## 2 Apr 2016

### Suppes (3.3) Introduction to Logic, “Predicates,” summary

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Summary of

Patrick Suppes

Introduction to Logic

Ch. 3 Symbolizing Everyday Language

§3.3 Predicates

Brief summary:
Predication can be expressed as a relation. In this text, we use lower-case letters for variables and upper-case letters for relations. We would write “For every x, x is red or x is not red” as: For every x, Rx ∨ –Rx. Predicates can be either one-place, two-place, or n-place, depending on the number of terms the predicate relates.

Summary

[Recall from the last section that we had the example sentence, “Everything is either red or not red”. We then used variables to write, “For every x, x is red or x is not red”. If we used other symbols for disjunction and negation, we might also write: “For every x, x is red ∨ x is –red”.] Last time we used lowercase letters for variables. We can also use capital letters for predicates. So if R can stand for “is red,” then we can write our example sentence:

For every x, Rx ∨ –Rx.
(45)

Here, –Rx is just the negation of the formula Rx, and thus means, “x is not red” (46).

In traditional grammar, we the predicate is the part of the sentence where we say something about the subject. But in our symbolic logical system, we do not have all the grammatical parts we have in English, like adverbs and so on. Our predicate will simply be a letter. And also, in logic, predicates have a broader role than they have in natural language. Suppes gives an example. In logic we do not distinguish common nouns. When we work with them, we need to use a relation. [The idea seems to be that to signify a common noun, we need to speak of a variable that has some property.]

Consider, for instance, the sentence:

(1) Every man is an animal.

We translate this:

(2) For every x, if x is a man then x is an animal.

In ordinary grammar ‘is an animal’ is the predicate of (1) . Its translation (2), has the additional predicate ‘is a man’ which replaces the common noun ‘man’ in (1). Using ‘M’ for the predicate ‘is a man’ and ‘A’ for the predicate ‘is an animal’, we may then symbolize (1):

(3) For every x, Mx Ax.

Sentence (3) in fact exemplifies the standard form for sentences of the type ‘ Every such and such is so and so’.
(Suppes 46)

Predicates can be one-place. Let e = Emma, k = Mr. Knightly, G stand for “was very gay”, and O stand for “was considerably older than”.  Now take the sentence “Emma was very gay”, which we can render as, Ge. This of course is a one-place predicate. Instead consider, “Mr. Knightly was considerably older than Emma.” We can write it, Oke. This would be a two-place predicate. There can be as many places as you want. An example of a three-place predicate could be one that expresses one thing being between two others. And,

An example of a four-place predicate is suggested by the relation of two objects being the same distance apart as two other objects. (Euclidean geometry, by the way, can be axiomatized in terms of just these two notions of betweenness and equidistance.)
(47)

When translating from English, if we have a compound predicate, we can choose to join two relations with a conjunction, or we can express both predicates with one relation. So consider:

Emma was very gay and beautiful
(47)

We can on the one hand say that B is “was beautiful” and write:

Ge & Be

Or, we can just make W be “was very gay and beautiful” and write,

We

What we choose depends on “questions of convenience and context” (47)

Suppes, Patrick. Introduction to Logic. New York: Van Nostrand Reinhold / Litton Educational, 1957.

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