paper. This system may play a positive role in the indirect justification of theorems in philosophy in a broad sense.

In this paper we study the algebraization of two Gentzen systems, both of them generating the implication-less fragment of the intuitionistic propositional calculus. We prove that they are algebraizable, the variety of pseudocomplemented distributive lattices being an equivalent algebraic semantics for them, in the sense that their Gentzen deduction and the equational deduction over this variety are interpretable in one another, these interpretations being essentially inverse to one another. As a consequence, the consistent deductive systems that satisfy the properties of Conjunction, Disjunction and Pseudo-Reductio ad Absurdum are described by giving apropriate Gentzen systems for them. All these Gentzen systems are algebraizable, the subvarieties of the variety of pseudocomplemented distributive lattices being their equivalent algebraic semantics respectively. Finally we give a Gentzen system for the conjunction and disjunction fragment of the classical propositional calculus, prove that the variety of distributive lattices is an equivalent algebraic semantics for it and give a Gentzen system, weaker than the latter, the variety of lattices being an equivalent algebraic semantics for it.

Two proofs of Fine's theorem

We introduce Boolean-like algebras of dimension n (n{mathrm {BA}}s) having n constants {{{mathsf {e}}}}_1,ldots ,{{{mathsf {e}}}}_n, and an (n+1)-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of n{mathrm {BA}}s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The n{mathrm {BA}}s provide the algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, n{mathrm {CL}}, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in n{mathrm {CL}}, and, via the embeddings, in all the finite tabular logics.

The starting point of this paper is the empirically determined ability to reason in natural language by employing probable sentences. A sentence is understood to be logically probable if its schema, expressed as a formula in the language of classical propositional calculus, takes the logical value of truth for the majority of Boolean valuations, i.e., as a logically probable formula. Then, the formal system P is developed to encode the set of these logically probable formulas. Based on natural semantics, a strong completeness theorem for P is proved. Alternative notions of consequence for logically probable sentences are also considered.

Propositional algorithmic logic PAL is a propositional counterpart of algorithmic logics. It investigates properties of program connectives: begin...end, while...do, if..then..else, or .... PAL supplies tools for reasoning about programs constructed from program variables by means of program connectives and about their algorithmic properties. Sound rules of inference and tautologies of PAL are as important in analysis of programs (e.g. verification) as tautologies of classical propositional calculus. On the other hand propositional algorithmic theories are of highest interest, since they can capture properties of data structures and also algorithmic properties of behaviours of concurrent systems. We proved that the semantical and the syntactical consequence operations coincide (completeness property) and that the model existence theorem holds.

In [1] and [2] D. Lewis formulates his counterfactual logic VC as follows. The language contains the connectives ∧, ∨, ⊃, ¬ and the binary connective ≤. A ≤ B is read as “A is at least as possible as B”. The following connectives are defined in terms of ≤.A < B: = ¬(B ≤ A) (it is more possible that A than that B).◊ A ≔ ¬(⊥ ≤ A) (⊥ is the false formula; A is possible).□ A ≔ ⊥ ≤ ¬A (A is necessary). (if A were the case, then B would be the case). (if A were the case, then B might be the case). and are two counterfactual conditional operators. (AB) iff ¬(A ¬B).The following axiom system VC is presented by D. Lewis in [1] and [2]: V: (1) Truthfunctional classical propositional calculus.

Some determinative sequent system (DS) for classical propositional calculus is introduced on the base of wellknown Tseytin's transformation. It is proved that the system DS is polynomial equivalent to the resolution system R and cut-free sequent system PK -. Then we define the system SDS (DS with a substitution rule) and the systems SkDS (DS with restricted substitution rules, where the number of connectives in substituted formulas is bounded by ). It is proved that for every the system Sk+1DS has an exponential speed-up over the system SkDS in the tree form, and the system SDS is polynomially equivalent to the Frege systems.

Introduced by Chang in the late fifties, MV-algebras stand to Łukasiewicz’s infinite-valued propositional logic as boolean algebras stand to the classical propositional calculus. As stated by Chang in his original paper, “MV is supposed to suggest many-valued logics … for want of a better name”. The name has stuck. After some decades of relative quiescency, MV-algebras are today intensely investigated. On the one hand, these algebras find applications in such diverse fields as error-correcting feedback codes and logic-based control theory. On the other hand, MV-algebras are interesting mathematical objects in their own right. The main aim of this paper is to show that the interaction between MV-algebras and lattice-ordered abelian groups — including the timehonored theory of magnitudes — has much to offer, not only to specialists in these two fields, but also to people interested in the fan-theoretic description of toric varieties, and in the K0-theory of AF C*-algebras.

On the Inconsistency of Classical Propositional Calculus

In the first of her papers on functional calculi based on strict implication, Ruth Barcan Marcus takes as her starting point the Lewis systems S2 and S4, supplemented by one of the normal bases for quantification theory, and by one special axiom for the mixture, asserting that if possibly something φ's then something possibly φ's. In the symbolism of Łukasiewicz, which will be used here, this axiom is expressible as CMΣxφxΣxMφx. In the present note I propose to show that if S5 had been taken as a startingpoint rather than S2 or S4, this formula need not have been laid down as an axiom but could have been deduced as a theorem.It has been shown by Gödel that a system equivalent to S5 may be obtained if we add to any complete basis for the classical propositional calculus a pair of symbols for ‘Necessarily’ and ‘Possibly,’ which here will be ‘L’ and ‘M’; the axiomsthe ruleRL: If α is a thesis, so is Lα;and the definitionDf. M: M = NLN.

In many of the standard formulations of the classical propositional calculus there are to found the theorems A ⊃ [B ⊃ A] and -7A ⊃ [,A ⊃ B]. The sense of the first is frequently expressed by saying that if a proposition is true then any proposition whatever implies it. The sense of the second is expressed by asserting that if a proposition is false then it implies any proposition. These theorems are often called the ‘paradoxes of (material) implication’. Certain related kinds of implication such as AA‘B and B ⟶ A v A’ are sometimes regarded as fallacious or paradoxical, even though classically correct, on the grounds that AA‘ (or A v A’) is irrelevant to the arbitrary proposition B. Much of the criticism of such implications has focused on the nature of disjunction.

at our disposal, and truth-tables tell us that (3) is a valid argument form of classical sentential calculus (CSC); hence, (2) is valid. Essentially the same procedure can be repeated for many other valid arguments; indeed, for all the valid arguments that can be so paraphrased as to fall under the scope of an extant (correct) logical system. No such procedure, however, is available to establish the invalidity of invalid arguments. Consider for example the argument