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Extensional semantics of nouns, verbs and adjectives

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In this article, we will examine what is known as formal semantics. "Formal semantics - the kind we're doing here - says that every word is a kind of name"(Steinhart, p. 86).

Languages have a lists of words called "vocabularies." Basically, a vocabulary is just a set of words. Steinhart defines a vocabulary formally as a series (w0, w1,...wn) "where each w1 [although the variables and numbers in the original text are subscripts] is a word"(Steinhart, p. 86). Some words are proper nouns or "names." These names refer to things. Nouns, more broadly speaking, refer to objects. These objects, as Steinhart notes, may be individuals or sets.

In addition to its vocabulary, a language has a reference function(Steinhart, p. 86).

"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 local functions are very similar (if they weren't, the members of that ocmmunity couldn't communicate). For the the sake of simplicity, we'll assume that all language users agree entirely on their reference functions"(Steinhart, p. 86).

Let's review. Vocabularies are lists of words. Each name in a vocabulary refers to a thing. "...the American writer Samuel Clements used the pen name "Mark Twain". So the reference function f maps the name "Mark Twain" onto the person Samuel Clemens. We can display this several ways:

"Mark Twain" refers to Samuel Clemens;

the referent of "Mark Twain" - Samuel Clemens;

f("Mark Twain") = Samuel Clemens;

"Mark Twain" --> Samuel Clemens"(Steinhart, pp. 86-87).

Ordinarily, we think of a noun as a person, place or a thing. This is fine so far as it goes, but we're going to need a more technically specialized understanding of a noun for our study of formal semantics. A common noun, for our purposes,

"refers to some one thing shared in common by all things named by that noun. You point to Rover and say "dog"; you point to Fido and say "dog"; so "dog" refer sto what they have in common. generally, "dog" refers to what all dogs have in common. Of course, you could point to arbitrary things and repeat the same name; but that name wouldn't be useful for communication - there could be no agreement about what it means, since the things it refers to have nothing in common. It would be a nonsense name. One hypothesis about commonality is that what things of the same tpye share inc ommon is membership in a single set. For example, what all dogs share in common is that every dog is a member of the set of dogs"(Steinhart, p. 87).

We follow Steinhart in assuming that "what all Ns have in common is membership in the set of things named by N"(Steinhart, p. 87). When we do this, "we can let the common noun N refer to that set"(Steinhart, p. 87). More specifically, "The noun N is a name that refers to the set of all Ns"(Steinhart, p. 87). Steinhart explains: "Thus f maps "man" onto the set of all men...The set of all things that are named by N is the extension of N. So every common noun refers to its extension"(Steinhart, p. 87).

In this study, we will likewise follow Steinhart in using all capital letters to designate the part of speech being used. NOUN, in his example, refers to a common noun. He articulates it formally:

"the referent NOUN = the set containing each x such that x is a NOUN;

the referent of "man" = the set containing each x such that x is a man;

f("man") = {x | x is a man}"(Steinhart, p. 87).

Now let's look at adjectives. "An adjective, like a common noun, refers to something shared by all the things shared by that adjective"(Steinhart, p. 87). Steinhart's example is that of red. ""...red" refers to what all red things have in common. An adjective refers to the set of all things that are truly described by the adjective"(Steinhart, pp. 87-88). Just as the reference function of the noun maps each noun onto its respective set, which is constituted by the extension of the noun (for example, the function of nouns maps "Junior" onto the set of all dogs), so also the reference function of the adjective maps each of its referents onto the set "of all things that are truly described by the adjective"(Steinhart p. 88).

Using a variation of Steinhart's example, suppose someone loves someone else. Felipe likes Gordon Clark and Sunny likes Junior. The pairs (Felipe, Gordon Clark) and (Sunny, Junior). These pairs have in common the relation of "love." "We can use this idea to define extensions for verbs. Verbs are relational terms. The extension of a verb is the set of all tuples of things that stand in that relation"(Steinhart, p. 88).

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

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