## 21 May 2016

### Agler (6.1) Symbolic Logic: Syntax, Semantics, and Proof, "The Expressive Power of Predicate Logic", summary

by Corry Shores
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[The following is summary. Boldface (except for metavariables) and bracketed commentary are my own. Please forgive my typos, as proofreading is incomplete. I highly recommend Agler’s excellent book. It is one of the best introductions to logic I have come across.]

Summary of

David W. Agler

Symbolic Logic: Syntax, Semantics, and Proof

Ch.6: Predicate Language, Syntax, and Semantics

6.1 The Expressive Power of Predicate Logic

Brief summary:

While everything in the language of propositional logic (PL) can be expressed in English, not everything in English can be expressed in PL. In PL, propositions are treated as whole units (symbolized as singular letters) without regard to logical properties internal to the sentences, as for example between subject and predicate and with respect to quantification. Thus we will examine a more expressive language of predicate logic (RL), which is a logic of relations.

Summary

We previously examined the language, syntax, and semantics of propositional logic (PL). Agler notes now two of PL’s strengths, namely, that we may apply it to English and there are decision procedures for testing for such logical properties as validity, consistency and so on.

The language, syntax, and semantics of PL have two strengths. First, logical properties applicable to arguments and sets of propositions have a corresponding applicability in English. So, if one is dealing with a valid argument in PL, then that argument is also valid for English. Second, the semantic properties of arguments and sets of propositions have decision procedures. That is, there are mechanical procedures for testing whether any argument is valid, whether propositions in a set have some logical property (e.g., they are consistent, equivalent, etc.), and whether any proposition is always true (a tautology), always false (a contradiction), or neither always true nor always false (a contingency).

(Agler 247)

But what is PL’s weakness? [It seems that the idea we noted above is that whatever is expressed in PL can be expressed in English. However, the idea now seems to be, not everything that we can express in English can be expressed in PL. We see this with quantification. When we use quantifiers in English, like “all” and “some”, the way such sentences relate can have certain logical properties that are not evident in PL. For, in PL we would not modify any proposition with a quantifier. All sentences, regardless of whether they have quantifiers – and if they do, regardless of which quantifier they have – are treated generically as units, symbolized by singular propositional letters. So inferences based on sentences using quantifiers can appear valid in PL, but were we to examine the sentences themselves, our intuition could tell us the inferences are invalid. What is needed, then, is a language that expresses the logical properties that a) hold between the parts within a proposition and b) are relevant to the logical properties a proposition might itself have or have in relation to others.] PL’s weakness is that it does not express certain kinds of internal logical properties of certain sentences. Instead we will discuss a more expressive language that takes into account the subjects and predicates of sentences, which is called the language of predicate logic, abbreviated RL, because it is a logic of relations.

The weakness of PL is that it is not expressive enough. That is, some valid arguments and semantic relationships in English cannot be expressed in propositional logic. Consider the following example:

All humans are mortal.

Socrates is a human.

Therefore Socrates is a mortal.

This argument is clearly valid in English but cannot be expressed as a valid argument in PL. Symbolically, the argument is represented as follows:

M

S

R

The above argument is clearly invalid. In order to bring English arguments like the one above into the domain of symbolic logic, it is necessary to develop a formal language that does not symbolize sentences as wholes (e.g., John is tall as ‘J’), but symbolizes parts of sentences. That is, a formal language whose basic unit is not a | complete sentence but the subject(s) and predicate(s) of the sentence such a language will be more expressive and able to represent the above argument as valid. This is the language of predicate logic (sometimes called the logic of relations). We’ll symbolize it as RL.

(Agler 247-248)

Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.