Arbitrary Sentences Research Articles

Rejoinder to George

Alexander George ('Reply to Weir on Dummett and Intuitionism', Mind, I987, pp. 404-6) raises two objections to my criticisms of Michael Dummett in part II of my 'Dummett on Meaning and Classical Logic', Mind, I986, pp. 465-77. The second criticism concerns my doubts as to the legitimacy of the intuitionist claim that it is decidable, of an arbitrary construction, whether or not it is an intuitionist proof of an arbitrary sentence. (I was not, as George seems to suggest, concerned with the problem, for a fixed sentence, of the decidability of proofhood for it.) The problem arises out of the intuitionist insistence that our store of axioms and rules of proof is not fixed once and for all but can always be extended by the 'intuiting' of new axioms and methods of proof. As George points out, I was mistaken to focus on the transformations figuring in the account of provability for conditionals and universal generalizations: the intuitionists recognize no restriction on the type of sentence for which new methods of proof may become available. However the problem is especially salient there since it thereby impinges on the intuitionist account of the logical constants: particularly, as Dummett notes (Truth and Other Enigmas, London, Duckworth, I978, bottom of the paragraph p. 242), the conditional, since we explain its meaning in terms of operations on proofs of antecedents and it seems we have no effective means of generating each one of these. George denies that intuitionists are faced with a problem here on the grounds that all acceptable extensions of the proof relation preserve the property of decidability. 'Although we may come to admit more and more constructions as proofs of a given sentence, it will always remain a decidable matter whether a given construction is a proof of that sentence or not' (p. 405; as remarked, he focuses on an overly restrictive form of the problem). My doubts remain. We have to ask whether the addition of new proof methods entails a change in meaning in the sentences affected by the addition. Dummett appears to think it does (cf. Elements of Intuitionism, Oxford, 1977, pp. 401-3). But then the doctrine that the methods of proof for a given language are not fixed is only trivially true; it is true of the sentences considered as uninterpreted formulae, something no one would deny. It is not true of the propositions expressed in the language at a given time. Alternatively, an ever-growing body of proof methods may be applicable to a fixed set of propositions, enabling more and more to be decided. But then it seems plausible to require of one who understands all these propositions that they be able to tell of any putative new method, applied to a sentence expressing one of these propositions, whether it is indeed a proof or not. Plausible, that is, on constructivist assumptions about the relation between meaning and proof. Thus the motivation for requiring the proof relation to be decidable. But in this second case, what matters is not whether it is decidable whether something is currently a proof of a sentence, but whether

A foundational approach to autonomous knowledge acquisition

A formal, foundational approach to autonomous knowledge acquisition is presented. In particular, “learning from examples” and “learning from being told” and the relation of these approaches to first‐order representation systems are investigated. It is assumed initially that the only information available for acquisition is a stream of facts, or ground atomic formulae, describing a domain. On the basis of this information, hypotheses expressed in set‐theoretic terms and concerning the application domain may be proposed. As further instances are received, the hypothesized relations may be modified or discarded, and new relations formed. The intent though is to characterize those hypotheses that may potentially be formed, rather than to specify the subset of the hypotheses that, for whatever reason, should be held.Formal systems are derived by means of which the set of potential hypotheses is precisely specified, and a procedure is derived for restoring the consistency of a set of hypotheses after conflicting evidence is encountered. In addition, this work is extended to where a learning system may be “told” arbitrary sentences concerning a domain. Included in this is an investigation of the relation between acquiring knowledge and reasoning deductively. However, the interaction of these approaches leads to immediate difficulties which likely require informal, pragmatic techniques for their resolution. The overall framework is intended both as a foundation for investigating autonomous approaches to learning and as a basis for the development of such autonomous systems.

A formal system for fuzzy reasoning

A formal system for fuzzy reasoning is described which is capable of dealing rationally with evidence which may be inconsistent and/or involve degrees of belief. The basic idea is that the meaning of each formal sentence should be given by a certain commitment or bet associated with it. Each item of evidence is first expressed in the form of such a (hypothetical) bet, which is then written as a formal sentence in a language related to Łukasiewicz logic. The sentences may be weighted to express the relative reliability of the various informants. A sentence is considered to “follow” from the evidence if the bet it represents can be offered by a speaker without fear of loss, on the assumption that the bets representing various items of evidence have been offered to him. A detailed account, illustrated by concrete examples, is given of the procedures by which an arbitrary sentence in common language can be translated into a formal sentence. The treatment of inconsistency, degrees of belief, and weights is illustrated by a practical example which is solved in full. It is shown that in most practical cases the computations involved in the process of formal reasoning reduce to a problem in linear programming. In the last section the relation between this system and the procedures advocated by Zadeh is examined. It is shown that, subject to certain modifications in formulas, there is general agreement in the region of overlap.

Logical and graph-theoretic characterizations of the self-dual sentences of projective geometry

The self-dual sentences of projective geometry are characterized in terms of their logical structure. This syntactical analysis reveals an unexpected relationship with graph theory, the self-dual sentences being precisely the geometric interpretations (in a reasonably natural sense) of arbitrary sentences of graph theory.

On the lattice of recursively enumerable sets

This paper presents some new theorems concerning recursively enumerable (r.e.) sets. The aim of the paper advance the search a decision procedure the elementary theory of r.e. sets. More precisely, an effective method sought deciding whether or not an arbitrary sentence formulated in the lower predicate calculus with sole relative symbol c true of the r.e. sets. The main achievement of the paper the characterisation of the hh-simple sets as those coinfinite r.e. sets whose r.e. supersets form a Boolean algebra. The reader referred Davis's book [1] basic information about the partial recursive (p.r.) functions and about r.e. sets. Other background material required a proper understanding of the present paper consists of [8], [3, Theorem 2], [10, Introduction and ?4], and [5] where the contributions have been listed in their natural order. We take the formulation of the lower predicate calculus given in Abraham Robinson [9]. Natural numbers are denoted by lower case Roman letters and sets of them by lower case Greek letters. The empty set denoted by 0 and the set of all natural numbers by v. The complement of any set a denoted by a'; a called cofinite or coinfinite just if a' finite or infinite respectively. For sets a, / we write aoc3 just if the set (a -3 u (3 a) finite; otherwise we write aZ 3. By function we mean a map of some subset of v x v x **. x v into v; functions will be denoted by upper case Roman letters as will relations on the natural numbers. The informal logical signs used are v &, ->, (x), (Ex), -,> which are be read as or , and , implies , not, for all x, exists x, is equivalent to respectively. Let ,' be a finite class of propositions; then s otherwise sup a be oo. The plan of the paper as follows. In the first section we give a brief discussion of the elementary theory of r.e. sets and prove that its decision problem of the same degree as that of the elementary theory of the lattice obtained by taking the equivalence classes of r.e. sets with respect -. In ?2 we prove the main theorem which states: if a an r.e. subset of an r.e. set / then either there exists a recursive subset 8 of:/ such that a u 8 = 3 or there exists a recursive sequence {8il of disjoint finite subsets of / such that 8i a nonempty all i. This theorem was inspired by