We move beyond truth-functional logic by recognizing complete expressions other than sentences and operations other than connectives. Our additions are motivated by a traditional description of grammatical subjects and predicates. The new complete expressions are individual terms, whose function is to name. Given this idea, we can define a predicate as an operation that forms a sentence from one or more individual terms.

A predicate corresponds to an English sentence with blanks that might be filled by terms. These blanks are the predicate’s places and the operation of filling them is predication.

We will maintain something analogous to truth-functionality by requiring that predicates be extensional. This means that all places of a predicate must be referentially transparent (rather than referentially opaque): when judging the truth value of a sentence formed by the predicate, we must be able see through the terms filling these places to what those terms refer to. Thus, just as a connective expresses a truth function, a predicate expresses a function that takes reference values as input and issues truth values as output. Such a function may be called a property if it has one place and a relation if it has 2 or more. In symbolic notation, it takes the form σ = τ and, in English notation, it takes the form σ is τ.

While recognizing quite a variety of non-logical vocabulary in our analyses, we recognize only one new item of logical vocabulary, the predicate identity. This is a 2-place predicate that forms an equation, which is true when its component terms have the same reference value.

In our symbolic notation, we use lower case letters to stand for unanalyzed individual terms, the equal sign for identity, and capital letters to stand for non-logical predicates. Non-logical predicates, both capital letters and predicate abstracts are written in front of the terms they apply to (with a predicate abstract enclosed in brackets), and = is written between the terms to which it applies. In English notation, predications other than equations are written as θ fits (series) τ₁, …, an’ τ_n.

In addition to proper names, the individual terms include definite descriptions and various non-anaphoric pronouns. They do not include certain other noun phrases, quantifier phrases in particular. We will speak of the person, place, thing, or idea referred to by an individual term by using such words as object, entity, individual, and thing, understanding these to apply to anything that might be named. Common nouns are also not individual terms. Indeed, they may be labeled general terms to distinguish their function of indicating a class of objects from the function of individual terms, also called singular terms, which is to refer to a single individual in a definite way. The word term will often be used as shorthand for individual term.

A functor is an operation that takes one or more individual terms as input and yields an individual term as output. Just like other operations, it expresses a reference function, which yields reference values when applied to reference values. Although a reference function is a particular sort of function we will use that term primarily for reference functions. The operation of combining a functor with input is application, and the individual term that is the output is a compound term. For any functor, there will almost always be some terms for which the application of the functor yields an undefined term. Although this problem can be reduced by limiting the input of functors to objects of certain types, we will not include this complication in our account of logical forms.

It can be difficult to recognize the individual terms that fill the places of a predicate or a functor. It is important in include in a definite description all the modifiers that are part of it. Some of these may be prepositional phrases or relative clauses which follow the common noun. In some cases, a prepositional phrase in this position might either be part of a definite or modify a verb; but such an ambiguity cannot arise with relative clauses so a prepositional phrase can be made into a relative clause in order to test what it modifies. Relative clauses must therefore be part of the definite description when they are restrictive; on the other hand, non-restrictive clauses (the sort set off by commas) are analyzed using conjunction.