Editor’s note: This is a guest post from Prof. Dr. Jürgen Lenerz from University of Cologne
Statements are propositions which are true with respect to a certain possible world, the discourse world. We have already discussed the concept of the “world’” in a previous post: “What is the world?”. What does that mean for the concept of truth?
Typically, a proposition is a combination of a subject and a predicate. The relation between both is called the predication. That means that there is supposed to exist an individual which is denoted by the subject and which is then said to possess the property which is expressed by the predicate. This, in fact, is the very core of human language. (No other animal can, to our present knowledge, do this. Non-human animals may express warnings or directives, but not propositions or statements! Even if there isn’t a subject-individual in the world, language seems to force us to assume a non-referential subject-expression like in It is raining.)
When we say that an individual x has the property P, this means that the individual x is an element of the set of all (and only) those individuals which share (and thus define) the property P. Typical (1-place-) properties are given by words such as be red, be sleeping, snores, be a horse etc. (Horses are in fact individuals, and all members of the set of horses are individuals (this horse, a horse, many horses etc.) which all share the property of being horses. Why we distinguish verbs, adjectives and nouns and what their properties are, is a long and winding story, the end of which we do not yet know.) There are also 2- and 3-place properties, which we will return to later.)
The core idea of predicate logic is that predicates are functions which apply to individuals. So, the predicate (being a) horse is represented as the function HORSE (x). This function, if applied to an individual like Leo (HORSE (Leo)), is TRUE for our discourse world if Leo indeed is a horse in our discourse world, FALSE otherwise. Thus, with respect to the discourse world, the predicate HORSE (x) yields the truth value TRUE (or “1″) for all the individuals which are horses, FALSE (or “0″) for all other individuals. The set of individuals which are horses may be different in different worlds. But in general it holds that the predicate HORSE (x) is a function from individuals to truth values. This is what it means when we talk about modern formal semantics as “truth functional semantics”.
If we know the meaning of the word horse, we are able to determine for any given individual in any possible world if it belongs to the set of horses or not.
“Truth” is here conceived of as a correspondence between propositions and (situations in) the (discourse) world. This view was developed by Tarsky. There are other conceptions of truth, of course, but they are not relevant for truth functional semantics.
Prof. Dr. Jürgen Lenerz, born 1945, was a professor for German Linguistics at the University of Cologne from 1985 until his retirement in 2011. His main interests are in the interaction of syntax, semantics, intonation and information structure in natural languages