Compositionality in Transparent Intensional Logic

Compositionalityin Transparent Intensional Logic Pavel Materna Institute of Philosophy, Academy of Sciences of Czech Republic Prague materna@lorien.site.cas.cz Compositionality

Compositionality: Definitions Compositionality is a condition. It is • “a condition on semantics for languages. A common statement of it is that the meaning of a complex expression is a function of the meanings of its immediate constituents and the grammatical rule which is used to combine them…” (Hodges 2001, 7) In other words, the meaning of a complex expression • “is determined by the meanings of the component expressions plus the way they are combined into the complex expression.” (Sandu, Hintikka 2001, 49) Compositionality

Formal definition of compositionality “Consider now F, a k-ary syntactic operation on E. m is F-compositional just in case there is a k-ary partial function G on Msuch that whenever F(e1,…,ek) is defined, then m(F(e1,…,ek)) = G(m(e1),…,m(ek)).“ (Szabó 2005, 5) E: set of expressions, m: a meaning-assignment, M: set of ‘available’ meanings • (Cf. also Stechow, Wunderlich 1991, 107.) Compositionality

Two problems A. • In which way does the syntactic function F break the given expression into (immediate) constituents (component expressions)? B. • What do we mean by ‘meaning’? Compositionality

Component expressions: syntactic task? “Autonomous syntax”: • “It would be…in vain to ask an autonomous syntactician what the term ‘constituent’ means. He certainly cannot say that a constituent is an expression which is complete in that it refers all by itself to a definite entity, in contrast to an incomplete expression which refers only in combination with some other expressions. For that… would amount to leaving the domain of autonomous syntax. The term ‘constituent’ (or ‘phrase’) is apparently not to be burdened with any pre-theoretical meaning at all: a constituent is simply whatever the grammarians’ theory brands as such in any particular case. (Tichý 2004, 807) • “…logical grammar, with its principle of compositionality of meaning, goes straight against the autonomy of syntax so cherished in the generative tradition. …And that means, at least in principle, that semantic considerations may influence the syntax, thus breaching the supposed autonomy of syntax.” (Gamut 1991, 141) Compositionality

Compositionality: logical languages vs. natural language • “The principle of compositionality of meaning has important consequences for the relationship between syntax and semantics. Usually in a logical system the definition of the semantic interpretation of expressions closely follows the lead of their syntactic construction. … The obvious way to proceed is to let the definition of the semantics parallel the finite, recursive definition of the syntax. Succinctly put, logical languages satisfy the following principle: the interpretation of a complex expression is a function of the interpretations of its parts.… every syntactic rule should have a semantic interpretation; and on the other hand, every aspect of the semantics which is not related to the interpretation of basic expressions should be linked to a syntactic operation. … But a natural language is not something we construct; it comes as given.” (Gamut 1991, 140) • Thus it seems that the Tarskian semantics of formal languages has got compositionality gratis. Whatever can be called ‘meaning’ in such languages is stipulated in such a way that the syntactic rules (determining, e.g., which concatenation of symbols counts as ‘well-formed formula’) select just those components of an expression that get due to the interpretation unambiguously a ‘meaning’, and derive unambiguously the ‘meaning’ of the whole expression from the ‘meanings’ of the components. Compositionality

Twoconsequences of compositionality Two simple consequences of compositionality are • a compositional semantics is Husserlian(see below), • synonyms are substitutable Ad i): The term ‘Husserlian’ has been introduced by Hodges in his (2001). Roughly, a semantics S is Husserlian if for every expressions E, E’ it holds that if E is synonymous with E’ in S then E belongs in S to the same (Tarskian) category as E’; i.e., the expression containing E as its subexpression is S-meaningful just if the expression containing E’ in the place of E is S-meaningful. Ad ii): If E is synonymous with E’ in S then so is any expression A containing E as its subexpression with the expression B that arises from A by substituting E’ for E Compositionality

Natural language: ambiguities • Simple examples show that both i) and ii) cannot be satisfied if applied to expressions of a natural language. The reason is that natural languages, which “come as given”, are replenished by ambiguities. • One kind of ambiguities comprises lexical ambiguities. These are rather simple and can be thought of as ‘corrigible’. They make it impossible to satisfy even the Husserlian condition. For example, consider the word means. To be Husserlian the semantics would have to admit that since “What Charles means is that…” is meaningful and we can (roughly) claim that means and resources are synonymous then “What Charles resources is that…”were likewise meaningful. Compositionality

