6.1.5. Analyzing predications
Apart from the special case of equations, our symbolic notation for predications will identify the predicate first followed by a list of the individual terms that are its input. This is a departure from English word order in most cases, but we can present analyses in this way even before we introduce symbols. The example below presents the analysis of a predication into a predicate and individual terms as a series of steps.
| Bill introduced himself to Ann | |
| Identify (referentially transparent) occurrences of individual terms within the sentence, making sure they are all independent by replacing pronouns by their antecedents | Bill introduced Bill to Ann |
| Separate the terms from the rest of the sentence |
Bill introduced Bill to Ann
Bill introduced Bill to Ann |
| Preserve the order of the terms, and form a predicate from the remainder of the sentence |
[ _ introduced _ to _ ] Bill Bill Ann
[ _ introduced _ to _ ] Bill Bill Ann |
| Write the terms in the places of the predicate | [ _ introduced _ to _ ] Bill Bill Ann |
Underlining will often be used, as it is here, to mark the places of predicates when they are filled by English expressions. In examples and answers to exercises, we will move directly from the second of these steps to the last, so the process can be thought of as one of removing terms, placing them (in order and with any repetitions) after the sentence they are removed from, and enclosing sentence-with-blanks in brackets.
In general, an application of an n-place predicate θ to a series of n individual terms τ1, …, τn takes the form
and our English notation is this:
fits (series) τ1, …, ən τn
The use of the verb fit here is somewhat artificial. It provides a short verb that enables θτ1…τn to be read as a sentence, and it is not too hard to understand it as saying that θ is true of τ1, …, τn. Another artificial aspect of this notation is the unemphasized form ən of and, which is designed to distinguish the use of and here to join the terms of a relation from its use as a truth-functional connective. The role of the term series, which will rarely be needed, is discussed in 6.1.7. We will use the general notation θτ1…τn when we wish to speak of all predications, so we will take it to apply to equations, too, even though the predicate = is written between the two terms to which it is applied.
In our fully symbolic analyses, unanalyzed non-logical predicates will be abbreviated by capital letters. This is consistent with our use of capital letters for unanalyzed sentences since predicates have sentences as their output. When we add non-logical operations that yield individual terms as output, they will be abbreviated by lower case letters just as unanalyzed individual terms are.
As was done in the display above, we will use the Greek letters θ, π, μ, and ρ to refer to stand for any predicates, so they may stand for single letters and for =. The may also stand for complex predicates whose internal structure has been analyzed, something we will go on to consider in 6.2.1. We will also go on to consider compound terms, and we will use the Greek letters τ, σ, and υ to stand for any terms, simple or compound.
If we complete the analysis of Bill introduced himself to Ann, carrying it into fully symbolic form and restating it in English notation, we would get the following:
fits b, b, ən a
Notice that the bracketed English sentence-with-blanks does not appear in the final analysis, but it does appear in the key.
When sentences contain truth-functional structure, that structure should be analyzed first; an analysis into predicates and individual terms should begin only when no further analysis by connectives is possible. Here is an example:
If either Ann or Bill was at the meeting, then Carol has seen the report and will call you about it
Either Ann or Bill was at the meeting → Carol has seen the report and will call you about it
(Ann was at the meeting
∨ Bill was at the meeting)
→ (Carol has seen the report
∧ Carol will call you about the report)
([ _ was at _ ] Ann the meeting
∨ [ _ was at _ ] Bill the meeting)
→ ([ _ has seen _ ] Carol the report
∧ [ _ will call _ about _ ] Carol you the report)
if either A fits a ən m or A fits b ən m then both S fits c ən r and L fits c, o, ən r
A: [ _ was at _ ]; L: [ _ will call _ about _ ]; S: [ _ has seen _ ]; a: Ann; b: Bill; c: Carol; m: the meeting; o: you; r: the report
When analyzing atomic sentences into predicates and terms, be sure to watch for repetitions of predicates from one atomic sentence to another—such as that of [ _ was at _ ] in this example. Such repetitions are an important part of the logical structure of the sentence.
Since the notation for identity is different from that used for non-logical predicates, you need to watch for atomic sentences that count as equations. These will usually, but not always, be marked by some form of the verb to be but, of course, forms of to be have other uses, too. Consider the following example:
If Tom was told of the nomination, then if he was the winner he wasn’t surprised
Tom was told of the nomination → if Tom was the winner he wasn’t surprised
Tom was told of the nomination → (Tom was the winner → Tom wasn’t surprised)
Tom was told of the nomination → (Tom was the winner → ¬ Tom was surprised)
[ _ was told of _ ] Tom the nomination
→ (Tom = the winner → ¬ [ _ was surprised] Tom)
if L fits t ən n then if t is r then not S fits t
L: [ _ was told of _ ]; S: [ _ was surprised]; t: Tom; n: the nomination
It is fairly safe to assume that a form of to be joining two individual terms indicates an equation, but it is wise to always think about what is being said: an equation is a sentence that says its component individual terms have the same reference value. A use of to be joining noun phrases will indicate an equation only when these noun phrases are individual terms; the conditions under which that is so are discussed in the next subsection. Finally, notice that no identity predicate should appear in the key to the analysis. That is because it is part of the logical vocabulary; as such, it is like the connectives, which also do not appear in keys.