Philosophers of language often assume that our understanding of language resides in our having an implicit semantic theory that enables us to compose sentences whose meanings are constructed out of the meanings of their component words. To account for the fact that we infer more from one another's utterances than is strictly said, consider these two examples:
Our lecturer was sober today or
The department chair has not yet been sent to prison.
They invoke a secondary device.
This device, also implicitly understood by us, is called implicature [26], and is based upon rules that are supposed to govern what we say and when we say it. In the former cited example, the words sober and today retain their fixed meanings; the fact that we infer that sobriety is not her usual state is accounted for by maxims of conversational propriety, not by some variability in the meanings of the sentence-elements. In particular, the so-called logical words and, or, not, are supposed to have fixed meanings that can be specified in truth-tables. These set out the conditions under which sentences containing them are true or false. In fact, the logical vocabulary of natural language has long been supposed to provide a sort of truth-conditional bedrock upon which a full semantic theory can eventually be built.
Even introductory logic texts say things that are in direct contradiction to the intuitive notion that words have single meanings, at least about the word or, even those written by eminent logicians, such as Tarski [59]. He says:
The word or in everyday language, possesses at least two different meanings. -Tarski, 1941, p.21-
This is an exclusive/inclusive distinction:
In the so-called non-exclusive sense, the disjunction of two sentences is true if at least one of the sentences is true...When people use or in the exclusive sense to combine two sentences, they are asserting that one of the sentences is true and the other false. -Suppes[58], 1957, p.5-6-
Almost invariably, the logic texts that make this point go on to claim that one of the meanings of or coincides with that of the familiar xor function. If we are all supposed to be possessed of a semantic theory, this is a very curious fact, for a string of sentences composed with xor will be true if and only if an odd number of its component sentences are true. So a sentence of the form A xor B xor C will be true if exactly one of its component sentences is true, but it will be true also if they all are. So it seems that our semantic theory is inconsistent. We can test our own semantic theory on the following sentence, adapted from a common type of example in the textbooks. Consider:
You can have soup or you can have juice.
Is the quoted sentence an inclusive or an exclusive disjunction? Most undergraduate logic students, asked this question, will respond that it is an exclusive disjunction, and most textbook authors will either agree or will argue that since it would not be false if you were allowed to have both, it must be an inclusive disjunction. Which is correct? In fact neither is, since the sentence is not a disjunction at all. If a waiter said this to you, you would be correct in inferring from what he said that you could have soup; you would also be correct in inferring that you could have juice. It cannot, therefore, be a disjunction, since from a disjunction neither disjunct can be correctly inferred. It must in fact be a kind of conjunction. If, as the customer might assume, it is taken to exclude one or the other starters, then it is being taken as the conjunction:
You can have soup; you can have juice; you cannot have both soup and juice.
We could add numerous independent uses to the list of uses of the word or, none of which is adequately represented by the disjunctive truth-functor. These conflicting prejudices - that or has just one meaning, that or has two meanings, and so on - and the confidence about our possession of an accessible semantic theory appear to have become prevalent only since the invention of the truth-table in the twentieth century. Earlier logical theorists were not so confident of their understanding. Venn [60], famous eponym of the diagrams, himself confessed bewilderment at ``the laxity, the combined redundancy and deficiency, of our common vocabulary (and and or)''. Our use of so-called "logical language" does not in general rest upon any underlying semantic theory, accessible or inaccessible, and our various uses of the word or do not rest upon an implicit understanding of truth-functions. Given that this is so, any explicitly formulated semantic theory must be regarded as suspect. The fact is, a semantic theory is not required in the transmission of language. Our daily use of language is not driven by an underlying semantic theory. Indeed, children use language long before they are introduced to any sort of linguistic theory and are able to use it adequately. In fact, the transmission of language requires very little theoretical understanding. If it did, languages would not change beyond recognition within so short a span as a thousand years.
But, we may ask, if the transmission of language does not require much understanding, even of the logical vocabulary, how does a language come to have any logical vocabulary at all? It is partly because so little understanding is necessary for the transmission of language that languages acquire the vocabulary that we think of as logical- likewise, the vocabulary that we think of as psychological, or ethical, or religious. In short, for certain kinds of vocabulary, we may say that in using it, we literally do not know what we are talking about. Its use does not require that we do. How is this possible?