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
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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 of the proper granularity of hierarchy of types, TIL is a logical framework structured meanings remains open. This is a within which such an extensional logic of 5 pressing issue, because in hyperintensional hyperintensions has been introduced. contexts only strictly synonymous expressions In an effort to solve the problem of the can be mutually substituted. We encounter the procedural identity we introduced the notion of problem of hyperintensional contexts in particular procedural isomorphism. Procedural isomorphism in sentences expressing propositional attitudes is a nod to Carnap’s intensional isomorphism and like believing, knowing, etc., and objectual Church’s synonymous isomorphism. Any two attitudes of seeking, wishing, solving, designing, terms or expressions whose respective meanings calculating, and the like. Substitution of (merely) are procedurally isomorphic are deemed equivalent expressions yields the proliferation of semantically indistinguishable, hence agent’s knowledge or abilities here, and the synonymous and substitutable in any context. problem of logical/mathematical omniscience The goal of this paper is to define the rule for crops up. substitution of identicals in hyperintensional In my background theory which is Tichý’s contexts. Since in such contexts only Transparent Intensional Logic (TIL), synonymous expressions can be mutually hyperintensionality is defined in a positive rather substituted, we need a criterion of synonymy. The than negative way.3 Any context in which the novel contribution of this paper is the proposal of meaning of an expression is displayed rather than a new criterion of synonymy. It is an adjustment of executed is hyperintensional. Moreover, we Church’s Alternative (A1). The adjustment explicate hyperintensions as abstract procedures consists in a more carefully formulated definition rigorously defined as TIL constructions which are of -conversion and the new definition of - assigned to expressions as their context-invariant conversion, to wit, the -conversion by value. meanings.4 The semantics is tailored to the The rest of the paper is organized as follows. hardest hyperintensional contexts, and Section 2 sets out the logical foundations of TIL. generalized from there to simpler intensional and The main results are introduced in Section 3 extensional ones. This entirely anti-contextual and where the relation of procedural isomorphism, compositional semantics is, to the best of my Alternative (A1’’), is defined. I prove that the so knowledge, the only one that deals with all kinds defined relation is a strict equivalence on the set of context, whether extensional, intensional or of constructions. Concluding remarks are in hyperintensional, in a uniform way. The same Section 4. extensional logical laws are valid invariably in all kinds of context. In particular, there is no reason why Leibniz’s law of substitution of identicals, and 2 Logical Foundations of TIL the rule of existential generalization were not valid. What differ according to the context are not The syntax of TIL is Church’s (higher-order) typed the rules themselves but the types of objects to -calculus the terms of which are procedurally which these rules are applicable. In an interpreted, which means that they denote structured modes of presentation (that is, TIL constructions) of functions rather than set- 3 See [12, §§ 2.6, 2.7], and also [11]. 4 Similar conception of meaning has been propagated also by Moschovakis. For details see [15]. 5 See, for instance, [6], [7], [8], [9].
Computación y Sistemas Vol. 18, No. 3, 2014 pp. 439–453 ISSN 1405-5546 DOI: 10.13053/CyS-18-3-2018 442 Marie Duží theoretic functions. Thus, lambda abstraction For the purposes of natural language analysis, transforms into the molecular procedure of we are assuming the following base of forming a function and application into the ground types: molecular procedure of applying a function to an argument. The identity of TIL constructions is ο: a set of truth-values {T, F}; rigorously determined by a definition. Yet we still ι: a set of individuals (constant universe of have to deal with the issue of granularity of discourse); meanings/procedures, because TIL constructions τ: a set of real numbers (doubling as temporal are a bit too fine-grained from the procedural continuum); point of view. The main issue here is the ω: a set of logically possible worlds (logical following. Constructions that differ at most by space). using different -bound variables of the same type differ so slightly that from the semantic point of Within this type system we define possible- view they should be treated as identical world intensions and extensions. An -intension is procedures, because in natural languages we do a function of type (ω), or frequently (()), the not express -bound variables and thus do not types that we abbreviate as ; an -extension is reflect such differences. an object of type , where (ω) for any . Constructions are the key entities of TIL. They Definition 2 (construction). are algorithmically structured procedures, of one (i) Variable x is a construction that v-constructs or multiple constituent parts and they serve to an object that a valuation v assigns to x. explicate linguistic meanings. Importantly, the (ii) The Trivialization 0X is a construction. Let X constituent parts of a construction C are its be any object whatsoever. Then 0X executed subconstructions rather than the constructs X without any change of X. product (if any) of C which is located beyond C. (iii) Composition [X Y …Y ] is a construction. Just to be clear, constructions are not set- 1 m theoretic mapping/functions, nor are they Let X v-construct a function f of type (α formulae or otherwise linguistic entities. Their β1…βm), and let Y1,…,Ym v-construct entities inductive definition below enumerates six different B1,…,Bm of types β1,…,βm, respectively. constructions, which is the logical core of TIL. The Then [X Y1…Ym] v-constructs the value (an stratified ontology of TIL is organized in the entity, if any, of type α) of f on the tuple ramified hierarchy of types. For the sake of argument B1,…,Bm. Otherwise the simplicity we first define simple types of order 1, Composition [X Y1…Ym] does not v- then constructions together with the types of their construct anything and so is v-improper. products, and finally the ramified hierarchy (iv) λ-Closure [λx1 … xm Y] (or Closure) is a of types. construction. It v-constructs the following function f of type (α β1…βm). Let variables x ,…x v-construct entities of types β ,…, Definition 1 (types of order 1). Let B be a base, 1 m 1 where a base is a collection of pair-wise disjoint, βm, and let Y v-construct an entity of type α. non-empty sets. Then Let v(B1/x1,…,Bm/xm) be a valuation identical with v at least up to assigning objects B1/β1, (i) every member of B is an elementary type of …, Bm/βm to variables x1, …, xm. If Y is order 1 over B; v(B1/x1,…,Bm/xm)-improper (see iii), then f is undefined on B1, …, Bm. Otherwise the (ii) let α, β1, ..., βm (m > 0) be types of order 1 value of f on B1, …, Bm is the α-entity over B. Then the collection (α β1 ... βm) of all v(B1/x1,…,Bm/xm)-constructed by Y. m-ary partial mappings from β1 ... βm 1 into α is a functional type of order 1 over B; (v) Single Execution X is a construction. Let X v-construct object o. Then 1X v-constructs o. (iii) nothing else is a type of order 1 over B. Let X be either a non-construction or a v-
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 443
