Thus, a characteristic such as “warm” (conventionally symbolized by a capital letter W) may be predicated of some singular subject, for example, a dish—symbolized by a small letter d, often called the “argument.”
The resulting statement is “This dish is warm”; i.e., Wd.
Using ∼ to symbolize “not,” the denial ∼Wd can also be predicated.
If that of which “warm” is predicated is indefinite, a blank may be left for the predicate, W—, or the variable x may be employed, Wx, thus producing the propositional function “x is warm” instead of a definite proposition.
By quantifying the function by (∀x), meaning “For every x … ,” or by (∃x), meaning “There is an x such that … ,” it is transformed into a proposition again, either general or particular instead of singular, which predicates warmness (or its negation) of several or many subjects of a kind.
Philosophers have long debated what predicates really are.
In the early Middle Ages, they were usually treated as having a being beyond all linguistic and mental entities and thus were viewed as metaphysical.
Garland the Computist, the author of an early system of logic, however, viewed predication as mere utterance (vox).
Peter Abelard, the foremost dialectician of the 12th century, amended this view to include significatio as well as vox.
Logicians have long distinguished the existential statement “x is” from the predicational statement “x is Y.”
Franz Brentano, a precursor of Phenomenology prior to World War I, argued that they are both existential, that “x is Y ” means “xY is”; e.g., “Some fish have four eyes” means “Four-eyed fish exist.”
An exactly opposite approach was taken by Alexander Bain, a Scottish philosopher and psychologist, who held that all existential statements have complex subjects from which a predicate can be extracted.
The limitations of predication as a logical form are increasingly evident.
The predicate logic is now seen to be but one species of the logic of terms—the others being the logic of classes, the logic of relations, and the logic of identity; and the entire logic of terms, in turn, is distinct from the propositional logic, which deals with whole or unanalyzed statements.
In the logic of relations, it is even questionable whether there is any predicate at all, since all of the terms can be regarded as subjects on the same footing (as in “Jane is the sister of Edith is the sister of Rachel”).
Moreover, logics that distribute the predicate (with the quantifiers “all,” “some,” etc.) have also been explored.