This paper develops a new proof method for two propositional paraconsistent logics: the propositional part of Batens' weak paraconsistent logic CLuN and Schütte's maximally paraconsistent logic Φv. Proofs are de.ned as certain sequences of questions. The method is grounded in Inferential Erotetic Logic.

We present a formalization of first-order predicate calculus with equality which, unlike traditional systems with axiom schemata or substitution rules, is finitely axiomatized in the sense that each step in a formal proof admits only finitely many choices. This formalization is primarily based on the inference rule of condensed detachment of Meredith. The usual primitive notions of free variable and proper substitution are absent, making it easy to verify proofs in a machine-oriented application. Completeness results are presented. The example of (...) Zermelo-Fraenkel set theory is shown to be finitely axiomatized under the formalization. The relationship with resolution-based theorem provers is briefly discussed. A closely related axiomatization of traditional predicate calculus is shown to be complete in a strong metamathematical sense. (shrink)

For every finite n ≥ 4 there is a logically valid sentence φ n with the following properties: φ n contains only 3 variables (each of which occurs many times); φ n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol): φ n has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n - 1 variables. This result was first proved using the machinery of (...) algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φ n has a proof with only n variables. To show that φ n has no proof with only n - 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic. (shrink)

The k-provability for an axiomatic system A is to determine, given an integer k 1 and a formula in the language of A, whether or not there is a proof of in A containing at most k lines. In this paper we develop a unification-theoretic method for investigating the k-provability problem for Parikh systems, which are first-order axiomatic systems that contain a finite number of axiom schemata and a finite number of rules of inference. We show that the k-provability problem (...) for a Parikh system reduces to a unification problem that is essentially the unification problem for second-order terms. By solving various subproblems of this unification problem , we solve the k- probability problem for a variety of Parikh systems, including several formulations of Peano arithmetic. Our method for investigating the k-provability problem employs algorithms that compute and characterize unifiers. We give some examples of how these algorithms can be used to solve complexity problems other than the k-provability problem. (shrink)

In the first two sections I present and motivate a formal semantics program that is modeled after the application of numbers in measurement (e.g., of length). Then, in the main part of the paper, I use the suggested framework to give an account of the semantics of necessity and possibility: (i) I show thatthe measurement theoretic framework is consistent with a robust (non-Quinean) view of modal logic, (ii) I give an account of the semantics of the modal notions within this (...) framework, and (iii) I defend the suggested account against various objections. (shrink)

There are different views of the logic of plurals that are now in circulation, two of which we will compare in this paper. One of these is based on a two-place relation of being among, as in ‘Peter is among the juveniles arrested’. This approach seems to be the one that is discussed the most in philosophical journals today. The other is based on Bertrand Russell’s early notion of a class as many, by which is meant not a class as (...) one, i.e., as a single entity, but merely a plurality of things. It was this notion that Russell used to explain plurals in his 1903 Principles of Mathematics; and it was this notion that I was able to develop as a consistent system that contains not only a logic of plurals but also a logic of mass nouns as well. We compare these two logics here and then show that the logic of the Among relation is reducible to the logic of classes as many. (shrink)