Natural language: non-lexical ambiguities • The non-lexical ambiguities are especially insidious. Here is an example of a sentence that seems not to contain any lexical ambiguity but does not admit an unambiguous interpretation (cf. also the famous Montague’s example with seeking unicorns): Charles wants to marry a princess. • “An expression may be ambiguous without having two distinct constituent structures. But compositionality simply requires that there be different ‘parts’ whenever there is non-lexical ambiguity, and if none of the known notions will do, the parts have to be ‘invented’. … compositionality demands a disambiguated level of representation in the syntax.” (Gamut 1991, 218) Compositionality

Logical analysis of natural language Summarising: • Identifying the component expressions of a complex NL expression presupposes a logical analysis of the natural language (LANL). This is because of the fact that natural expressions develop spontaneously so that the logical structure underlying the ‘natural encoding’ given by the language (NB at the given stage of its development) remains hidden and has to be discovered. The degree of the adequateness of such discovery should be dependent on the degree in which the respective LANL could ensure that compositionality holds. • As for the disambiguation necessary according to the above quotation, LANL should be capable to offer to each reading (i.e., to each analysis tree) of an ambiguous expression a separate logical construction (indeed, under the assumption that the component expressions occurring in the tree have been determined as meaningful expressions and the dependencies between the particular terminals are mediated via rules admitting semantic interpretation). Compositionality

Horwich: compositionality independent of the definition of meaning? • No great problems with compositionality are admitted from a deflationary viewpoint. Horwich formulates the problem as follows: “The issue of how the meanings of sentences are built out of the meanings of their constituents words” (Horwich 1997, 503) • Further he claims that “the compositionality of meaning imposes no constraint at all on how the meaning properties of words are constituted.”(Ibidem) • If this Horwich’s claim means that compositionality holds independently on how the notion meaning is defined then it can be easily demonstrated that this claim is wrong. Compositionality

Reference, denotation, meaning I: TIL • The main principles of TIL can be found in the work of the founder of TIL Pavel Tichý (see his Collected papers (2004), and his monograph (1988)); further information is offered, e.g., by Materna, P., Duží, M. (2005) and the important definitions in Duží, M., Materna, P. (2005).Since the apparatus is available in the literature just mentioned I will try to get along without formal definitions, and since the principles have been sufficiently defended in that literature I will only mention them and show how they help to elucidate the problems of compositionality. • Let us return to various definitions of compositionality. In all of them what is said to be ‘a function of’ or ‘determined by’ is called meaning. We can therefore ask “what sort of stuff meaning is” (Horwich) or simply state that meaning is in such definitions “a generic rather than specific term” (Sandu, Hintikka). In the latter text the authors propose to speak “of the different semantic attributes of an expression”. Now we will examine three such semantic attributes and show that only one of them can guarantee such a disambiguation of a natural language which makes it possible to let compositionality hold. Compositionality

Reference, denotation, meaning II: Frege’s “sense” Tichý’s criticism of Frege’s semantic schema: a) “Sense”: • Frege has characterized his sense as a mode of presentation of the object denoted (“die Art des Gegebenseins”), which is, of course, no definition. No wonder that, e. g., Bertrand Russell refused to accept such an indefinite notion. In any case, the lack of a satisfactory definition led to never ending discussions concerning necessity and character of sense, or, as we use to say nowadays, meaning. Alternatively, meaning is conceived of as having “at least two components: the sense and the reference” (Kirkham 1992/1997, 4). • Since the term sense is no more frequently used we will use the term meaning as one of the levels that are connected with semantics and let it play the role that corresponds to Frege’s characterization of sense as a mode of presentation. Compositionality

Reference, denotation, meaning III: Frege’s denotation b) Denotation: • If the object denoted were unambiguously determined by the sense then Frege’s notion of denotation (Bedeutung) as illustrated by his famous example with morning star vs. evening star does not satisfy this requirement. • For Frege the expression morning star as well as evening star denotes Venus, the particular celestial body. Imagine however a well thinkable (in any case possible) situation when the role played by the morning star (being the brightest celestial body in the morning sky and suchlike) begins to be played by another body, say, Mars. A natural way to state this change is to say “Now Mars became the morning star”. The sense – be it anything – connected with the term morning star did not change but the Fregean denotation did. This is a consequence of what Tichý calls – e. g., in (2004, 825) – • “Frege’s Thesis”, viz. that “an expression … is not a name of the determiner itself but rather of the object, if any, determined by it”. Compositionality

