On the Triviality of the Principle of Compositionality
Tolgahan ToyThe principle of compositionality states that the meaning of an expression is a function of the meanings of its parts. The principle is based on Gottlob Frege’s works that consider linguistic meaning in the function-argument structure. Through Richard Montague, this principle has taken a central place in formal semantics. According to the principle, a homomorphic relationship exists between linguistic and semantic structures. Proponents of the principle argue that compositionality makes an important contribution to the systematic and productive structure of language. Contrary to this view, Wlodek Zadrozny has argued compositionality to not contribute to the systematic structure of language. To justify this claim, Zadrozny put forward proof showing that any semantic structure can be encoded compositionally. Zadrozny also attributed the existence of such a semantic function for any semantic structure to Peter Aczel’s non-well-founded set theory. The solution lemma informs one that such a meaning function exists. However, an important objection to Zadrozny’s claim is found. The obtained semantic structure has been claimed to be insufficient for encoding the original semantic structure. The basis of this claim involves the phenomenon of synonymity. Non-compositional semantic structures in which synonyms do not follow the principle of substitution have been argued as being unable to be represented by compositional semantics. This work examines Zadrozny’s proof and answers the objection by claiming synonymity, being the subject of this objection, is not a property that needs to be preserved.
Bileşimsellik İlkesinin Önemsizliği Üzerine
Tolgahan ToyAnlambilimsel bileşimsellik ilkesi, herhangi bir ifadenin anlamının o ifadenin parçalarının anlamlarının bir fonksiyonu olduğunu bildirmektedir. İlke Gottlob Frege’nin dilsel anlamı fonksiyon argüman uygulaması olarak ele alan çalışmalarına dayanmaktadır. Bileşimselliğin, Richard Montague ile birlikte biçimsel anlambilim içerisinde merkezi bir rol aldığı görülmektedir. Bileşimsellik ilkesi, dilsel yapı ve anlamsal yapı arasında homomorfik bir ilişki bulunduğunu bildirmektedir. İlkenin savunucuları bileşimselliğin dilin sistematik ve üretken yapısına önemli katkılar sağladığını iddia etmektedirler. Wlodek Zadrozny, bu görüşe karşı olarak, bileşimselliğin dilin sistematik yapısına herhangi bir katkı sağlamadığını iddia etmektedir. Zadrozny, iddiasını temellendirmek için, herhangi bir anlambilimsel yapının bileşimsel olarak kodlanabileceğini gösteren bir ispat ortaya atmıştır. Zadrozny herhangi bir anlambilimsel yapı için böyle bir anlam fonksiyonunun var olduğunu ise Peter Aczel’in iyi-yapılanmamış kümeler kuramına dayandırmaktadır. Çözüm Lemması bize böyle bir anlam fonksiyonunun var olduğunu bildirmektedir. Zadrozny’nin iddiasına karşı önemli bir itiraz bulunmaktadır. Elde edilen anlamsal yapının asıl anlamsal yapıyı kodlamak için yeterli olmadığı iddia edilmektedir. Bu iddianın temelinde eş anlamlılık olgusu bulunmaktadır. Eş anlamlı ifadelerin yer değiştirme ilkesine uymadığı, bileşimsel olmayan anlambilimsel yapıların elde edilen bileşimsel yapılar tarafından temsil edilemeyeceği iddia edilmektedir. Bu çalışmada, Zadrozny’nin ispatı incelenerek, eş anlamlılık üzerinden yapılan itiraz cevaplandırılmaktadır. İtirazın konusu olan eş anlamlılığın, anlambilimsel olarak korunması gereken bir özellik olmadığı iddia edilmektedir.
According to the principle of compositionality, the meaning of any expression is determined by the meaning of its parts. The principle is based on Gottlob Frege’s analysis of linguistic meaning in the function-argument structure.1 The principle has taken a central role in formal semantics with Richard Montague.2 , 3 The principle of compositionality states that a homomorphism exists between syntactic structure and semantic structure. Let A be a set of expressions and F a set of syntactic operations. Similarly, let B be a set of meanings and G be a set of semantic operations. The principle of compositionality states a homomorphism to exist between the syntactic algebra, , and the semantic algebra, . Therefore, the meaning of an expression that results from applying the syntactic operation4 Fγ on expressions α1 , …, αn is computed as follows:5
Meaning (Fγ (α1 , …, αn )) = Gγ (meaning[α1 ), …, meaning(αn ))
The proponents of the principle claim that compositionality is the key mechanism behind the systematic and productive structure of languages.6 , 7 Contrary to this view, Wlodek Zadrozny has argued that the compositionality principle does not contribute to the systematic structure of language.8 To justify his claim, Zadrozny put forward proof showing that any semantic structure can be encoded compositionally. Against Zadrozny’s claim is the claim that the obtained compositional semantics cannot preserve the synonymity in the original semantics. This study examines Zadrozny’s proof and answers the objection made over synonymy.
In order to express Zadrozny’s proof, one first needs to construct an arbitrarily chosen semantic structure where:
M = an arbitrary set of meanings
A = an arbitrary alphabet
. = a binary operator
S = a set closure of A under the operator (.)
m = an arbitrary function from the set S to the set M
Zadrozny proposed the existence of a set M* and a unique μ function that is defined from S to M*, where μ has the following two properties:
1. ∀s ∀t μ (s.t) = μ(s) (μ(t))
2. ∀s μ(s) (s) = m(s)
The first property shows that for any two expressions s and t, the meaning of their syntactic combination can be computed through these two expressions’ meanings. The meaning of the expression s.t is the function application of s to t. Therefore, μ is a compositional semantic function. The second property shows that the original meaning function m can never recovered. The proof shows that compositionality is a trivial principle. Therefore, compositionality does not make any contribution to the systematicity of languages.
An important objection is found against Zadrozny’s claim, one that argues the μ function to be insufficient for encoding the original semantic structure. Therefore, although the μ function is a compositional one, the claim is made that it cannot encode the semantic structure of the original structure. The basis of this claim involves synonymity: Semantic structures that do not comply with the substitution principle cannot be encoded with the μ function.9 , 10, 11 One example of such a semantic structure is Frege’s referential semantics. According to referential semantics, the meanings of expressions are the objects to which they refer. For example, the expressions “the morning star is bright” and “the evening star is bright” refer to the same object:
m(“the morning star is bright”) = m(“the evening star is bright”)
However, when one of these two expressions replaces the other in any expression S, the object that S refers to may change.12
S1 : According to John, the morning star is bright.
S2 : According to John, the evening star is bright.
Although the meanings of these expressions “the morning star is bright” and “the evening star is bright” are the same, the references of the expressions S1 and S2 may be different. As a result, the compositional semantics need to assign different values to the expressions “the morning star is bright” and “the evening star is bright.” As such, it does not capture the synonymy relation in the original semantics.
This paper claims that this notion of synonymity is not a precise one but a pre-theoretical one. It does not need to be reflected in the lexical structure of the encoded semantics. What is expected from encoded semantics is to derive synonymity claims in the original semantics as theorems. In other words, what is expected is not the following identity:
μ (“morning star is bright”) = μ (“evening star is bright”)
Instead, what is expected is to represent the sameness of their meanings when they are used in extensional contexts.
