2: Propositional Calculus: Language

Sentence-Functors

The sentence,

            Mary is English and John is Welsh

may be thought of as consisting of two constituent sentences

            Mary is English

            John is Welsh

together with the matrix,

            j and y.

A matrix, in this sense, is something with spaces marked with variables which accept appropriate fillers, and it yields something new when the spaces are appropriately filled. In the case of the matrix above, the variables j and y mark spaces which  accept declarative sentences (the constituent sentences); and when the variables are replaced by declarative sentences, the result is itself a declarative sentence. A matrix of this sort is a sentence functor.

A piece of terminology: "j and y" is a 2-place sentence-functor, because it contains two different variables (two different shaped spaces). "j and j", by contrast is a 1-place sentence functor. "j and y, or c;" is a 3-place sentence-functor. And so on. (If you run out of Greek letters, you can always add subscripts: j₂, for example.)

Here is a point to watch. Consider,

But it doesn't. "Mary took from the library" can indeed be used as a declarative sentence (as something which can be true or false) - as meaning Mary took things from the library. But that is not how it is being used in the sentence in question, which cannot be paraphrased as

            John read the book which Mary took things from the library.

In fact "j which y" is not a sentence-functor at all.

Similarly "John arrived when Mary left" does not consist of the declarative sentences "John arrived" and "Mary left" together with the sentence-functor, "j when y". To see this, notice that "John arrived", when used as a declarative sentence, can be paraphrased as "John arrived at a certain time"; and "Mary left", when used as a declarative sentence, can be paraphrased as "Mary left at a certain time". But "John arrived when Mary left" cannot be paraphrased as "John arrived at a certain time when Mary left at a certain time". Can "j when y" ever be used as a sentence-functor?