Reference, denotation, meaning IV: Criticism of ‘Frege’s Thesis” • The relation of denoting, if conceived of as a semantic relation (as it should be, at least if semantics of NL is construed as LANL), is, of course, a necessary relation. • Thus what is denoted by an empirical expression is always just a condition to be fulfilled by an object of the given type. Such conditions (called “determiners” in the above quotation) are best modeled as intensionsin the sense of P(ossible) W(orld) S(emantics). Intensions are in this sense functions mapping possible worlds to objects of the given type, mostly to chronologies of such objects, i.e., to functions from time moments. • Objects that are not intensions are extensions. • Empirical expressions denote intensions due to a linguistic convention (which is given and which, therefore, enables us to say that from the viewpoint of LANL the denotation of an empirical expression is given a priori). • Now whereas empirical expressions denote intensions independently of their instantaneous actual ‘population’ and are so immune to empirical facts (so that the relation of denoting is necessary) we can consider the value of any such intension in the actual world-time. This is our opportunity to distinguish: • denoting as a necessary, independent of empirical facts relation, and referring (or reference) as a contingent relation that is irrelevant – as being contingent – for LANL. Compositionality

Reference, denotation, meaning: Examples • “the capital city of Poland” denotesthe intension called individual role(Church has called it individual concept), which returns as its value in W at T at most one individual; its referencein the actual world-time is Warsaw, the town; • “The capital city of Poland is Warsaw” denotesthe proposition (a function from possible worlds to chronologies of truth values); its reference in the actual world-time is the truth-value True; • “star” denotesa property; its reference in the actual world-time is the set of all actual stars; • etc. etc. Compositionality

Actual world. Non-empirical expressions • One misunderstanding should be avoided: we can speak about the actual world but logically it is inaccessible. Saying, e.g., “I am actually hungry” we do not say anything more than “I am hungry”. A systematic explanation of this frequently ignored fact can be found in Jespersen (2005). • As for the non-empirical, in particular mathematical expressions, to distinguish between denotation and reference is inoperative: • semantics of mathematical expressions does not need any possible worlds or times, and • non-empirical (analytic) expressions containing empirical subexpressions denote constant intensions, i.e., functions whose value in all worlds-times is the same. Compositionality

‘Sense’ is not an intension • A frequent interpretation of Frege’s sense in the post-Fregean literature consists in construing it as intension • At least two essential objections to this identification of senses with intensionsare: • Let the sense of an (empirical) expression be the intension of the given type. Then the only object that would correspond to the denotation could be the value of the intension in the actual world. Then the link between the sense defined in this way and the denotation would be contingent, not unambiguous.For example, if the sense of the sentence “There are exoplanets where there live mammals” were its truth-condition, i.e., the respective proposition, then what would be denoted would be a truth-value. But the proposition itself does not possess the force of determining the truth-value. We need empirical methods to verify/falsify empirical sentences. • Mathematical expressions would not possess any sense, since they are independent of intensions. Compositionality

Meaning (‘sense’) as an abstract procedure • Let us consider some mathematical expression, say, 3 – 2  0. • The most natural answer to the question what this expression denotes is probably that it is the truth-value True. What would be the way that links the expression with this truth-value, playing thus the role of the Fregean sense? • The general form of the answer has been given by Tichý as early as in (1968) and (1969). The Fregean sense is best construed as an abstract procedure.Later – during developing the system of TIL – Tichý has formulated exact definitions of such procedures: they are what is called in TIL constructions and formally are influenced by the ingenious Church’s idea that two operations/procedures are the core of handling functions: creating functions via abstraction and applying functions to arguments, which led to -calculi. What the typed -calculus formulates on the level of formal languages TIL interprets objectually: Compositionality

