Elementary Logic Chapter 7 Propositional Logic
Biconditional
A compound statement consisting of two conditionals--one indicated by the word "if" and the other indicated by the phrase "only if." The triple bar symbol is used to translate a biconditional statement
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
substitution instance
occurs when a uniform substitution of statements for the variables results in a statement, or when a uniform substitution of statements for the variable results in the argument
simple statement
one that does not have any other statement or logical operator as a component
logical operators
special symbols that can be used as part of ordinary language statement translations
contingent statements
statements that are sometimes true or sometimes false
logically equivalent
statements that have the same truth values
consistent statements
Two (or more) statements that have at least one line on their respective truth tables where the main operators are true
necessary condition
Whenever one thing is essential, mandatory, or required in order for another thing to be realized. in other words, the falsity of the consequent ensures the falsity of the antecedent
Inclusive disjunction
a compound statement in which both disjuncts can be true at the same time
exclusive disjunction
a compound statement in which both disjuncts cannot be true at the same time
disjunction
a compound statement that has two distinct statements (called disjuncts) connected by the wedge symbol (V)
conjunction
a compound statement that has two distinct statements called conjuncts connected by the dot symbol (.)
Compound statement
a statement that has at least one simple statement and at least one logical operator components
tautology
a statement that is necessarily true
modus pones
a valid argument form, affirming the antecedent
Modus tollens
a valid argument form, denying the consequent
well formed formula (wwf)
an arrangement of operator symbols such that the resulting symbolic expressions are grammatically correct
Statement form
an arrangement of statement variables and operators such that the uniform substitution of statements in place of the variables results in a statement
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 simple components
Fallacy of Affirming the Consequent
an invalid argument form, it is a formal fallacy
fallacy of denying the antecedent
an invalid argument form, it is a formal fallacy
Statement variable
can stand for any statement, simple or compound
conditional statement
in ordinary language, the word "if" typically precedes the antecedent of a conditional and the statement that follows the word "then" is referred to as the consequent
Argument form
refers to the structure of an argument, not to its content.In propositional logic, argument form is an arrangement of logical operators and statement variables.
Propositional Logic
the basic components in propositional logic are statements
main operator
the operator that has the entire well-formed formula in its scope
order of operations
the order of handling the logical operators within a proposition, it is a step by step method of generating a complete truth table
scope
the statement or statements that a logical operator connects
Truth-functional proposition
the truth value of a compound proposition that uses one of the five logical operators can be determined solely on the basis of the truth value of its components
non-contingent statements
the truth values in the main operator column do not depend on the truth values of the components parts
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 (~)
contradictory statements
two statements that have opposite truth values on every line of their respective truth tables
sufficient condition
whenever one event ensures that the other event is realized. in other words, the truth of the antecedent guarantees the truth of the consequent