## 31 Mar 2016

### Agler (2.1) Symbolic Logic: Syntax, Semantics, and Proof, "Truth Functions," summary

by Corry Shores
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[The following is summary. Boldface, underlining, 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.2: Language, Syntax, and Semantics

2.1 Truth Functions

Brief Summary:
We will examine the language of propositional logic (PL). Propositions with no truth functional operators are are atomic propositions. A propositional operator makes a proposition more complex. If it does so by combining propositions, then it is a propositional connective. Propositional operators are truth-functional if the value of the more complicated proposition they make is entirely dependent on the truth value of the component parts.

Summary

Ch.2: Language, Syntax, and Semantics

In this chapter, we will examine the formal language of propositional logic (PL).

2.1 Truth Functions

In English, propositions can be built up from propositional connectives like and, or, if ... then, and if and only if (25). Agler provides this definition:

Propositional connective: A propositional connective is a term (e.g., and, or, if ... then..., ... if and only if ...) that connects or joins propositions to create more complex propositions.
(26)

Agler then notes how we can modify sentences in ways that make more complex propositions but without connecting one proposition with another. He offers these examples:

(1) John went to the store.
(7) It is not the case that John went to the store.
(8) It is known that John went to the store.
(9) It is suspected that John went to the store.
(Agler 26)

This other kind of modification are called propositional operators:

Propositional operator: A propositional operator is a term (e.g., and, or, it is not the case that) that operates on propositions to create more complex propositions.
(26)

In this book, we are concerned mostly with propositional operators that are used truth-functionally. He explains:

A propositional operator is used truth-functionally insofar as the truth value of the complex proposition is entirely determined by the truth values of the propositions that compose it. Propositional operators that are used truth-functionally are called truth-functional operators.

Truth-functional operator: A truth-functional operator is a propositional operator (e.g., and, or, it is not the case that) that is used in a truth-functional way.
(26)

Agler now distinguishes two kinds of propositions. The first kind are atomic propositions.

Atomic proposition: An atomic proposition is a proposition without any truth-functional operators.
(26)

Atomic propositions do not have any logical operators. The second kind are complex propositions.

Complex proposition: A proposition is complex when it has at least one truth-functional operator.
(27)

We may understand how truth-functional operators work by noting how logical functions work: “In general, a function associates a value or values (known as the input) with another value (known as the output)” (27). The specific sort of value that concerns us is are truth values, and thus we are interested in truth-value functions.

Truth function: A truth function is a kind of function where the truth-value output is entirely determined by the truth-value input.
(28)

One example where “The truth value of the output proposition is entirely determined by the truth values of the input propositions” is conjunction (28).

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