Constructions I. • Variables, compositions (-applications), closures (-abstractions), trivializations (0X constructs X without any change) and doubleexecutions(where C is a construction that constructs a construction D the double execution 2C constructs what constructs D) are extra-linguistic procedures, and the ‘language of constructions’ is no formal language in the standard sense but only a direct code that enables us to deal with the abstract procedures. • Writing [X X1…Xm] for applying what is constructed (maybe dependently on valuations v) by X to what is constructed by X1,…,Xm, and [x1…xm X]for constructing a function (technically just as it is done in -calculus) we can logically handle procedures ‘working’ in the area of (partial) functions over some simple typesand even in the area of constructions themselves, which get their types and become thus a kind of objects to be mentioned (not only used) within a ramified hierarchy of types. Compositionality

Constructions II. • Constructions as abstract procedures embody an important property which a meaning of an expression should possess: they are (algorithmically) structured.Tichý’s and Cresswell’s (see his (1985) ) idea of structured meanings (see also the more recent Moschovakis (1994)) has been realized in TIL in a systematic way: • meaning of an expression is a construction, and LANL tries to find such a construction that would obey compositionality. (See Materna, Duží (2005).) • A special kind of construction can be defined: roughly, a closed construction, i.e., a construction not involving free variables. Such a construction has been called concept (see Materna (1998, 2004)). • The meaning of any non-indexical expression is accordingly a concept. Compositionality

Denotation, reference, meaning Summarising: • Fregean levels (sense, denotation/reference) have been revised in TIL as follows: i) Empirical expressions of a natural language denote intensions, never their values in the actual world-time. ii) Reference, as the value of the denotation in the actual world-time, is logically (and thus for LANL) inaccessible, hence it cannot be dealt with within LANL. iii) Meaningof an expression E is a construction; if E is a non-indexical expression its meaning is a concept. Meaning in this sense constructs what the expression denotes, which is as it should be. Compositionality

Constructions: Example I. • Returning to our example we can write down the concept that is the meaning of this mathematical sentence (whose denotation is True): [0 [0– 03 02] 00] • i.e.: the function  constructed by trivialization is applied to the pair M, N of procedures, where M is application of the subtraction function (constructed by trivialization) to the pair of numbers 3, 2 (constructed by trivialization) and N is trivialization of 0. Compositionality

Constructions: Example II. To show an example of an empirical sentence let us consider the sentence Charles believes that the Moon is bigger than the Earth. Reading:“Charles has the property of believing that the Moon is bigger than the Earth”. • ‘Charles’, ‘Moon’ and ‘Earth’ names of definite individuals (a simplification, of course), • ‘bigger than’ name of a binary relation-in-intension between individuals, • ‘believe that’ a relation-in-intension between an individual and a proposition. • wis a variable ranging over the type of possible worlds, • t ranges over the type of time moments (= real numbers), • x ranges over the type of individuals. Abbreviating [[Xw]t] as Xwtand omitting brackets where there is no danger of confusion we analyze our sentence as follows: wt [wtx [0Belwt x [wt [0Biggerwt0Moon 0Earth]]]wt0Charles] Compositionality

Compositionality w.r.t. reference, denotation and meaning I. Once more, let compositionality be defined like in Szabó: m(F(e1,…,ek)) = G(m(e1),…,m(ek)). We will try to examine what happens if mis interpreted as: a) reference, b) denotation, c) meaning (defined as above) • Let us recollect that synonymy is defined as follows: The expression E is synonymous withthe expression E’ iff m(E) = m(E’). • Finally, an easily derivable consequence of compositionality is: If e is a constituent of E, and if E’ is like E with the only distinction that e is replaced by e’, then m(e) = m(e’) implies m(E) = m(E’). (Principle of substitutability, PS) Compositionality

Compositionality w.r.t. reference, denotation and meaning II: a) m is reference Obviously, if reference is defined as above then we cannot expect that the m in definition of compositionality were reference. Indeed, we can easily prove that the PS does not hold for reference. Let our Charles believe that the following sentence is true: The Moon is bigger than the Earth. • Now the truth-value of the proposition denoted by this sentence in the actual world-time and so the reference of the sentence  is False. Hence the sentence is synonymous (w.r.t. reference) with the sentence Every woman has two wings. • If PS held for reference Charles would have to believe that this sentence too were true. But he does not have to, of course. In the semantics for which the meaning of E equals the reference of E the PS does not hold so that compositionality does not hold either. Compositionality

