# 6 Information-theoretic logic6信息论逻辑

Corcoran, J. 1998. Information-theoretic logic, in Truth in Perspective edited by C. Martínez, U. Rivas, L. Villegas-Forero, Ashgate Publishing Limited, Aldershot, England (1998) 113-135. 6 Information-theoretic logic John Corcoran In memory of Alfred Tarski 1901-1983 and Alonzo Church 1903-1995 on the fortieth anniversary of their classic works: Logic, Semantics, Metamathematics and Introduction to Mathematical Logic. 6.1. Introduction Information-theoretic approaches [114] to formal logic analyse the “common intuitive“ concept of prepositional implication (or argumental validity) in terms of information content of propositions and sets of propositions: one given proposition implies a second if the former contains all of the information contained by the latter; an argument is valid if the conclusion contains no information beyond that of the premise-set. This paper locates information-theoretic approaches historically, philosophically and pragmatically. Advantages and disadvantages are identified by examining such approaches in themselves and by contrasting them with standard transformation-theoretic approaches. Transformation-theoretic approaches analyse validity (and thus implication) in terms of transformations that map one argument onto another: a given argument is valid if no transformation carries it onto an argument with all true premises and false conclusion. Model-theoretic, set-theoretic, and substitution-theoretic approaches, which dominate current literature, can be construed as transformation-theoretic, as can the so-called possible-worlds approaches. Ontic and epistemic presuppositions of both types of approaches are considered. Attention is given to the question of whether our historically cumulative experience applying logic is better explained from a purely information- theoretic perspective or from a purely transformation-theoretic perspective or whether apparent conflicts between the two types of approaches need to be reconciled in order to forge a new type of approach that recognizes their basic complementarity. 6.2. Preliminaries The information-theoretic viewpoint dominated logic in the period during which the seeds of mathematical logic were being sown by Boole, De Morgan, Jevons, Venn and others. In fact, the writings of the logicians who succeeded and worked in the shadow of Boole and De Morgan show almost no trace of awareness of any other viewpoint. It is worthwhile to review some of the relevant passages in order to identify our topic and in order to confirm the pervasiveness of this mode of thought. The two . logical inferences . from the original [set of] propositions . give us all the information which it contains respecting the class .(George Boole, 1847, p.75). . it is the office of a conclusion not to present us new truth, but only to bring into explicit form some portion of that truth which was implicitly involved in the premises . [some portion of] the particular information conveyed in the premises . (George Boole 1856?, p. 239). Every collective set of premises contains all its valid conclusions; . speaking objectively, the assumption of them [the premises] is the assumption of the conclusion; though, ideally speaking, the presence of the premises in the mind is not necessarily the presence of the conclusion (Augustus De Morgan, 1847, p. 254). All the propositions of pure geometry, which multiply so fast that only a small .class . among mathematicians . know all that has been done . , are certainly contained in a very few notions . .[The] consequences are virtually contained in the premises (Augustus De Morgan, 1847, p. 45). The very purpose of syllogism is to deduce a conclusion which will be true when the premises are true. The syllogism enables us to restate in a new form the information . contained in the premises, just as a machine may deliver to us in a new form the material . put into it (W. Stanley Jevons, 1870, p. 149). We extract out of the premises all the information . useful for the purpose in view-and this is the whole which reasoning accomplishes (W. Stanley Jevons 1870,p.15). [To deduce is] . to draw . propositions as will necessarily be true when the premises are true. By deduction we investigate and unfold the information contained in the premises . (W. Stanley Jevons, 1879, p. 49). These . [consequences] . contain every particle of information yielded by the original [premise] . , or in any way deducible from it (John Venn, 1881, p. 296). That is, [in making this inference] we have had to let slip a part of the information contained in the data (John Venn, 1881, p. 362). [115] . logicians in overwhelming majority maintain that every conclusion is implicitly contained in the premises (John Venn, 1889, p. 42). Information-theoretic approaches to logic may be characterized to some extent by six remarks. All information and all propositions mentioned in these remarks are assumed to pertain to some one, limited, and coherent “domain of investigation“ established in advance and remaining fixed throughout (Corcoran, 1995, §4.3). The purpose of this assumption is to limit the scope of the inquiry in order to avoid incoherent pseudo- questions, in order to circumvent extraneous issues, and in order to put a bound on “some“ and “all“ as applied to information and propositions. Thus, in particular, every proposition is to be elliptical for every proposition pertaining to the domain of investigation and all information is to be elliptical for all pertinent information . In view of the role of domains of investigation it is natural to refer to them also as “informational domains“. First, a given proposition follows from, is a consequence of, a given postulate set if all of the information contained in the proposition is contained within the set. Second, a given proposition is independent of, not a consequence of, a given postulate set if the proposition contains any information outside of the information content of the set. Third, a proposition is tautological if it is devoid of information; accordingly a sentence that expresses a tautology conveys no information. A tautology is thus implied by every pertinent proposition, and is thus useless as a postulate, whether postulate sets are intended as presentations of given information or whether they are intended as characterizations of a given subject matter. Fourth, a proposition is contradictory if it contains all information (pertaining to the domain of investigation); accordingly, a sentence that expresses a contradiction conveys all such information. A contradiction thus implies every pertinent proposition and is thus useless as a postulate if postulate sets are intended as characterizations of a given subject matter. No subject matter is accurately described or characterized by a contradictory proposition. Fifth, no proposition has any information in common with its own negation, although a proposition and its negation need not (but may) divide all pertinent information between them. Sixth, the disjunction of one given proposition with a second contains exactly the information that the first has in common with the second, i.e. the information that the two share. As usual two propositions that imply each other are said to be logically equivalent (to each other). “No prime number exceeding two is even“ is logically equivalent to “No even number exceeding two is prime“ and to “No number exceeding two is both even and prime“. The first