1 improper construction. Then X is v- unless it follows from Cn (i)-(iv). improper. 2 T (types of order n+1) Let be a collection (vi) Double Execution X is a construction. Let X n+1 n of all constructions of order n over B. Then v-construct a construction Y and let Y v-
construct an object Z (possibly itself a i) n and every type of order n are types of 2 construction). Then the Double Execution X order n+1 over B; 2 v-constructs Z. Otherwise X is v-improper. ii) if m > 0 and , 1,...,m are types of order (vii) Nothing else is a construction. n+1 over B, then ( 1 ... m) (see T1 ii)) is Here are some explicative remarks. A variable a type of order n + 1 over B; constructs an object by having that object as its iii) nothing else is a type of order n + 1 over B. value dependent on a valuation function v arranging variables and objects in a sequence. Logical objects like truth-functions and Trivialization is TIL’s objectual counterpart of a quantifiers are extensional: (conjunction), non-descriptive constant term, which simply (disjunction) and (implication) are of type (), provides a particular object. Variables and and (negation) of type (). The quantifiers , Trivializations are the atomic constructions of TIL, are type-theoretically polymorphous, total Composition, Closure and Executions are the functions of type (()), for an arbitrary type , molecular constructions. An atomic construction is defined as follows. The universal quantifier is a a structured whole with but one constituent part, function that associates a class A of -elements namely, the construction itself. A molecular with T if A contains all elements of the type , construction is a structured whole with more parts otherwise with F. The existential quantifier is a than just itself. Importantly, the only part of 0X is function that associates a class A of -elements 0X and not X, which is located beyond 0X: the with T if A is a non-empty class, otherwise with F. product of a procedure is no part of Below all type indications will be provided outside the procedure. the formulae in order not to clutter the notation. The definition of the typed universe of TIL Furthermore, ‘X/’ means that an object X is (a amounts to a definition of the ramified hierarchy of member) of type . ‘X ’ means that the type types which is divided into three parts: firstly, v simple types of order 1, which were already of the object valuation-constructed by X is . This defined by Definition 1; secondly, constructions of holds throughout: the variables w v and t v order n; thirdly, types of order n+1. . If C v then the frequently used Composition [[Cw] t], which is the intensional Definition 3 (ramified hierarchy of types). descent (a.k.a. extensionalization) of the - T1 (types of order 1). See Def. 1. intension v-constructed by C, will be encoded as Cn (constructions of order n). ‘Cwt’. When using constructions of truth-functions, i) Let x be a variable ranging over a type of we often omit Trivialization and use infix notation order n over B. Then x is a construction of to conform to standard notation in the interest of order n over B. better readability. Also when using constructions ii) Let X be a member of a type of order n of identities of -entities, =/(), we omit over B. Then 0X, 1X, 2X are constructions Trivialization, the type subscript, and use infix of order n over B. notion when no confusion can arise. Moreover, the outermost brackets of Closure will be iii) Let X, X1,..., Xm (m > 0) be constructions of occasionally omitted. order n over B. Then [X X1... Xm] is a construction of order n over B. Definition 4 (subconstruction). Let C be a iv) Let x1,...xm, X (m > 0) be constructions of construction. Then order n over B. Then [x1...xm X] is a i) C is a subconstruction of C; construction of order n over B. ii) if C is 0X, 1X or 2X and X is a construction v) Nothing is a construction of order n over B then X is a subconstruction of C;
Computación y Sistemas Vol. 18, No. 3, 2014 pp. 439–453 ISSN 1405-5546 DOI: 10.13053/CyS-18-3-2018 444 Marie Duží iii) if C is [X X1…Xn] then X, X1, …, Xn are intensional context: construction C occurs subconstructions of C; executed in order to produce a function f but iv) if C is [x1…xn Y] then Y is a subconstruction not the value of f; moreover, the executed of C; construction C does not occur within another v) if A is a subconstruction of B and B is a hyperintensional context. (Hence the entire subconstruction of C then A is a function v-constructed by C serves as a func- subconstruction of C; tional argument to be operated on in D); vi) a construction is a subconstruction of C only extensional context: construction C occurs if it so follows from (i) – (v). executed in order to produce a particular There are two modes in which a value of the function v-constructed by C; subconstruction D of a construction C may occur, moreover, the executed construction does not to wit, displayed and executed. If the latter, then occur within another intensional or we say that D is a constituent of C. The hyperintensional context. (Hence the value of Trivialization 0C of a construction C displays the the function v-constructed by C serves as a construction C and all the subconstructions of C. 0 functional argument to be operated on in D). Hence C is not a constituent part of C; it is not executed, and so does not obtain an object Higher context is dominant over a lower one. It beyond it. We say that C occurs means that all the subconstructions of a displayed hyperintensionally. It is however important to construction occur hyperintensionally as well, and realize that Double Execution executes a all the subconstructions of an executed construction twice over. Thus in 20C the 20 construction that occurs intensionally occur subconstruction C is a constituent part of C. intensionally as well. If C is an executed constituent of D then C can Example. Below I analyze three sentences in occur intensionally or extensionally. In principle, which the meaning of the term ‘temperature in constituent C occurs in D intensionally if C is not Prague’ denoting a magnitude of type occurs composed with a construction of the argument of extensionally, intensionally and hyper- the function f v-constructed by C. Hence the intensionally, respectively. Note that there is no whole function f is an object of predication within reference shift and the meaning and denotation of D. As a limiting case, a constituent C that this term is the same in all three types of context. constructs an atomic entity, which is a 0-ary It is the construction of the denoted magnitude, to function without arguments, occurs intensionally. wit, the Closure On the other hand, a constituent C of D occurs extensionally if it is composed with a construction 0 0 wt [ Temperature_inwt Prague] . of an argument of the function f, and C does not occur within an intensional context of D. Hence a) Extensional context: the value (if any) of the function f is an object of 0 predication within D. “The temperature in Prague is 30 Celsius.” These three ways in which a subconstruction The types of the objects that the sentence talks C of a construction D can occur give rise to three 6 about are these: Temperature_in/(): attribute; kinds of context within D: 0 0 Prague/; wt [ Temperature_inwt Prague] hyperintensional context: a construction C : magnitude; 30C/. occurs displayed and serves itself as a func- The whole sentence denotes a proposition of tional argument to be operated on in D type constructed by this Closure: (though a construction one order higher 0 0 0 needs to be executed in order to produce the wt [wt [ Temperature_inwt Prague]wt = 30C]. displayed construction); In the Composition 6 Here I present just a summary. For exact definitions see [12, 0 0 0 [wt [ Temperature_inwt Prague]wt = 30C], §2.6].
