A structure which generalizes formulas by including substitution terms is used to represent proofs in classical logic. These structures, called expansion trees, can be most easily understood as describing a tautologous substitution instance of a theorem. They also provide a computationally useful representation of classical proofs as first-class values. As values they are compact and can easily be manipulated and transformed. For example, we present an explicit transformations between expansion tree proofs and cut-free sequential proofs. A theorem prover which represents (...) proofs using expansion trees can use this transformation to present its proofs in more human-readable form. Also a very simple computation on expansion trees can transform them into Craig-style linear reasoning and into interpolants when they exist. We have chosen a sublogic of the Simple Theory of Types for our classical logic because it elegantly represents substitutions at all finite types through the use of the typed -calculus. Since all the proof-theoretic results we shall study depend heavily on properties of substitutions, using this logic has allowed us to strengthen and extend prior results: we are able to prove a strengthen form of the firstorder interpolation theorem as well as provide a correct description of Skolem functions and the Herbrand Universe. The latter are not straightforward generalization of their first-order definitions. (shrink)

(1976). PROBABILITY, LIKELIHOOD AND SUPPORT: A METAMATHEMATICAL APPROACH TO A SYSTEM OF AXIOMS FOR UPPER AND LOWER DEGREES OF BELIEF. Philosophical Papers: Vol. 5, No. 2, pp. 153-161.

In order that a degree-of-belief function be coherent it is necessary and sufficient that it satisfy the axioms of probability theory. This theorem relies heavily for its proof on the two-valued sentential calculus, which emerges as a limiting case of a continuous scale of truth-values. In this "continuum of certainty" a theorem analogous to that instanced above is proved.

In an article “Countering a Counter-intuitive Probability” [4], Lynn E. Rose discusses a question in conditional probability, claiming that the following problem posed by Copi [1] is usually incorrectly solved:Remove all cards except aces and kings from a deck, so that only eight cards remain, of which four are aces and four are kings. From this abbreviated deck, deal two cards to a friend. If he looks at his cards and announces that his hand contains an ace, what is the (...) probability that both his cards are aces? If he announces instead that one of his cards is the ace of spades, what is the probability then that both his cards are aces? (shrink)

Book reviewed in this article: Patient Encounters: The Experience of Disease. By James H. Buchanan. Becoming a Good Doctor. By James F. Drane.

A solution to the sorites paradox is obtained by distinguishing three formats of the sorites argument and appraising them in the light of four fundamental considerations: (i) the appropriate notion of truth for the application of vague predicates to their borderline cases, (ii) a certain construal of borderline cases, (iii) a certain freedom of use of vague terms not enjoyed by non-Vague terms and (iv) the revocation of that freedom by deductive contexts.

Interest in the Keynesian concept of evidential weight has led to divergent views concerning the burden of proof in adjudication. It is argued that Keynes's concept is properly engaged only in the context of one special kind of decision, the decision whether or not the evidence is ripe for a decision on the underlying merits, whether the latter decision is based on probability, relative plausibility, coherence or otherwise. As a general matter, this question of ripeness is appropriately assigned to the (...) judiciary for resolution as part of the burden of production, rather than to the jury or other factfinder as part of the burden of persuasion. (shrink)

Critical Legal Studies poses a direct and expressed challenge to the basic tenets of American legal education and scholarship. Critical Legal Studies postulates that law is not a scientific exercise involving the application of objective principles, but rather a creative process involving the selection of conflicting rules which has the effect of reinforcing the existing political order. In an effort to explain the contribution of Critical Legal Studies to argumentation theory, this essay briefly discusses the role of legal reasoning in (...) the American legal system, describes and critiques Legal Positivism, lays the intellectual foundation for Critical Legal Studies, and considers the implications that this conception of jurisprudence has for argumentation theory. (shrink)