Compositionality w.r.t. reference, denotation and meaning III: b)m is denotation. We can immediately show that in this case doxastic or epistemic contexts make PS fail. Indeed, consider the sentences A. Some dogs are dangerous. B. Some dogs are dangerous and the only even prime is two. • The sentence A. is synonymous with the sentence B. w.r.t. denotation: The point is that the proposition denoted by “The only even prime is two” is true in all worlds-times. Thus the conjunction of the sentence A. with the sentence B. denotes one and the same proposition. If PS held for m = denotation then the following two sentences would be synonymous: Charles believes that some dogs are dangerous. Charles believes that some dogs are dangerous and (that) the only even prime is two. • This is however evidently not the case. In the semantics for which the meaning of E equals the denotation of E PS does not hold so that compositionality does not hold either. Compositionality

Compositionality w.r.t. reference, denotation and meaning IV: c)m is a concept First of all, this case is immune to counterexamples like those ones that were adduced in a) or b). For example, the sentences A. and B. above are not synonymous w.r.t. meaning: the construction (concept) underlying A. differs from the construction (concept) underlying B. This does not mean, however, that other counterexamples cannot be found. Consider the following ones: • 0yellowness constructs the same property as 0yellow. (Properly speaking, these are two ways of encoding in TIL one and the same construction.) Yet we cannot say ”This house is yellowness” – it seems that our semantics is not even Husserlian! Solution: The distinction between yellowness and yellow is in fact no distinction of meaning. Yet a distinction is present: the ness in the first expression signalizes that this name of a property can be used only if the property is mentioned (i.e., is in the supposition de dicto) rather than used to be predicated of something. (Thus there is a constraint to compositionality, which could be classified as compositionality in the sense iii) in Sandu, Hintikka (2001, 50).) There are more such signals in natural languages, cf. bravery vs. brave, in German Tugend vs. tugendhaft, schön vs. Schönheit etc. As soon as such a signal occurs in the respective analysis tree compositionality is no more jeopardized in this respect. Compositionality

Compositionality w.r.t. reference, denotation and meaning IV: c)m is a concept ii) Without any lexical change the sentence Charles wants to marry a princess. allows for two readings: a) There is a princess and Charles wants to marry he b) Charles wants that there were a princess whom he would marry. So the expression “to want to marry a princess” as if denoted two properties. Indeed, let Charles have the property according the reading a). Peter might have the property according to reading b). A simple substitution of the expression “wants to marry a princess” into both the statements that claim the possessing of a property by Charles and Peter would not result in the claim that Peter has the same property as Charles. Solution:The possible readings a) and b) show that 1. disambiguation is possible, 2. the ambiguous sentence is something like an abbreviation (where semantic distinctions are lost). There are two meanings (concepts) belonging to the sentence, each of them is the meaning of one of the two readings. (See Appendix.) No problem with compositionality arises after the disambiguation has been realized. Compositionality

Compositionality w.r.t. reference, denotation and meaning V: c)m is a concept iii)Suppose that Charles seeks the murderer of his father. Analyzing simple words as expressing trivialized objects (‘simple concepts’, Materna (2004)) can we really say that in any context 0seek is sufficient? Consider the situation when Charles says: “Success, it is Peter X!” and another situation when he says: “Success, he was in his flat!” Solution: This is again a problem of disambiguation. In a disambiguated language we would have: seek1 as look for the place where X occurs…and seek2 as investigate who plays the role of… or so. The resulting ambiguity is, as a matter of fact, a lexical ambiguity and should be signalized in the linguistic resources on which the analysis trees are based. Compositionality

Appendix We will show the two meanings that correspond to the two readings of the sentence Charles wants to marry a princess. Simple types: atomic … {True, False}  … individuals  … time moments / real numbers  …possible worlds (logical space) functional (1…m) … partial functions from 1  … m to  Types of objects: Charles …  Want … ((((())))) abbr.: (  ()) (relates an individual X with a property, viz. which X wants to possess) Marry … () Princess … () in general:  abbreviates (()) Types of logical functions:  … ()  … (()) Variables: w , t , x, y  First reading: There is a princess and Charles wants to marry her. wt [0x [0 [0Princesswtx] [0Wantwt0Charles wt y[0Marrywty x]]]] Second reading: Charles wants that there were a princess and he would marry her. wt [0Wantwt0Charles wty [0x [0 [0Princesswtx] [0Marrywty x]]]] Compositionality

