Alan Ross Anderson

The Pure Calculus of Entailment

The “implicational paradoxes” are treated by most contemporary logicians somewhat as follows: “The two-valued prepositional calculus sanctions as valid many of the obvious and satisfactory inferences which we recognize intuitively as valid, such as (A→.B→C)→.A→B→.A→C, 2 and A→B→.B→C→.A→C; it consequently suggests itself as a candidate for a formal analysis of implication.

The Formal Analysis of Normative Concepts

ence between visual images and external facts. The mental image or idea was thought to be like a mirror image or portrait or possibly a map which reproduces exactly or in some proportion the features of the object represented. The idea was true in proportion as it reproduced the objected represented.... The naive pictorial theory of correspondence was accordingly rejected by the more critical mechanists, and numerous attempts have been made to deal with the problem in terms of a symbolic theory of correspondence. The hypothesis of symbolic correspondence, however, creates a transition which has led many mechanists to suggest that it is not correspondence that is important in the truth of a sentence or formula, but the predictive power of these to produce expected results. The truth of a formula is its workability.29

Modalities in Ackermann's "Rigorous Implication"

Following a suggestion of Feys, we use “rigorous implication” as a translation of Ackermanns strenge Implikation ([1]). Interest in Ackermanns system stems in part from the fact that it formalizes the properties of a strong, natural sort of implication which provably avoids standard implicational paradoxes, and which is consequently a good candidate for a formalization of entailment (considered as a narrower relation than that of strict implication). Our present purpose will not be to defend this suggestion, but rather to present some information about rigorous implication. In particular, we show first that the structure of modalities (in the sense of Parry [4]) in Ackermanns system is identical with the structure of modalities in Lewiss S4, and secondly that (Ackermanns apparent conjecture to the contrary notwithstanding) it is possible to define modalities with the help of rigorous implication.

A Simple Treatment of Truth Functions

In this note we present an axiomatization of the classical two-valued propositional calculus, for which proofs of decidability, consistency, completeness, and independence, are almost trivial (given an understanding of truth tables).

Improved decision procedures for Lewis's calculus S4 and von Wright's calculus M

The decision procedures to be described in this paper constitute simplifications of methods previously developed by G. H. von Wright. The decision problems for both the systems under consideration have already been solved, but the known solutions are hopelessly impractical. Our methods for S4 and M , while not “practical” in the sense that they can be applied quickly and easily to any formula, do constitute a vast improvement over any other known methods. It appears, for example, that our methods (with the aid of very little ingenuity) will yield decisions readily when applied to modal functions of degree two (or less) with three (or fewer) distinct propositional variables.

Toward a formal analysis of cultural objects

In this essay we hope to make some progress toward an explication o r rational reconstruction of the concept culture, which has been of interest to philosophers at least since the time of Hegel 8, and also to social scientists since the 1870s, when Tylor z5 explicitly introduced the concept as an analytic category. Our aim is not only to clarify the concept, but also to indicate some unfortunate effects of philosophical naivet6 on the part of some social scientists, who have in fact been embroiled in problems calling for careful philosophical analysis, without recognizing the importance of philosophical techniques. We hope also to show the relevance of our analysis to some traditional philosophical problems, such as the realist-nominalist ontological debate, and the continuing critique of Aristotelian essentialism. And we remark finally that the problem is by no means artificial: we feel that the issues we hope to clarify have been genuinely perplexing both to philosophers and to social scientists. As will become clear, the things we construe as cultural are abstract, or conceptual (depending on whether one wants to take a Platonic or a Kantian approach). As we shall try to point out at the end of the paper, the issues posed here seem to require a much more adequate philosophical analysis than they have yet received. We hope also to show, in an incidental way, that the developments in modal logic, beginning with C. I. Lewis 13, have a direct bearing on our topic. Since his pioneering work, dozens of systems have been developed, and within the past few years a satisfactory uniform treatment has been effected, largely due to the efforts of Kripke 10 and Hintikka (some unpublished work). We shall not be concerned to discuss formal questions

Conditional permission in deontic logic

ConclusionIn the face of the considerations set forth above, I submit that Andersons proposed definition (D) does not represent an acceptable construction of the concept of conditional permission. This, in turn, suggests that conditional permission must be viewed as a viable deontic relationship in its own right, and is not definable in terms of unconditional deontic concepts. It appears, then, that a reduction of conditional to unconditional deontic logic is not warranted.

Modal Logics II: Toward a Formal Analysis of Cultural Objects

In this essay we hope to make some progress toward an explication or rational reconstruction of the concept culture, which has been of interest to philosophers at least since the time of Hegel 8, and also to social scientists since the 1870’s, when Tylor 25 explicitly introduced the concept as an analytic category. Our aim is not only to clarify the concept, but also to indicate some unfortunate effects of philosophical naivete on the part of some social scientists, who have in fact been embroiled in problems calling for careful philosophical analysis, without recognizing the importance of philosophical techniques. We hope also to show the relevance of our analysis to some traditional philosophical problems, such as the realist-nominalist ontological debate, and the continuing critique of Aristotelian essentialism. And we remark finally that the problem is by no means artificial: we feel that the issues we hope to clarify have been genuinely perplexing both to philosophers and to social scientists.

INDEPENDENT AXIOM SCHEMATA FOR VON WRIGHT'S M

In this paper we show how a modification of results due to Simons ([6]) yields a set of independent axiom schemata for von Wrights M ([8], p. 85), with a single primitive rule of inference. We first describe a system M *, then show its equivalence with M , and finally show that our schemata are independent. 1. Axiomatization of M *. We adopt the notational conventions of McKinsey and Tarski ([4], p. 2), as amended by Simons ([6], p. 309), except that we take “(α ⊰ β)” as an abbreviation for “˜◇˜(α 0→ β)”, rather than for “˜◇(α ∧ ˜β)”. Our only rule of inference is: Rule. If ⊦ α and ⊦ (˜◇˜α → β). then ⊦ β. We have six axiom schemata: We require a number of theorems for the proof of equivalence of M * with M . Theorem 1. If ⊦ α and ⊦ (α → β), then ⊦ β. Theorem 2. If ⊦ α and ⊦ (α ⊰ β), then ⊦ β. Proof by hypothesis, A 5, and Theorem 2 (twice). Theorem 3. If ⊦ (α ⊰ β), then (˜◇β → ˜ ◇α). Proof by hypothesis, A6, and Theorem 2. Theorem 4. If ⊦(α ⊰ β), then ⊦[˜˜(γ ∧ α) ⊰ ˜˜(β ∧ γ)].