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 445
0 0 0 0 the construction wt [ Temperature_inwt Prague] Hence wt [ Temperature_inwt Prague] of the magnitude occurs extensionally, because occurs hyperintensionally in the Composition the value of this magnitude in a given world w at 0 0 time t of evaluation is constructed, and this value [ Believewt Tilman 0 0 0 0 is an object of predication, to wit, this value [ w t [ Temperature_inwt Prague]wt = 30C]]. equals 30C. Within this Composition the 0 This is due to the fact that the mode of construction Temperature_in of the attribute presentation, or conceptualization, or construction occurs extensionally as well. 0 0 0 [wt [ Temperature_inwt Prague]wt = 30C] b) Intensional context: of the proposition that the temperature in Prague “The temperature in Prague is rising.” is 300C is an object of Tilman’s belief rather than Additional type. Rising/(): the property of the proposition itself. Thus all the a magnitude. subconstructions of this construction occur The analysis of the sentence is the following: hyperintensionally as well. In brief, the left most 0 0 Trivialization in [wt [ Temperature_inwt 0 0 0 0 0 w t [ Risingwt w t [ Temperature_inwt Prague]]. Prague]wt = 30C] raises the context up to the 0 0 hyperintensional level. Now, wt [ Temperature_inwt Prague] occurs intensionally in the Composition
0 0 0 [ Risingwt wt [ Temperature_inwt Prague]]. 3 Procedural Isomorphism The function (magnitude) rather than its value is an object of predication that this function is As mentioned at the outset, TIL can be viewed as rising. Due to the dominancy of a higher context an extensional logic of hyperintensions. By over a lower one, the construction ‘extensional’ I mean that the extensional rules like 0Temperature_in of the attribute occurs existential generalization and Leibniz’s laws of intensionally as well, though it is extensionalized substitutivity of identicals are validly applicable in (composed with w and t) and composed with the all contexts, whether extensional, intensional or construction 0Prague of its argument. hyperintensional. The rules of existential quantification in intensional and hyperintensional c) Hyperintensional context: contexts have been specified by Duží & Jespersen in [8], [9] and [11]. “Tilman believes that the temperature in 0 As for the substitution of identicals in an Prague is 30 Celsius.” extensional or intensional context, there is no Additional type. Believe/(n): relation-in- problem. In an intensional context analytically intension between an individual and a equivalent terms are mutually substitutable, while hyperproposition. in an extensional context also co-referring terms The analysis of the sentence is this: are substitutable.
0 0 Terms are co-referring if they refer to the same wt [ Believewt Tilman 0 0 0 0 object in a given world w and time t of evaluation. [wt [ Temperature_inwt Prague]wt = 30C]]. More generally, terms are co-referring if their Here we construe Believe as a relation-in- meanings are v-congruent, that is v-construct intension of an individual to a hyperproposition. functions that happen to have the same value at The reason is this: the proposition that the the same argument. Terms are analytically temperature in Prague is 300 Celsius can be equivalent if their meanings v-construct the same equivalently expressed, e.g., that the temperature object for every valuation v. Obviously, equivalent in Prague is 860 Fahrenheit. Yet one can believe terms are co-referring, but not vice versa. the former without believing the latter, and vice Thus, for instance, the following argument is versa. valid.