References I. • Cresswell, M.J. (1985) Structured Meanings, MIT Press, Mass. • Duží, M., Materna, P. (2005): “Logical Form”. In: Sica, G., ed.: Essays on the Foundations Of Mathematics and Logic, Polimetrica International Scientific Publisher Monza/Italy • Gamut, L.T.F. (1991): Logic, Language and Meaning II., Intensional Logic and Logical Grammar. Chicago University Press. • Hodges, W. (2001): “Formal Features of Compositionality”. Journal of Logic, Language, and Information 10, 7-28 • Horwich, P. (1997): “The Composition of Meanings”, The Philosophical Review, Vol 106 No 4, 503-532 • Jespersen, B. (2005): “Explicit Intensionalization, Anti-Actualism, and How Smith’s Murderer Might Not Have Murdered Smith” Dialectica 59/3, 285-314 • Kirkham, Richard L. (1992/1997): Theories of Truth. The MIT Press, Cambridge, Mass., London 1992, 4th printing 1997. • Materna, P. (1998): Concepts and Objects. Acta Philosophica Fennica 63, Helsinki Compositionality

References II. • Materna, P. (2004): Conceptual Systems. Logos Verlag, Berlin • Materna, P., Duží, M. (2005): “ ‘Parmenides Principle’ (The analysis of aboutness)” Philosophia 32, Nos 1-4, 155-180 • Moschovakis, Y. N.(1994): Sense and denotation as algorithm and value, Postscript file, 39 pages Published in Lecture Notes in Logic, #2 (1994), Springer J. Oikkonen and J. Vaananen, eds. pages 210-249. • Sandu, G., Hintikka, J. (2001): “Aspects of Compositionality”, ”. Journal of Logic, Language, Information 10, 49-61 • Szabó, Z.G. (2005): “Compositionality”. The Stanford Encyclopedia of Philosophy <http://plato.stanford.edu/archives/spr2005/entries/compositionality/>. • Stechow, A.v. and Wunderlich, D., eds. (1991): Semantik/Semantics. De Gruyter, Berlin, NewYork • Tichý, P.(1968): “Sense and Procedure”. In Tichý, P.(2004, 77-92) • Tichý, P.(1969): “Intension in Terms of Turing Machines”. In: Tichý (2004, 94-109) • Tichý, P.(1988): The Foundations of Frege’s Logic. De Gruyter, Berlin, New York • Tichý, P.(1994): “The Analysis of Natural Language.” In Tichý, P.(2004, 803-841) • Tichý, P.(2004): Pavel Tichý’s Collected Papers in Logic and Philosophy. (Svoboda, Vl., Jespersen, B., Cheyne, C., eds.) Filosofia, Prague & University of Otago Press, Dunedin Compositionality

Appendix II. Let us return to the sentence: Charles believes that the Moon is bigger than the Earth. In any analysis tree the following components (constituents) will occur: a) ”Charles believes that the Moon is bigger than the Earth.” ….t b) “Charles”…………………………………………………………..t’1 c) “believe”………………………………………………………..... t’21 d) “the Moon”……………………………………………………….t’221 e) “(is) bigger than”………………………………………………..t’222 f) “the Earth”……………………………………………………….t’223 g) “that the Moon is bigger than the Earth”………………………t’22 h) “believe that the Moon is bigger than the Earth”………………t’2 A probable structure of the respective analysis tree can be suggested as follows: a) = t(t’1, t’2(t’21, t’22 (t’221, t’222, t’223))) We can see that the particular steps (subconstructions, subconcepts) of the meaning of the sentence correspond to the constituents of the suggested tree above and that the syntactic dependencies that make up this tree correspond to the objectual functions constructed by the particular subconstructions. wt [wtx [0Belwt x [wt [0Biggerwt0Moon 0Earth]]]wt0Charles] C3(e) t’222 C4(d) t’221 C5(f) t’223 C1(c)t’21 C6(g) t’22 C8(h) t’2 C2(b) t’1 C7(a) t Compositionality

Compositionality in Transparent Intensional Logic