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A brief list of important functions in semantics

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Below, I will provide an account of the functions use in the chapter on semantics of Steinhart's book. Functions are extremely important, and the functions for specific procedures and concepts should be memorized. Before we delve into our list of functions, let's briefly go over Steinhart's definition of the reference function, of which each of the following is an example:

"Just as a language has a vocabulary it also has a reference function. The reference function maps each word in the vocabulary onto the object to which it refers. It maps the word onto its referent. A language can have many reference functions. Every competent user of some language has his or her own local reference function encoded in his or her brain. In any language community, these functions are very similar (if they weren't the members of that community couldn't communicate"(Steinhart, p. 86).

We would also do well to keep in mind Steinhart's understanding of the relation of a language's vocabulary to its reference function. Keeping in mind that each number associated with its word w is a subscript:

"A model for the language is an image of its vocabulary under its reference function. We can display a model by listing each term in a vocabulary on the left side and its referent on the right...More formally suppose our vocabulary is (w0, w1,...wn) where each wi is a word. The image of the vocabulary under the reference function f is (f(w0), f(w1).,,,f(wn)).

1) Nouns and adjectives - our formal definition for the reference functions of both nouns and adjectives are the same. "The reference function f maps each ADJ [or NOUN] onto a set (the extension of ADJ [or NOUN]"(Steinhart, p. 88). In each case, the extension of the set onto which the reference function is mapping the word is "the set of all things that are truly described by the adjective [or noun]"(Steinhart, p. 88).

3) Names - a name is a kind of noun. For specifically, it is a proper noun. For a name, "the reference function f maps the name "Mark Twain" onto the person Samuel Clemens"(Steinhart, p. 86). Written formally, Steinhart notes that this can be represented f("Mark Twain") = Samuel Clemens or "Mark Twain" ---> Samuel Clemens(Steinhart, p. 87).

4) Intensions

a) Intension of a word - The intension of a word is a function that associates every world with the referent of that word at that world"(Steinhart, p. 100). Thus, in Steinhart's example, keeping in mind that each number associated with a world w is a subscript, and letting IN represent "intension," IN("Allan") = {(w1, A), (w2, A), (w3, A), (w4, A)}, where "A" refers to Allan as existing in each of the worlds under consideration.

b) Intension of a sentence - In possible-world semantics, "the intension of a sentence [is] a function that associates every world with the truth-value of the sentence in that world"(Steinhart, p. 101). This is a form of "characteristic function" in which "the set of characteristic functions over W is { f | f: W --> {0, 1}}. Each of these characteristic functions is an intension"(Steinhart, p. 101).

5) Propositions - A proposition "is a function that associates each world with a truth-value. If a proposition associates a world w with 1, then it is true at w; if 0, it is false at w"(Steinhart, p. 101). Thus, using a variation of Steinhart's example, if Charlie is happy at world 1, but he is not happy at worlds 2, 3 and 4, then we have:

[Charlie is happy] = {(w1, 1), (w2, 0), (w3, 0), (w4, 0)}.

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009

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