Computación y Sistemas Vol. 18, No. 3, 2014 pp. 439–453 ISSN 1405-5546 DOI: 10.13053/CyS-18-3-2018 446 Marie Duží
The temperature in Amsterdam is 300 Celsius. 0 0 wt [x [[ Happywt x] [ Childwt x]]], The temperature in Amsterdam is the same as in Prague. 0 0 w t [ y [[ Happywt y] [ Childwt y]]], 0 0 0 The temperature in Prague is 30 Celsius. wt [z [[ Happywt z] [ Childwt z]]], …, The ‘temperature in Amsterdam’ as well as the are by Def. 2 different constructions of the ‘temperature in Prague’ occur extensionally both property of being a happy child. Yet from the in the premises and the conclusion. Moreover, the procedural point of view they are isomorphic. second premise establishes their co-reference, as They consist of these procedural steps: the analysis reveals: in any world w (w) at any time t (t) do this: 0 0 [wt [[ Temperature_inwt Amsterdam]wt = take any individual x (or y, or z, …), 0 0 0 wt [ Temperature_inwt Prague]wt]. take the property of being happy (by Happy), On this assumption the extensional apply the (extensionalized) property of being 0 occurrences of these two terms are mutually happy ( Happywt) to the chosen individual x 7 0 substitutable. However, this argument is invalid: ([ Happywt x]), or to y, or z, …, take the property of being a child (by 0Child), The temperature in Amsterdam is rising. The temperature in Amsterdam is the same as in Prague. apply the (extensionalized) property of being 0 a child ( Childwt) to the chosen individual x (or The temperature in Prague is rising. y, or z, …), The reason is this: in the first premise ‘the check whether the chosen individual has both 0 temperature in Amsterdam’ occurs intensionally. the properties ( ), Particular value of the magnitude cannot be abstract over the chosen individual (x, or rising; rather, being rising is a property of the y, or z, …). whole function, magnitude in this case. However, the second premise establishes only co-reference Using the terminology of -calculus, the above of the two terms rather than equivalency. Hence Closures are -equivalent. Church’s Alternative the substitution is not valid, because in such an (A0) includes -conversion and meaning intensional context only equivalent terms can postulates for atomic constants such as ‘bachelor’ be substituted. and ‘fortnight’. Of course, we need meaning However, substitution of (analytically) postulates to specify synonymy of semantically equivalent terms is not valid in hyperintensional simple terms. This is a matter of building up contexts. From a linguistic point of view, in a linguistic ontology. Now we are, however, hyperintensional context only synonymous interested in the synonymy of molecular terms, expressions can be substituted, because the very which is structural isomorphism. meaning of expressions is displayed. Our thesis is -conversion is certainly the rule that must be that synonymous expressions have structurally included. The question is, which other rules (and isomorphic meanings. And since meaning is a whether any other rules) could be included as procedure, we need to define the relation of well. Church’s (A0) and (A1) leave room for procedural isomorphism between constructions, additional Alternatives. To this end we consider - because constructions are a bit too fine-grained and -conversion. Yet, as explained above, we from the procedural point of view. The main issue are not content with Church’s Alternatives (A1) or here is the following. Constructions that differ at (A1’) due to their non-equivalency. most by using different -bound variables of the In this paper I suggest a new definition of the same type differ so slightly that we wish to say criterion of structured synonymy, (A1). This that such constructions are one and the same variant is close to Church’s (A1) and includes an procedure. For instance, the Closures adjusted versions of - and -conversion. Thus we exclude -conversion, and introduce a new version of -conversion, to wit, the conversion by 7 I use a variant of Barbara Partee’s example. value. In this proposal I follow two necessary
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 447 conditions that the meanings of synonymous not contain -expanded subconstructions. For expressions should meet. They must be (a) instance, the literal analysis of “The Pope is wise” strictly equivalent in the sense that in every is the Closure possible world at any time they either denote the 0 0 same entity or lack a denotation, and (b) their w t [ Wisewt Popewt] meanings must have the same constituents. rather than There are two reasons for not including - 0 0 conversion. First, it does not satisfy the condition wt [wt [x [ Wisewt x]]wt Popewt], (b). The -expanded construction of the form because the literal analysis of the predicate ‘is λx [F x] has two more constituents than the - wise’ is the Trivialization 0Wise rather than the 0 reduced construction F, because the former adds Closure wt [x [ Wisewt x]]. The types are the steps of applying the function v-constructed Wise/(); Pope/; x . by F to the value v-constructed by the variable x followed by the abstraction over the value of x. 3.1 -Conversion: ‘by Name’ vs. ‘by Value’ The second reason is the fact that -conversion does not preserve logical equivalence in a logic of The reasons for excluding unrestricted - partial functions such as TIL. Hence it does not conversion are similar to the above. Though it is satisfy the condition (a). To see this, consider the the fundamental computational rule of the - function F (( ) ) that v-constructs a function calculi, it is underspecified by the commonly that is not defined at the argument v-constructed acknowledged rule by A . Then the Composition [F A] () is v- 9 improper. However, the -expanded construction [[x C(x)] A] |– C(A/x). λx [[F A] x] (), x , v-constructs a The procedure of applying the function v- degenerate function, which is a function constructed by [x C(x)] to the argument v- undefined at all its arguments. To be sure, due to constructed by A can be executed in two different the v-improperness of [F A], the Composition ways: by value or by name. If by name then the [[F A] x] is also v-improper. But λ-abstraction procedure A is substituted for all the occurrences raises the context to an intensional one, hence of x into C. In this case there are two problems. the Closure λx [[F A] x] v-constructs a degenerate First, conversion by name is not guaranteed to function, which is an object, if a bizarre one. be a strictly equivalent transformation as soon as Hence the constructions [F A] and λx [[F A] x] are 8 partial functions are involved. This is due to the not strictly equivalent. fact that A occurs in an extensional context of the In practice the exclusion of -conversion from left-hand side construction, whereas when the definition of procedural isomorphism is going dragged into C its occurrence may become to be harmless. When analyzing expressions in intensional or hyperintensional provided the TIL we apply our method of literal analysis, which context in which x occurs in C is intensional or consists of three steps: (i) assigning types to the even hyperintensional. For instance, objects mentioned by the sub-terms of the the Composition analyzed expression E; (ii) combining the 0 0 0 0 Trivializations of the objects mentioned by the [x [y [ + x y]] [ : 3 0]] semantically simple sub-terms of E in order to 0 0 0 is improper, because [ : 3 0] is improper. This is obtain the construction of the object (if any) as it should be, for there is no value that might be denoted by E; (iii) checking whether the resulting substituted for the formal parameter x, because construction is type-theoretically coherent. Due to the Composition [0: 03 00] is improper by failing to step (ii) the application of this method yields a construction (namely the meaning of E) that does
8 I am grateful to J. Raclavský for calling our attention to this 9 For the sake of simplicity, we now consider a unary problem. See also Raclavský (2010). function. Generalization for n-ary functions is obvious.
