## 4 Apr 2016

### Agler (3.1) Symbolic Logic: Syntax, Semantics, and Proof, "Valuations (Truth-Value Assignments)", summary

by Corry Shores
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[The following is summary. Boldface and bracketed commentary are my own. 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.3: Truth Tables

3.1 Valuations (Truth-Value Assignments)

Brief Summary:
We can find the truth values of complex propositions on the basis of the value assignments for the atomic formula. We begin with the operators with the least scope and work progressively toward the main operator, whose truth value gives us that of the whole proposition.

Summary

Ch.3: Truth Tables

We will more fully articulate the semantics of the language of predicate logic (PL) by examining true-value assignment (valuation) and a mechanical method for determining truth values for complex formulations. Using this method, we will also determine if certain properties belong to propositions and arguments by means of a decision procedure (65).

3.1 Valuations (Truth-Value Assignments)

Agler writes:

The key semantic concept in PL is a valuation (or truth-value assignment). A valuation in PL is an assignment of a truth value (‘T’ or ‘F’) to a proposition.

Valuation: A valuation in PL is an assignment of a truth value (‘T’ or ‘F’) to a proposition.
(65)

Agler then notes two things about this definition. (a) “we stipulate that a valuation can only assign a value of ‘T’ or ‘F’ to a proposition” (65). But if we wanted, we could assign other values, such as ‘I’ for indeterminate (65). (b) Before we were writing

‘A’ is true

or

‘A’ is false

to state that proposition ‘A’ has a certain truth value (65d). What we write now will be formulations of this form:

v(A) = T

or

v(A) = F

These formulations say that ‘A’ is assigned the truth value of true (T), in the first case, and false (F) in the second (66). As we noted in section 2.3.3, we can use the formation rules to construct any proposition in PL. “In addition, the truth value of any complex proposition in PL is determined by the truth value of the propositional letters that make it up” (66).

Let us take an example sentence:

Mary is a zombie, and John is not a mutant.
(66)

We will translate it thus:

Z∧¬J

We will assume that the atomic formulas have these truth value assignments:

v(Z) = T
v(J) = F

To evaluate the sentence, Agler gives two steps in the procedure:

(1) Write the appropriate truth value underneath each proposition. |
(2) Starting with the truth-functional operator with the least scope and proceeding to the truth- functional operator with the most scope, use the appropriate truth-functional definition to determine the truth value of the complex proposition.
(66-67)

So we are working with this complex proposition.

We then begin with step 1 and write the truth values below the atomic formulas.

The truth operator with the least amount of scope is the negation. So we find its value.

Then we work onward to the operator with the next least scope, which in this case happens to be the main operator. So when we evaluate it, we also find the value for the whole complex proposition.

[Agler than gives another example, which further illustrates the procedure. See pp.67-68.]

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

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