of the above remarks characteristic of information-theoretic approaches to logic entails [116] that two propositions are logically equivalent if and only if they contain exactly the same information. In particular, every two tautologies, each being devoid of information, are logically equivalent and every two contradictions, each containing all pertinent information, are logically equivalent. For example, “Zero is zero“ is logically equivalent to “One is one“ and to “Every even number is either even or prime“; similarly “Zero isn t zero“ is logically equivalent to “One isn t one“ and to “Some even number is neither even nor prime“. Having the same (information) content neither entails nor precludes having the same (logical) form (Corcoran, 1989, p. 27-31 and Cohen and Nagel, 1993, pp. XXI, XXXI-XXXVII). Conversely, having the same form neither entails nor precludes having the same content: “Zero is odd“ has the same form as “Zero is even“; “Some prime number is odd“ has the same form as “Some prime number is even“. Every proposition has a unique logical form and a unique information content; but a logical form per se does not have content and an information content per se does not have form. One might say that it is the amorphous character, the formlessness, of information content that enables it to take on various forms (Cohen and Nagel, 1993, pp. XXV-XXIX). The amorphous character of every information content, in and of itself, neither entails nor precludes a kind of discreteness or atomicity. An earlier paper (Corcoran, 1995) pointed out the existence of informational atoms in one of the most important and best known informational domains, that of the 1931 Godel incompleteness paper. A proposition is an informational atom if it is informative (i.e. non-tautological) but it implies no weaker informative propositions. In other words, a proposition is an informational atom (of a given informational domain) if it is informative but there is no way to drop information from it without rendering it devoid of information. Besides tautologies, an informational atom implies only its own logical equivalents. The negation of the conjunction of the 1931 Godel axioms is informationally atomic. As is well known, the Godel axiom set is semantically complete or complete with respect to consequences in the sense of Church (1956, p. 329). This means that the conjunction of the axioms implies every pertinent proposition that it does not contradict. Thus, this conjunction is a consistent proposition to which no pertinent information can be consistently added. Such propositions can be called (informational) saturations. Every informational atom is equivalent to the negation of an informational saturation and every informational saturation is equivalent to the negation of an informational atom. The particulate, or atomic, character of the propositions noted above does not entail “an atomic theory of information“; even though the Godel domain [117] contains infinitely many informational atoms it is not the case that each of its propositions is logically equivalent to a set of informational atoms. In fact the Godel Axiom Set is not equivalent to a set of informational atoms (Corcoran, 1995, p. 75). Information seems to straddle “the continuous“ and “the discrete“, to share some aspects with “magnitudes“ and some with “multitudes“, to have some affinity with the category indicated by “mass nouns“ and some affinity with the category indicated by “count nouns“. It is worth making explicit the fact that information-theoretic approaches to logic extend to propositions as abstract individuals a kind of “hylomorphism“, or matter-and- form analysis, similar to that attributed by traditional Aristotelian ontology to concrete individuals. Just as an individual brass sphere involves brass as its matter and sphericity as its form, an individual arithmetic identity, say “One plus two is three“, involves arithmetic information as its content and the logical form of the identity as its form. Just as the same brass admits of being contained in infinitely many geometrically dissimilar brass objects, as indicated above, the information content of the identity is contained in each of infinitely many formally dissimilar propositions. The hylomorphic analogy that naturally accompanies information-theoretic approaches helps to make logic accessible to beginning students, it helps to make logic more useful to those who apply it, and it helps to make logic more exploitable to researchers. Far from being a crude metaphor, as Morris Cohen once called it (Cohen 1944, p. 194) , the hylomorphic aspect of the information-theoretic viewpoint has pedagogical, practical, and heuristic benefits that can be enjoyed even by persons not ready to accept the viewpoint philosophically. The expressions information content and logical form are far from self explanatory. Both are composed of notoriously ambiguous words and the ranges of senses of these words suitable for information-theoretic logic are severely restricted. In particular, the range of suitable senses of information is limited by the formal properties of information content required by the six characteristic remarks given above. For example, every false proposition is informative. Thus, in the senses required here, “misinformation“ is information, not all information is accurate. Moreover, the information content of a proposition is not to be measured by the number of its non-logical, or content, concepts: fewer non-logical concepts does not necessarily correlate with less information content. The proposition “Every number is inductive“ contains much more information than “Every even number is [118] inductive“. Of course, having the exact same content concepts does not necessarily correlate with having the exact same information content: in fact, “Every perfect number is even“ has no information in common with “Some non-even number is perfect“. In none of the senses of the expression information content suitable for use with information-theoretic logic can a concept be said to have information content: information content belongs exclusively to propositions and sets of propositions. As usual, (written) sentences, which are made up of conventional characters, or symbols, are distinguished from propositions, which are made up of concepts. One and the same proposition may be expressed by different sentences in different languages, or even in the same language. For example, “One plus two is three“, the proposition that one plus two is three, is expressed by each of many sentences: (1+2)=3 , One plus two is three , One and two are three , Uno y dos son ires , and so on. This article follows the increasingly widespread convention of indicating sentences (and other expressions) by single quotes while indicating propositions (and other meanings) by double quotes. A sentence may be said to convey the information contained in a proposition that it expresses. Our primary purpose is to locate information-theoretic conceptions of logic historically, pragmatically, and philosophically. Secondary purposes are to contribute to the dialogue concerning the nature of our “common intuitive“ notion of consequence (or argument validity) and to explore the ontic and epistemic basis of logical judgements both of consequence (or argument validity) and of independence (or argument invalidity). Attention is given to the question of whether our cumulative experience in applying logic is better