Computación y Sistemas Vol. 18, No. 3, 2014 pp. 439–453 ISSN 1405-5546 DOI: 10.13053/CyS-18-3-2018 448 Marie Duží produce a result. However, the ‘by-name’ - is larger than a”. Yet the two sentences are not reduced construction strictly synonymous, because in the former the property of being larger than itself is applied to a 0 0 0 0 [ y [ + [ : 3 0] y]] while in the latter the binary relation larger than is 12 is not improper as it constructs a degenerated applied to the pair (a,a). function undefined at all its arguments. The The idea of conversion by value is simple. improper construction [0: 03 00] has been drawn Execute the procedure A first, and only if A does into the intensional context of the Closure not fail to produce an argument value on which C [y [0+ x y]]. is to operate, substitute (Trivialization of) this The same problem crops up in the analysis of value for x. The solution preserves strict logical de re attitudes. For instance, the de re attitude equivalence, avoids the problem of the loss of expressed by analytic information, and moreover, in practice it is more efficient. The efficiency is guaranteed by “The Pope is believed by a to be wise” the fact that the procedure A is executed only obtains on its intensional reading the analysis10 once, whereas if this procedure is substituted for 0 0 0 all the occurrences of the -bound variable it can wt [x [ Believewt a wt [ Wisewt x]] Popewt]. subsequently be executed more than once. To elucidate the problem even more, In a world w and time t when the Pope does comparison with programming languages might 0 not exist, the Composition Popewt is v-improper be useful. Imagine one has a procedure and the so constructed proposition lacks a truth- (embodied as a program) C(x) with a ‘hole’ x (i.e., value, because there is no individual to whom the an unsaturated procedure with a formal property of being believed by a to be wise would parameter x), and a subprogram A that specifies be ascribable. the material (argument value) to be filled into the This is as it should be, because in the de re hole x. There are two ways of going about filling x: case there is an existential presupposition that the (1) (by name) inserting into the hole x the Pope exists. Yet the -reduction by name whole subprogram A and then computing C(A/x) transforms this analysis into the de dicto attitude: (2) (by value) computing A first in order to 0 0 0 obtain the argument value a, and then inserting a wt [ Believe a w’t’ [ Wise Pope ]]. wt w’t’ wt into the hole x and computing C(a/x) The so constructed proposition can be true In case (1) there may be undesirable side even in a w,t-pair in which the Pope does not effects. Imagine that the subprogram A is exist, which is not correct. somehow garbled and as a result the whole The second reason for refuting an unrestricted procedure C gets garbled as well by the insertion, -reduction by name is this: even in those cases damage being propagated upwards. Moreover, when -reduction by name is an equivalent instead of the hole x one gets A, and A may transformation, it may yield loss of analytic conflict with C. This is a case of invalid - information about which function has been reduction that fails to preserve equivalence. applied to which argument.11 For instance, Furthermore, even if A does not damage C when the Composition computing C(A), after the execution of C(A) one 0 will have lost track of A. The two procedures have wt [[ x [ Largerwt x x]] a] been merged together. Suppose one wants to which is the meaning of “a is larger than itself” compute another procedure E(x) and to supply 0 the same material for x. Even if the execution of reduces to wt [ Largerwt a a], the meaning of “a C(A) turns out to be successful, A may have been
10 changed by the execution. There is no guarantee In this case a hyperintensional analysis would be also the that the same material will be supplied for x into choice. For the sake of simplicity we consider the intensional one. 11 For the notion of analytic information see [5]. 12 See [17].
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E(x). This is the case of -reduction preserving types. First we need to define the equivalence but yielding loss of substitution function: analytic information. n n Definition 5 (Sub ). Function Sub /(nnnn) In case (2) we keep C(x), E(x), and A operates on constructions in this way: let separate. Procedure A is evaluated only if constructions C , C , C v-construct constructions needed, and if so, only once. Everything is as it 1 2 3 (of order n) D1, D2, D3, respectively. Then the should be: no loss of analytic information arises 0 n Composition [ Sub C C C ] v-constructs the and equivalence is preserved. 1 2 3 construction D that results from D3 by collisionless In programming languages the difference substitution of D for all occurrences of D in D . between cases 1 and 2 concerns evaluation 1 2 3 Occasionally we also need the polymorphic strategy and is often called passing by reference function Tr defined as follows. vs. passing by value, respectively. Historically, call-by-value and call-by-name date back to Algol Definition 6 (Tr ). The function Tr /(n ) returns 60, a language designed in the late 1950s. Only as its value the Trivialization of its -argument. purely functional languages such as Clean and Haskell use call-by-name. For instance, Java In what follows I simply write ‘Sub’ and ‘Tr’ manipulates objects by reference that is by name. omitting thus the type-superscripts whenever no However, Java does not pass arguments by confusion arises. reference, but by value. Call-by-value is not a For instance, let variable y v-construct single evaluation strategy, but rather a cluster of numbers of type such as . Then [0Tr y] v(/y)- evaluation strategies in which a function’s 0 constructs . Therefore, the Composition argument is evaluated before being passed to the calling procedure. In call-by-reference evaluation [0Sub [0Tr y] 0x 0[0Sin x]] (also referred to as call-by name or pass-by- 0 0 reference), a calling procedure receives an v( /y)-constructs the Composition [ Sin ]. implicit reference to the argument sub-procedure. Note that there is a substantial difference between the construction