Philosophy structure definition

Advancements in mathematical logic in late nineteenth century proved to have an important impact on the concept of structure as it was applied within philosophy. First order logic offered philosophers with a rich formal language in which to represent the logical forms of various entities, ranging from sentences to scientific theories, thus allowing reasoning involving these entities to be represented within a logic.

In the philosophy of language, the importance of the concept of structure can be seen in the importance attributed to the representation of sentences by their logical form. Such representation was suggested by Bertrand Russell in his classic paper "On Denoting," which proved to have an enormous impact on the subsequent practice of philosophy.

Russell's paper was motivated by concerns regarding the truth values of sentences that contain empty definite descriptions. As an example, consider the following sentence:

(1) The present king of France is bald.

The truth or falsity of this sentence would appear to depend upon the properties of the referent of the definite description `the present king of France.' The logical structure of the sentence appears to be that of

(2) Bk

where 'B' is a once place predicate signifying baldness and 'k' is a constant referring to the present King of France. But of course France does not have a King, and so the referent of 'k' is null. If (2) is the logical form of (1) it thus follows that the truth value of (1) is undecided. This means that if we propose to determine the meaning of a sentence in terms of its truth conditions, the sentence is meaningless. And this then results in a number of problems, discussed at length by Russell in his paper.

In "On Denoting," Russell offers a way to avoid these problems through the assignment of meanings to sentences such as (1) via the the assignment of logical forms to these sentences that differ from that of (2). That is, Russell proposes that the logical form of (1) is not indicated by its surface grammar. Instead, Russell proposes that the logical form of (1), and of other sentences that include definite descriptions, is instead

(3) For all x (Kx and For all y (Ky then y = x) and Bx)

where 'K' and 'B' are one place predicates stating that an entity is the King of France and is bald, respectively, and 'For all y (Ky then y = x)' states that any individual that satisfies the 'K' predicate is unique (as required of a definite description). According to Russell, the logical form of (1) thus differs dramatically from the surface grammar of the sentence. Rather than simply assume the existence of an individual 'k', the sentence instead takes the form of an existence claim. And as there is no individual that satisfies this claim, the sentence is thus false.

Within philosophy, the structure of an entity can thus be thought of as its logical form. This form offers a distillation of the fundamental semantic properties of the entity, determining both its meaning and the role that it plays in reasoning.

This concept of logical form found within philosophy thus appears to resemble quite closely the concept of structure proposed by Koppel, who suggested that the structure of a sequence, S, is a function, F, such that, when F is input particular data D,

(4) F(D) = S.

Considering (4) in the context of the above discussion, we can see that (3) plays the role of 'F', the interpretations of 'K' and 'B', and the specification of the domain of quantification in (3) play the role of 'D', and 'S' plays the role of (1).

The above discussion also offers some indication of the function of structure determinations, at least in philosophy.