PCT 5e C6 – DEDUCTIVE REASONING – PROPOSITIONAL LOGIC

CHAPTER SUMMARY

Connectives and Truth Values

- In propositional logic we use symbols to stand for the

relationships between statements—that is, to indicate the form of an

argument. These relationships are made possible by logical connectives

such as conjunction (and), disjunction (or), negation (not), and

conditional (If…then…). Connectives are used in compound statements, each

of which is composed of at least two simple statements. A statement is a

sentence that can be either true or false. - To indicate the possible truth values of statements and arguments,

we can construct truth tables, a graphic way of displaying all the truth

value possibilities. - A conjunction is false if at least one of its statement components

(conjuncts) is false. A disjunction is still true even if one of its

component statements (disjuncts) is false. A negation is the denial of a

statement. The negation of any statement changes the statement’s truth

value to its contradictory (false to true and true to false). A

conditional statement is false in only one situation—when the antecedent

is true and the consequent is false.

Checking for Validity

- The use of truth tables to determine the validity of an argument

is based on the fact that it’s impossible for a valid argument to have

true premises and a false conclusion. A basic truth table consists of two

or more guide columns listing all the truth value possibilities, followed

by a column for each premise and the conclusion. We can add other columns

to help us determine the truth values of components of the argument. - You can check the validity of arguments not only with truth tables

but also with the short method. In this procedure we try to discover if

there is a way to make the conclusion false and the premises true by

assigning various truth values to the argument’s components.

Proof of Validity

- The method

of proof

is a way to confirm the validity of an argument by deducing its conclusion

from its premises using simple, valid argument forms. Most valid complex

arguments consist of several of these valid sub-arguments (most of which

you may already know). Determining the validity of the larger argument

then is a matter of moving step by step from premises to conclusion,

identifying the valid, component arguments along the way.

- The method of proof

uses nine rules of inference and nine rules of replacement. By properly

applying them, you can confirm an argument’s validity.