Some theorems on structural consequence operations

Two characterizations are given of those structural consequence operations on a propositional language which can be defined via proofs from a finite number of polynomial rules.     

Some theorems in haploscopic vision

Three formal hypotheses are specified concerning the combination of neural signals from the two eyes. The hypotheses are (1) that a large uniform light presented to the opposite eye has no effect (independence), (2) that all information from one eye relevant to a given percept (for example, brightness) is encoded by a single neural signal (isolation), and (3) the mutually exclusive and exhaustive alternative to isolation (interaction). Though independence or isolation often has been claimed or simply assumed to hold, these hypotheses imply specific empirical relationships. These relationships are derived for brightness and for equilibrium colors. In addition, one model consistent with the interaction hypothesis is developed. The isolation hypothesis is important for generalizing results from monocular experiments to normal binocular vision. If it is false, monocular results can reflect a combination of one eye's neural signals that never occurs with binocular stimuli.     

Choice: Some quantitative relations

Six pigeons responded in fifty-six conditions on a concurrent-chains procedure. Conditions included several with equal initial links and unequal terminal links, several with unequal initial links and equal terminal links, and several with both unequal initial and terminal links. Although the delay-reduction hypothesis accounted well for choice when the initial links were equal (mean deviation of .04), it fit the data poorly when the initial links were unequal (mean deviation of .18). A modification of the delay-reduction hypothesis, replacing the rates of reinforcement with the square roots of these rates, fit the data better than either the unmodified delay-reduction equation or Killeen's (1982) model. The modified delay-reduction equation was also consistent with data from prior studies using concurrent chains. The absolute rates of responding in each terminal link were well described by the same hyperbola (Herrnstein, 1970) that describes response rates on simple interval schedules.     

Game theoretical semantics and entailment

The essence of the meaning of a declarative sentence is given by stating its truth conditions, and consequently semantics, the study of meaning, must include a theory of truth conditions. Such a theory must not only describe accurately the truth conditions of declarative sentences, it must also answer the question of when two sentences have the same truth conditions. The fundamental semantic relation of having the same truth conditions cannot be ignored by any reasonable theory.This paper is an attempt to find a partial account of this relation by using game theoretical semantics as developed by Hintikka and his followers. The account given will establish a connection between this approach to semantics and the theory of firstdegree entailment formulated by Anderson and Belnap.     

Perfect validity,entailment and paraconsistency

This paper treats entailment as a subrelation of classical consequence and deducibility. Working with a Gentzen set-sequent system, we define an entailment as a substitution instance of a valid sequent all of whose premisses and conclusions are necessary for its classical validity. We also define a sequent Proof as one in which there are no applications of cut or dilution. The main result is that the entailments are exactly the Provable sequents. There are several important corollaries. Every unsatisfiable set is Provably inconsistent. Every logical consequence of a satisfiable set is Provable therefrom. Thus our system is adequate for ordinary mathematical practice. Moreover, transitivity of Proof fails upon accumulation of Proofs only when the newly combined premisses are inconsistent anyway, or the conclusion is a logical truth. In either case Proofs that show this can be effectively determined from the Proofs given. Thus transitivity fails where it least matters — arguably, where it ought to fail! We show also that entailments hold by virtue of logical form insufficient either to render the premisses inconsistent or to render the conclusion logically true. The Lewis paradoxes are not Provable. Our system is distinct from Anderson and Belnap's system of first degree entailments, and Johansson's minimal logic. Although the Curry set paradox is still Provable within naive set theory, our system offers the prospect of a more sensitive paraconsistent reconstruction of mathematics. It may also find applications within the logic of knowledge and belief.