Trivialization and the This typically means that the calling procedure 0 can modify the argument sub-procedure. A call- function Tr . Whereas y constructs just the variable y regardless of valuation, y being 0-bound by-reference language makes it more difficult for 0 0 a programmer to track the effects of a procedure in y, [ Tr y] v-constructs the Trivialization of the object v-constructed by y. Hence y occurs free call, and may introduce subtle bugs. 0 The notion of reduction strategy in the -calculi in [ Tr y]. is similar to the evaluation strategy in programming languages. My proposal amounts to Definition 7 (-conversion by value) Let Y v a specification of an evaluation strategy by-value ; x1, D1 v 1,…, xn, Dn v n, [x1…xn Y] v as adapted to TIL that is to hyperintensional, (1...n). Then the conversion partial typed -calculus. Similar work has been [[x …x Y] D …D ] done since the early 1970s, but merely for simple- 1 n 1 n 2 0 0 0 0 0 0 0 typed or untyped -calculi. For instance, Plotkin in [ Sub [ Tr D1] x1 … [ Sub [ Tr Dn] xn Y]] [16] proved that the two strategies are not operationally equivalent. Hence the call-by-name is -conversion by value. strategy should not be used in hyperintensional - Note that in the converted construction Double calculi such as TIL due to operational non- Execution is necessary. The function Sub equivalence. Our substitution method that I am operates on argument constructions, and as a going to define below is similar to Chang & result it produces again a construction; to wit, the Felleisen call-by-need reduction by value (see construction in which all occurrences of the formal [2]). But their work is couched in an untyped - parameters x1, …, xn are replaced by the calculus. TIL, by contrast, is a hyperintensional, Trivializations of the respective argument values. partial -calculus based on ramified theory of In order to obtain an -value (if any) produced by
Computación y Sistemas Vol. 18, No. 3, 2014 pp. 439–453 ISSN 1405-5546 DOI: 10.13053/CyS-18-3-2018 450 Marie Duží the so pre-processed construction Y the resulting previous paragraph I summarized objections construction must be executed. Hence, against Church’s Alternatives (A1) or (A1’), and it Double Execution is performed. should be clear now that -conversion should not be included. An obvious plausible candidate is - Claim 1: Constructions [[x1…xn Y] D1… Dn] and 2 0 0 0 0 0 0 0 conversion, which would yield Alternative (A0). [ Sub [ Tr D ] x … [ Sub [ Tr D ] x Y]] are 1 1 n n strictly equivalent in the sense that for any In my opinion, we also need -conversion, valuation v they either v-construct one and the because application of a function to an argument same entity or are both v-improper. is the very fundamental computational procedure. However, the objections against unrestricted - Proof. Let C be [[x1…xn Y] D1…Dn] and let D be conversion (by name) are serious. Fortunately 2 0 0 0 0 0 0 0 [ Sub [ Tr D1] x1 … [ Sub [ Tr Dn] xn Y]]. there is a way out. The above defined - If some of the constructions D1,…,Dn is v- conversion by value is immune against those improper then so are both C and D, according to objections. It is the correct way of applying the Def. 2, iii) and vi). Otherwise, let D1,…,Dn all be v- function constructed by the Closure [x1…xn Y] to proper, v-constructing the objects d1,…,dn, the arguments constructed by D1,…,Dn. According respectively. Then by Def. 2, iv) the Closure to Claim 1, it is a strictly equivalent conversion, [x1…xn Y] v-constructs the following function f. If unlike -conversion by name. Moreover, it does Y is v(d1/x1,…,dn/xn)-improper, then f is undefined not yield loss of analytic information, because the on d1,…,dn and thus C is v(d1/x1,…,dn/xn)- calling procedure Y and argument procedures improper according to Def. 2, iii). Otherwise the D1,…,Dn are kept separated. value of f on d1,…,dn is the -entity Now I will demonstrate the thesis that -value v(d1/x1,…,dn/xn)-constructed by Y. equivalent constructions should be considered as Let the entity v(d1/x1,…,dn/xn)-constructed by Y procedurally isomorphic. For the sake of simplicity be e. Then by Def. 2, iii) of Composition, the I will again consider a one-argument Composition construction C v-constructs e. We are to show of the form [x [F x] A]. It is the specification of that the construction D also v-constructs e. The 0 0 calling the procedure x [F x] with the formal first Execution of D constructs Y(x1/ d1,…, xn/ dn), parameter x at the argument provided by the i.e., the construction Y where according to the procedure A. My thesis is that the correct way of definition of the functions Sub and Tr all the executing this procedure consists of these occurrences of variables x1,…,xn are replaced by 0 0 execution steps: d1,…, dn, respectively. Since the Trivializations 0 0 execute A in order to obtain the argument d1,…, dn construct the entities d1,…,dn, value a; if A fails to v-construct anything (is v- respectively, the second Execution improper) then abort the execution; else: v(d /x ,…,d /x )-constructs the entity e, or else 1 1 n n take the argument value a (by the Trivializa- nothing in case Y is v(d /x ,…,d /x )-improper. 1 1 n n tion of a) and substitute it for all the occur- Hence C and D come out strictly equivalent. rences of the variable x in the procedure body F, and finally: 3.2 Alternative (A1’’) Execute the result of the substitution. At the outset of this paper I formulated the The Composition 2[0Sub [0Tr A] 0x 0F] has problem of granularity of individuation meanings. exactly the same constituents. These are Any individuation that is finer than necessary or logical equivalence qualifies as hyperintensional. A: execute A in order to obtain the argument Since we explicate hyperintensions procedurally, value a; if A is v-improper then the entire the problem transforms into the issue of Composition is v-improper; else: 0 individuation of procedures. In other words, under [ Tr A]: obtain the Trivialization of a, 0 0 0 0 which conditions are we ready to consider two [ Sub [ Tr A] x F]: substitute the Trivializa- constructions as procedurally isomorphic? Church tion of a for x in F, considered -, - and -conversion. In the 2[0Sub [0Tr A] 0x 0F]: execute the result.