In classical Greek poetry there is a familiar distinction between verse which repeats line upon line, and that which forms patterns liable to closure at intervals, in stanzas or lyric sections. This is often equated with the distinction between spoken and sung verse, but the equation is only approximate. At an earlier stage all verse had some musical accompaniment—so much can be deduced from a number of passages in Homer, and is in any case implicit in the nature of quantitative (...) verse. By Hellenistic times it seems that, in new composition, the more complex lyric structures had died out, while of the simpler ones more and more were being taken over as spoken verse. In the fifth century—and this can probably be extended to at least the late sixth and most of the fourth—only two kinds of verse are known to have dispensed with music altogether in performance: the heroic hexameter and the iambic trimeter. (shrink)

What kind of Theory of Music and Theory of Metric was taught to the young Pindar or the young Sophocles? So far are we from an answer to this question that we do not even know how far extra study was necessary, or usual, for the professional poet as compared with the ordinary educated Greek citizen. The interdependence of music and metric in lyric poetry gave complexity to the word-rhythms but kept the study of music, the subordinate partner, theoretically simple. (...) Doubtless by the time the young poet had learnt by heart the words of past masters of lyric verse, sung or accompanied them on the lyre, and danced them in choir or κ⋯μος, he had absorbed by practice and somewhat rule-of-thumb methods of training a great deal of the τ⋯χενη which his calling would require. He would have to learn, in common with the χοροδιδ⋯σκαλος, how to read and write a simple score, and so he must know the symbols of pitch in such scales as were then in use. The notation of arsis and thesis may have been required too, and if we knew that this was so, and how he was taught to apply it, we might be in a much better position to assess the kind of theory he learnt. But of one thing we may be sure: genius apart, it was his practical training, as performer and spectator, rather than theoretical teaching which gave his ear its cunning and technical skill to compose new rhythms. (shrink)

In order that a degree-of-belief function be coherent it is necessary and sufficient that it satisfy the axioms of probability theory. This theorem relies heavily for its proof on the two-valued sentential calculus, which emerges as a limiting case of a continuous scale of truth-values. In this "continuum of certainty" a theorem analogous to that instanced above is proved.

Text, interpretation and metre present a tangled problem in this threnody, and the solutions of editors differ widely. The chief function of detailed metrical study in such corrupt passages of lyric is to weight the scales in favour of—or more often against—certain methods of handling the text. The positive results of this present attempt to apply metrical criteria are necessarily modest and tentative; negatively they are, I think, sometimes decisive.

The relationship between vague statements and fuzzy sets is examined. It is shown that the probability of vague statements may be defined in a manner analogous to that discussed in Reichenbach's logic of weight.

Dactylo-Epitrite has established itself in a privileged position among choral lyric metres, since the greater volume of material and the greater regularity of its component units have encouraged a more careful study, and closely reasoned controversies have arisen and been resolved over its structure. Other kinds of Pindaric metre have for the most part been analysed in a hit-or-miss spirit, and arbitrary schemes have been produced of which rarely are two alike for the same poem, yet little attempt has been (...) made to base any preference upon a theory. (shrink)

I Am not proposing in this essay to treat at length and in detail of the metric of other lyric poets. In most cases questions of metre are intimately involved with questions of text, into which so many other considerations enter that in dealing with them proportion would be lost, while metrical analysis of such material would still remain largely speculative. What follows is therefore little more than a general account of the principles of composition which these poets appear to (...) me to follow; I have, however, indicated a few passages where emendations made purely on metrical grounds and widely accepted seem to me, in the absence of responsion, based on too ready a desire to reduce all poems to the best-known formulas of versification. (shrink)

In classical Greek poetry there is a familiar distinction between verse which repeats line upon line, and that which forms patterns liable to closure at intervals, in stanzas or lyric sections. This is often equated with the distinction between spoken and sung verse, but the equation is only approximate. At an earlier stage all verse had some musical accompaniment—so much can be deduced from a number of passages in Homer, and is in any case implicit in the nature of quantitative (...) verse. By Hellenistic times it seems that, in new composition, the more complex lyric structures had died out, while of the simpler ones more and more were being taken over as spoken verse. In the fifth century—and this can probably be extended to at least the late sixth and most of the fourth—only two kinds of verse are known to have dispensed with music altogether in performance: the heroic hexameter and the iambic trimeter. (shrink)