Logic and Truth: Some Logics without Theorems

Abstract

Two types of logical consequence are compared: one, with respect to matrix and designated elements and the other with respect to ordering in a suitable algebraic structure. Particular emphasis is laid on algebraic structures in which there is no top-element relative to the ordering. e signi cance of this special condition is discussed. Sequent calculi for a number of such structures are developed. As a consequence it is re-established that the notion of truth as such, not to speak of tautologies, is inessential in order to de ne validity of an argument.Keywords: logical truth, logical consequence, lattice . Introduction: Logical consequences vis a vis truth e notion of truth serves various explanatory purposes. One of these is found in the attempted explanations of validity of deductive arguments. An argument is valid if and only if the conclusion of an argument is true whenever all the premises of the argument are also true. e context, here, is taken as classical and two-valued. When consequence is understood as a relation preserving truth, the notion of truth is taken to be that which relates semantics to states of a airs. Set theoretic models of well-formed formulae of st order languages are the formal representation of states of a airs. Sometimes formulae (propositions) are modeled as sets of possible worlds, logical connectives by set theoretic operations and logical consequence by set theoretic relation. ough apparently there is no mention of 'truth' in this approach, the underlying intention is that a formula be associated with the set of those worlds (states of a airs) where the formula is true. Long back in 's manyvaluedness (i.e., allowing sentences to have values other than true and false) was introduced and gradually accepted within the discourse of logic and

Negation-Free Definitions of Paraconsistency

“…Due to this, while in our case, every α ∈ L is a theorem, i.e. ∅ |= V α, the logics in [11] equipped with such consequence relations are without any theorems.…”

“…While for certain logics, such as classical propositional logic, the two approaches lead to the same consequence relation, such is not the case for other logics. A comparison between the two approaches has been discussed in [11]. The consequence relation above is, however, slightly different from the one defined using the order relation in the semantic structure in [11].…”

“…A comparison between the two approaches has been discussed in [11]. The consequence relation above is, however, slightly different from the one defined using the order relation in the semantic structure in [11]. There Γ |= V α iff for all v ∈ V, v(Γ) ≤ v(α), where Γ ∪ {α}, V are as in 2.11 and v(Γ) denotes some kind of a composition of the values in {v(β) | β ∈ Γ}, the composition being an operation in the semantic structure.…”