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We can see that -conversion by value is an Definition 8 (-equivalence) Let C, D be explicit specification of the procedure of applying constructions. Then C, D are -equivalent, if the function constructed by x [F x] to the either C, D differ at most by using different - argument constructed by A. Yet one might object bound variables (of the same type), or their - that the above execution steps are not explicitly value equivalent forms differ at most by using specified by the term ‘[x [F x] A]’. This is true but different -bound variables (of the same type). this term can be taken as an abbreviation of the full-fledged application specification provided by Claim 2. -equivalent constructions are strictly 2 0 0 0 0 ‘ [ Sub [ Tr A] x F]’. In other words, we can equivalent being either v-improper or v- consider the left-hand side of the -value rule as constructing one and the same entity. definiendum and the right-hand side as definiens. For this reason it is reasonable to claim the Proof. Due to Claim 1 it suffices to prove that 2 0 0 0 0 two terms ‘[x [F x] A]’ and ‘ [ Sub [ Tr A] x F]’ Closures of the form [ x1…xn Y(x1,…,xn)], [ y1…yn ex definitione synonymous and thus substitutable Y(y1,…,yn)], where Y(x1,…,xn) differs from in all contexts, including hyperintensional ones. Y(y1,…,yn) only by collisionless substitution of variables x ,…,x for y ,…,y , respectively, v- In order to define procedural isomorphism on 1 n 1 n construct one and the same function. But this the set of constructions, we still need another immediately follows from Def. 2, iv). definition, to wit, the definition of -conversion. The standard definition that defines -equivalent Definition 9 (procedural isomorphism) Let C, D constructions as those that differ at most by using be constructions. Then C, D are procedurally different -bound variables does not do. The isomorphic iff either C and D are identical or there 0 reason is this: if -value equivalent constructions are constructions C1,…, Cn (n > 1) such that C = 2 0 0 0 0 0 0 0 of the form [x [F x] A] and [ Sub [ Tr A] x F(x)] C1, D = Cn, and for each Ci, Ci+1 (1 i < n) it are procedurally isomorphic, then it does not holds that Ci, Ci+1 are either - or - matter which particular variable of the respective value equivalent. type, whether x, or y, or z, …, has been used as a formal parameter. Hence since Corollary. Procedural isomorphism is an equivalence relation defined on a set of [x [F x] A], [y [F y] A], … constructions such that procedurally isomorphic constructions are strictly equivalent in the sense are -equivalent, so should be that for any valuation v they either v-construct one 2[0Sub [0Tr A] 0x 0F(x)], 2[0Sub [0Tr A] 0y 0F(y)], … and the same entity or they are v-improper. For instance, constructions Proof. It follows immediately from Claims 1 and 2.
[x [0+ x 01] 05] Having defined procedural isomorphism, we [y [0+ y 01] 05] can now specify the rule of substitution in hyperintensional contexts. are -equivalent according to the standard In a hyperintensional context only procedurally definition. Yet their -reduced forms isomorphic constructions are mutually substitutable. 2[0Sub [0Tr 05] 0x 0[0+ x 01]] 2 0 0 0 0 0 0 0 Example. Consider the analysis of the [ Sub [ Tr 5] y [ + y 1]] sentence “Tilman is calculating the value of the would not be -equivalent. But they should be, function sin(x):cos(x) at the argument equal to the because from the procedural point of view it is (principal) square root of 2” that comes down to irrelevant which variables are used as formal 0 0 wt [ Calculatewt Tilman parameters for which the respective argument is 0 0 0 0 0 0 (1) [x [ : [ Sin x] [ Cos x]] [ 2]]]. substituted. Thus we define the following:
Computación y Sistemas Vol. 18, No. 3, 2014 pp. 439–453 ISSN 1405-5546 DOI: 10.13053/CyS-18-3-2018 452 Marie Duží
Types. Calculate/(n); Tilman/; :/(); Sin, Obviously, (4) follows from (2) but not vice versa. Cos/(); /(): the principal, non-negative square root. Yet there are other interesting issues Since the construction concerning the interplay between Trivialization and Double Execution. While Trivialization raises 0 0 0 0 0 [x [ : [ Sin x] [ Sin x]] [ 2]] the context up to the hyperintensional level, Double Execution decreases it back to the is procedurally isomorphic with its -value intensional one. Hence for any construction C it equivalent form 20 holds that C is logically equivalent to C. The 20 2[0Sub [0Tr [0 02]] 0x 0[0: [0Sin x] [0Sin x]]], question is whether the two constructions, C and C, are not procedurally isomorphic as well. In it follows that Tilman calculates this Composition our opinion they are not. The former contains two as well as its -value and -equivalent additional executive steps, to wit, Double procedurally isomorphic variants: Execution and Trivialization. Though in an wt [0Calculate 0Tilman ordinary vernacular this slight difference would wt (2) 02[0Sub [0Tr [0 02]] 0x 0[0: [0Sin x] [0Cos x]]]], most probably not matter, in the semantics of a programming language it does matter. wt [0Calculate 0Tilman Thus we are considering whether it is wt (3) 02[0Sub [0Tr [0 02]] 0y 0[0: [0Sin y] [0Cos y]]]]. philosophically wise to adopt several notions of procedural isomorphism. The definition I pro- But it does not follow that Tilman calculates posed in this paper is a very strict criterion of the ratio of Sine at 2 and Cosine at 2: synonymy, and it is not improbable that several degrees of hyperintensional individuation are 0 0 wt [ Calculatewt Tilman called for, depending on