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Computación y Sistemas ISSN: 1405-5546 [email protected] Instituto Politécnico Nacional México Duží, Marie Structural Isomorphism of Meaning and Synonymy Computación y Sistemas, vol. 18, núm. 3, julio-septiembre, 2014, pp. 439-453 Instituto Politécnico Nacional Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=61532067003 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Structural Isomorphism of Meaning and Synonymy Marie Duží VSB-Technical University Ostrava, Czech Republic [email protected] Abstract. In this paper I am going to deal with the questions like what the meaning of an expression phenomenon of synonymy from the logical point of is and how fine-grained meanings should be. view. In Transparent Intensional Logic (TIL), which is This is a pressing issue, because many my background theory, the sense of an expression is paradoxes and invalid inferences arise from a too an algorithmically structured procedure detailing what coarse-grained analysis of premises. The purpose operations to apply to what procedural constituents to arrive at the object (if any) denoted by the expression. of logic is to differentiate between valid and Such procedures are rigorously defined as TIL invalid arguments so that all valid arguments be constructions. In this new orthodoxy of structured provable and invalid ones rejected. This in turn is meanings and procedural semantics we encounter the closely connected with the granularity of problem of the granularity of procedure individuation. individuation of meaning. Though the identity of TIL constructions is rigorously Possible-world semantics (PWS) that arose defined, they are a bit too fine-grained from the around 1960 models meanings as mappings procedural point of view. In an effort to solve the defined on a domain of logical possibilities problem we introduced the notion of procedural isomorphism. Any two terms or expressions whose (‘possible worlds’). Possible-world semantics respective meanings are procedurally isomorphic are gained respectability and proved to be highly deemed semantically indistinguishable, hence deployable in various areas, in particular, in synonymous and thus substitutable in any context, modal logic and its variants like temporal and whether extensional, intensional or hyperintensional. deontic logics, by solving the granularity issue. The novel contribution of this paper is a formally Co-intensionality is tantamount to necessary co- worked-out, philosophically motivated criterion of extensionality. Hence, mappings that take the hyperintensional individuation, which is defined in terms same worlds to the same extensions come out of a slightly more carefully formulated version of - identical. Formally, let the variables f, g range conversion and -conversion by value, which amounts over intensions and w over possible worlds: to a modification of Church’s Alternative (A1). Keywords. Procedural semantics, -conversion by fg (w (f(w) = g(w)) f = g) value, procedural isomorphism, transparent intensional logic, synonymy. The basic thing to understand about this individuation of possible-world intensions is that it offers an extensional account of intensional 1 Introduction entities. If A, B are sets of possible worlds then if A, B share exactly the same elements then A is The phenomenon of synonymy has been of a identical to B. This principle of individuation is central interest for both linguists and logicians, generalized to properties, relations-in-intension, and though it is an important theoretical relation etc. The possible-world semantics for modal logic existing in language, a satisfactory criterion of is firmly and comfortably immersed in set theory. synonymy is still a hot issue. A seemingly simple Once necessary equivalence between intensions definition of synonymy as the identity of meaning has been established, lots of valid inferences can evokes many problems including, inter alia, be drawn, and many invalid inferences are blocked. Various mathematical properties, or their Computación y Sistemas Vol. 18, No. 3, 2014 pp. 439–453 ISSN 1405-5546 DOI: 10.13053/CyS-18-3-2018 440 Marie Duží absence, of various formal systems of intensions substituends are hyperintensions. Indeed, any can also be established, like soundness and hyperintensional logic and formal semantics worth completeness. its name must be able to block various invalid So far so good; however, already in 1947 [3, inferences. But there is the other side of the coin, §§13ff] Carnap pointed out that there are which is the positive topic of which inferences attitudinal contexts that are neither extensional should be validated. That is, how hyper are nor intensional, because the substitution of hyperintensions? If there is one central question logically equivalent expressions for the permeating hyperintensional logic and semantics complement of an attitude fails here. For instance, then that is this one. one can easily believe that it is false that A The problem how fine-grained ‘intensional implies B without believing that A is true and B entities’ hence meanings should be was of the false. Yet the truth-conditions of these two utmost importance to Church who considered 1 embedded clauses are identical, hence they are several alternatives of constraining these entities. indiscernible in possible-world semantics. Senses are identical if the respective expressions are (A0) ‘synonymously isomorphic’, (A1) Moreover, possible-world semantics is out of 2 place in case of mathematics; all true mutually -convertible, (A2) logically equivalent. mathematical statements are necessarily (A2), the weakest criterion, was refuted already equivalent, hence identical from the PWS point of by Carnap, and was not acceptable to Church as view. For instance, the set of equilateral triangles well. The alternative (A0) arose from Church’s is identical with the set of equiangular triangles, criticism of Carnap’s notion of intensional but the property of being equilateral is not isomorphism, and it is synonymy resting on - identical with the property of being equiangular. equivalence and meaning postulates for Drawing an equilateral triangle is a task different semantically simple terms. from drawing an equiangular triangle, and one (A1) is presumably considered to be the right can be able to do the former without being able to criterion of synonymy. Yet it was subjected to a do the latter. fair amount of criticism in particular due to the Ludwig & Ray said: involvement of -reduction. For instance, Salmon "In general, one term can be substituted for in [17] adduces examples of expressions that another in 'that'-clauses salva veritate only if they should not be taken as synonymous yet their are synonymous. Perhaps the most popular meanings are mutually -convertible. Moreover, solution to the problem of providing a partiality throws a spanner in the works; - compositional semantics for natural languages reduction is not guaranteed to be an equivalent aims to exploit this fact by treating 'that'-clauses transformation as soon as partial functions are as referring to intensional entities – entities (at involved. Though Church’s formal apparatus in least) as finely individuated as the meanings which meanings are rendered is the typed - of sentences." calculus of total functions, we cannot avoid the Thus since the late 60s, the issue of work with partial functions, because there are structured, hyperintensional meanings has been meaningful terms like the ‘King of France’ that studied. The topic of hyperintensionality was born lack a reference in the actual possible world and out of negativity, as it were. As mentioned above, time. Or, in mathematics there are terms like ‘the Carnap noticed that there are attitudinal contexts that are neither extensional nor intensional, 1 The concept of hyperintensionality is not exactly new; the because the substitution of logically equivalent terms ‘hyperintensionality’ and ‘fine-grained intensionality’ expressions fails here. Cresswell defines any are. The standard term for this concept was simply individuation as hyperintensional that is finer ‘intensionality’ (maybe coined by Leibniz), and is still in use. than logical/necessary/strict equivalence. However, possible-world semantics has used the terms Hyperintensionality became originally a matter ‘intensionality’, ‘intensional logic’, etc., for its own concept of blocking unwanted and unwarranted of intensionality. inferences, by pointing out that the correct 2 For details see [1] and [4]. Computación y Sistemas Vol. 18, No. 3, 2014 pp. 439–453 ISSN 1405-5546 DOI: 10.13053/CyS-18-3-2018 Structural Isomorphism of Meaning and Synonymy 441 greatest prime’ that lack a reference regardless of extensional context they are values of the possible worlds and times, they do not denote functions denoted by the respective expression; in anything. Yet they have meaning. an intensional context they are the denoted Church also considered Alternative (A1’) that functions themselves; finally in a hyperintensional is (A1) plus -convertibility. Yet similar defects of context they are the displayed procedural -convertibility as those connected with - meanings themselves. Due to its stratified convertibility are evincible. ontology of entities organized in a ramified Thus the problem