Logic Chapter 7
truth table
an arrangement of truth values for a truth-functional compound proposition that displays for every possible case how the truth value of the proposition is determined by the truth values of its single components
exclusive disjunction
an exclusive disjunction is where both disjuncts cannot be true at the same time
inclusive disjunction
an inclusive disjunction is where both disjuncts can be true at the same time
main operator
the operator that has in its range the largest component or components in a compound statement
Rule 1 of a WFF
1) the dot, wedge, horseshoe, and triple bar symbols must go between two statements
Rule 2 of a WFF
2) the tilde goes in front of the statement it is meant to negate
biconditional
a compound statement consisting of two conditionals-- one indicated by the world "if" and the other indicated by the phrase "only if". the triple bar symbol is used to translate a biconditional statement.
modus ponens
a valid argument form (also referred to as affirming the antecedent)
simple statement
one that does not have any other statement as a component
logical operators
special symbols that can be used as part of ordinary language statement translations
noncontingent statements
statements such that the truth values in the main operator column do not depend on the truth values of the component parts
contingent statements
statements that are neither necessarily true nor necessarily false (they are sometimes true, sometimes false)
order of operations
the order of handling the logical operators within a proposition; it is a step-by-step method of generation a complete truth table
rule 3 of a WFF
3) the tilde cannot, by itself, go between two statements
rule 4 of a WFF
4) parentheses, brackets, and braces are required in order to eliminate ambiguity in a complex statement
conditional
in ordinary language, the word "if" typically precedes the antecedent of a conditional. the horseshoe symbol is used to translate a conditional statement. if=antecedent. only if=consequent
fallacy of denying the antecedent
invalid argument form, it is a formal fallacy
negation
(~) the word "not" and the phrase "it is not the case that" are used to deny the statement that follows them, and we refer to their use as negation
conjunction
a compound statement that has two distinct statements (called conjuncts) connected by the dot (.) symbol. and, but still, moreover, while, however, also, although, yet.
disjunction
a compound statement that has two distinct statements (called disjuncts) connected by the wedge (v) symbol
statement form
a pattern of statement variables and logical operators
compound statement
a statement that has at least one simple statement as a component
self-contradiction
a statement that is necessarily false (p . ~p)
tautology
a statement that is necessarily true (p v ~p)
statement variable
a statement variable can stand for any statement, simple or complex
modus tollens
a valid argument form (denying the consequent)
argument form
an arrangement of logical operators and statement variables in which a consistent replacement of the statement variables by statements results in an argument
fallacy of affirming the consequent
an invalid argument form; it is a formal fallacy
well-formed formulas
compound statements forms that are grammatically correct
propositional logic
the basic components in propositional logic are statements
truth function
the truth value os a truth-functional compound proposition is determined by the truth values of its components and the definitions of the logical operators involved. any truth-functional compound proposition that can be determined in this manner is said to be a truth function.
inconsistent statements
two (or more) statements that do not have even one line on their respective truth tables where the main operators are true (but they can be false) at the same time.
consistent statements
two (or more) statements that have at least one line on their respective truth tables where the main operators are true
contradictory statements
two statements that have opposite truth values on every line of their respective truth tables
logically equivalent
two truth-functional statements may appear different but have identical truth tables. when this occurs, they are logically equivalent