which sort of discourse 0 0 0 0 0 0 0 0 [ : [ Sin [ 2]] [ Cos [ 2]]]]. happens to be analyzed. Thus we admit that 0 0 0 0 0 0 0 slightly different definitions of procedural Note that [ : [ Sin [ 2]] [ Cos [ 2]]] is the isomorphism are still thinkable. What appears to result of -reduction by name of the original be synonymous in an ordinary vernacular might 0 0 0 0 0 Composition [ x [ : [ Sin x] [ Cos x]] [ 2]]. While not be synonymous in a professional language in the latter the square root of 2 is calculated only like the language of, for instance, logic, once, in the reduced construction it is calculated mathematics or physics. twice. Hence the two constructions are not procedurally isomorphic. Since Calculate is a relation(-in-intension) of 4 Conclusion an individual to a construction, and the application of Sub produces a construction, a question arises here. Could we omit in (2) or (3) the Trivialization I demonstrated how to validly apply Leibniz’s Law and Double Execution preceding the application of the substitution of identicals in hyperintensional of the function Sub? In other words, are the contexts. Since in such a context the meaning of constructions (2) and (3) equivalent with this one: an expression is displayed, only synonymous 0 0 expressions with the same meaning can be wt [ Calculatewt Tilman mutually substituted. Our criterion of synonymy is 0 0 0 0 0 0 0 0 0 (4) [ Sub [ Tr [ 2]] x [ : [ Sin x] [ Cos x]]]] ? procedural isomorphism of the constructions Our answer is no, it is not. The reason is this. expressed by the respective expressions. The In (4) Tilman is related to the product of paper offers a formally worked-out and philosophically motivated criterion of [0Sub [0Tr [0 02]] 0x 0[0: [0Sin x] [0Cos x]]] hyperintensional individuation, which is the relation of procedural isomorphism. The definition which is the following Composition: of procedural isomorphism includes a slightly 0 0 0 0 0 [ : [ Sin 1.4142135…] [ Cos 1.4142135…]]. more carefully stated version of -conversion and
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-conversion by value, which amounts to a 10. Duží, M. & Jespersen, B. (2013). Procedural modification of Church’s Alternative (A1). isomorphism, analytic information, and - conversion by value. Logic Journal of the IGPL, 21(2), 291–308. Acknowledgements 11. Duží, M., Jespersen, B. (to appear). Transparent quantification into hyperintensional objectual This work has been supported by the internal attitudes. Synthese, special issue on grant agency of VSB-Technical University Hyperintensionality Ostrava, Project No. SP2014/157, “Knowledge 12. Duží, M., Jespersen, B., & Materna, P. (2010). Procedural Semantics for Hyperintensional Logic: modeling, process simulation and design”. Foundations and Applications of Trasnsparent Versions of this paper have been presented at Intensional Logic. Dordrecht: New York: Springer. Logica 2013, Czech Republic, and CICLing 2014, 13. Ludwig, K. & Ray, G. (1998). Semantics for Nepal. opaque contexts. Philosophical Perspectives, 12(S12), 141–166. 14. Materna, P. (1998). Concepts and Objects. Acta References Philosophica Fennica, vol. 63. Helsinki: Edidit Societas Philosophica: Distribuit Akateeminen 1. Anderson, C.A. (1998). Alonzo Church’s Kirjakauppa. contributions to philosophy and intensional logic. 15. Moschovakis, Y.N. (1993). Sense and denotation The Bulletin of Symbolic Logic, 4(2), 129–171. as algorithm and value. Lecture Notes in Logic, 2, 2. Chang, S. & Felleisen, M. (2012). The call-by- 210–249. need lambda calculus, revisited. Programming 16. Plotkin, G.D. (1975). Call-by-name, call-by-value, Languages and Systems. Lecture Notes in and the lambda calculus. Theoretical Computer Computer Science, 7211, 128–147. Science, 1, 125–159. 3. Carnap, R. (1947). Meaning and necessity. 17. Salmon, N. (2010). Lambda in sentences with Chicago: Chicago University Press. designators: an ode to complex predication. 4. Church, A. (1993). A revised formulation of the Journal of Philosophy, 107(9), 445–468. logic of sense and denotation. Alternative (1). 18. Tichý, P. (1968). Smysl a procedura. Filosofický Noûs, 27(2), 141–157. časopis, 16, 222–232. Translated as ‘Sense and 5. Duží, M. (2010). The paradox of inference and the procedure’ in (Tichý 2004: 77-92). non-triviality of analytic information. Journal of 19. Tichý, P. (1969). Intensions in terms of Turing Philosophical Logic, 39(5), 473–510. machines. Studia Logica, 24(1), 7–21. Reprinted 6. Duží, M. (2012). Towards an extensional calculus in (Tichý 2004: 93–109). of hyperintensions. Organon F, 19, supplementary 20. Tichý, P. (2004). Collected Papers in Logic and issue 1, 20–45. Philosophy. Prague: Filosofia; Dunedin, N.Z.: 7. Duží, M. (2013). Deduction in TIL: From simple to University of Otago Press. ramified hierarchy of types. Organon F, 20, supplementary issue 2, 5–36. Marie Duží received her CSc. degree (roughly 8. Duží, M. & Jespersen, B. (2010). Transparent equivalent to Ph.D.) in 1992 from the Czech Quantification into Hyperintensional Contexts. In M. Peliš and V. Punčochář (eds.), The Logica Academy of Sciences in Logic. Since 2001 she Yearbook 2010 (81–98). London: College has been an Assistant Professor of Computer Publications. Science in VSB-Technical University Ostrava, 9. Duží, M. & Jespersen, B. (2012). Transparent Czech Republic. quantification into hyperpropositional contexts de re. Logique & Analyse, 55(220), 513–554. Article received on 08/01/2014 on 06/02/2014.
Computación y Sistemas Vol. 18, No. 3, 2014 pp. 439–453 ISSN 1405-5546 DOI: 10.13053/CyS-18-3-2018