phi_270_text_6_1

In our symbolic notation for predications other than equations, the predicate will come first followed by the individual terms that are its input. So we might begin an analysis of Bill introduced himself to Ann as follows:

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, …, τ_n takes the form

The use of the verb fit here is somewhat artificial. It provides a short verb that enables θτ₁…τ_n to be read as a sentence, and it is not too hard to understand it as saying that θ is true of τ₁, …, τ_n. Another artificial aspect of this notation is the unemphasized form an’ 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…τ_n 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 = and also for predicates, which we will consider in the next subsections, whose internal structure has been analyzed. For the time being, all terms will be single letters in our symbolic notation; but in the next section we will consider compound terms, so we will use the Greek letters τ, σ, and υ to stand for any terms, simple or compound.

If we continue the analysis of Bill introduced himself to Ann into fully symbolic form, we would get the following:

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)

(Aam ∨ Abm) → (Scr ∧ Lcor)
if either A fits a an’ m or A fits b an’ m then both S fits c an’ r and L fits c, o, an’ 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 like 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)

Ltn → (t = r → ¬ St)
if L fits t an’ 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 to 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. Notice also that identity does not appear in the key to the analysis. That is because it is part of the logical vocabulary; that is, it is like the connectives, which also do not appear in keys.

	`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`