and I CaPI e D, then I Pl e D for all similar assignments. (2) For all values of P and q, I CPCNPql e D. (3) For all values of the variables in a, if la( e U then INal e D. (4) The F,P are constant functions such that, for all values of P, ~ FIP~ = 1, I F, Pl = 2,..., I Fât I = m.
A bivalent valuation is snt iff sound (standard PC inference rules take truths only into truths) and non-trivial (not all wffs are assigned the same truth value). Such a valuation is normal iff classically correct for each connective. Carnap knew that there were non-normal snt valuations of PC, and that the gap they revealed between syntax and semantics could be jumped as follows. Let VAL snt be the set of snt valuations, and VAL nrm be the set of normal ones. (...) The bottom row in the table for the wedge is not semantically determined by VAL snt, but if one deletes from VAL snt all those valuations that are not classically correct at the aforementioned row, one jumps straights to VAL nrm and thus to classical semantics. The conjecture we call semantic holism claims that the same thing happens for any semantic indeterminacy in any row in the table of any connective of PC, i.e., to remove it is to jump straight to classical semantics. We show (i) why semantic holism is plausible and (ii) why it is nevertheless false. And (iii) we pose a series of questions concerning the number of possible steps or jumps between the indeterminate semantics given by VAL snt and classical semantics given by VAL nrm. (shrink)
Gupta’s Rule of Revision theory of truth builds on insights to be found in Martin and Woodruff (1975) and Kripke (1975) (who in turn build on Tarski) in order to permanently deepen our understanding of truth, of paradox (and of the absence of it), and of how we work our language while our language is working us. His concept of a predicate deriving its meaning by way of a Rule of Revision ought to impact significantly on the philosophy of language. (...) Still, fortunately, he has left me something to.. (shrink)
1. Rescher 1964 — henceforth HR — proposes a way of reasoning from a set of hypotheses which may include both some of our beliefs and also hypotheses contradicting those beliefs. The aim of this paper is to point out what I take to be a fault in Rescher’s proposal, and to suggest a modification of it, using a nonclassical logic, which avoids that fault. The paper neither attacks nor defends the broader aspects of Rescher’s proposal, but merely assumes that (...) it is at least prima facie worthwhile and therefore worthy of amendment; consequently, I shall try to tinker as little as possible. In particular, the use of a nonclassical logic which I propose does not replace any use by HR of classical logic — in those places where Rescher is classical, I shall be classical, too. (Instead, the amendment introduces a nonclassical logic at a point where HR uses no logic at all.). (shrink)
According to John A Barker, whether an argument begs the question is purely a matter of logical form (Dialogue, 1976). According to me, it is also a matter of epistemic conditions; some arguments which beg the question in some contexts need not beg the question in every context (Analysis, 1972). I point out difficulties in Barker's treatment and defend my own views against some of his criticisms. In the concluding section, "Alleged difficulties with disjunctive syllogism," I defend the validity of (...) disjunctive syllogism against the views of Alan Ross Anderson and Nuel D Belnap, Jr., ask pointed questions about the notion of intensional disjunction, and suggest how my treatment of begging the question can be extended to deal with the so-called paradoxes of strict implication. (shrink)
A bivalent valuation is snt iff sound (standard PC inference rules take truths only into truths) and non-trivial (not all wffs are assigned the same truth value). Such a valuation is normal iff classically correct for each connective. Carnap knew that there were non-normal snt valuations of PC, and that the gap they revealed between syntax and semantics could be "jumped" as follows. Let $VAL_{snt}$ be the set of snt valuations, and $VAL_{nrm}$ be the set of normal ones. The bottom (...) row in the table for the wedge 'v' is not semantically determined by $VAL_{snt}$ , but if one deletes from $VAL_{snt}$ all those valuations that are not classically correct at the aforementioned row, one jumps straights to $VAL_{nrm}$ and thus to classical semantics. The conjecture we call semantic holism claims that the same thing happens for any semantic indeterminacy in any row in the table of any connective of PC, i.e., to remove it is to jump straight to classical semantics. We show (i) why semantic holism is plausible and (ii) why it is nevertheless false. And (iii) we pose a series of questions concerning the number of possible steps or jumps between the indeterminate semantics given by $VAL_{snt}$ and classical semantics given by $VAL_{nrm}$. (shrink)
Tableau formulations are given for the relevance logics E (Entailment), R (Relevant implication) and RM (Mingle). Proofs of equivalence to modus-ponens-based formulations are vialeft-handed Gentzen sequenzen-kalküle. The tableau formulations depend on a detailed analysis of the structure of tableau rules, leading to certain global requirements. Relevance is caught by the requirement that each node must be used; modality is caught by the requirement that only certain rules can cross a barrier. Open problems are discussed.
