Chapter 7 Flashcards
Logical operators:
Special symbols that are used to translate ordinary language statements.
Simple statement:
One that does not have any other statement or logical operator as a component.
Compound statement:
A statement that has at least one simple statement and at least one logical operator as components.
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.
Disjunction:
A compound statement that has two distinct statements (called disjuncts) connected by the wedge symbol.
Inclusive disjunction:
When we assert that at least one disjunct is true, and possibly both disjuncts are true. Given this, an inclusive disjunction is false when both disjuncts are false, otherwise it is true.
Exclusive disjunction:
When we assert that at least one disjunct is true, but not both. In other words, we assert that the truth of one excludes the truth of the other. Given this, an exclusive disjunction is true when only one of the disjuncts is true, otherwise it is false.
Conditional statement:
In ordinary language, the word “if” typically precedes the antecedent of a conditional statement, and the statement that follows the word “then” is referred to as the consequent.
Sufficient condition:
Whenever one event ensures that another event is realized.
Necessary condition:
Whenever one thing is essential, mandatory, or required in order for another thing to be realized
Biconditional:
A compound statement made up of two conditionals—one indicated by the word “if” and the other indicated by the phrase “only if.”
Sufficient condition:
Whenever one event ensures that another event is realized. In other words, the truth of the antecedent guarantees the truth of the consequent.
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.
Well-formed formula:
Any statement letter standing alone, or a compound statement such that an arrangement of operator symbols and statement letters results in a grammatically correct symbolic expression.
Scope:
The statement or statements that a logical operator governs.
Main operator:
The operator that has the entire well-formed formula in its scope.
Truth-functional proposition
The truth value of any compound proposition using one or more of the five operators is a function of (that is, uniquely determined by) the truth values of its component propositions.
Statement Variable:
A statement variable can stand for any statement, simple or compound.
Statement form:
In propositional logic, an arrangement of logical operators and statement variables such that a uniform substitution
Argument form:
In propositional logic, an argument form is an arrangement of logical operators and statement variables such that a uniform substitution of statements for the variables results in an argument.
Substitution instance:
A substitution instance of a statement occurs when a uniform substitution of statements for the variables results in a statement. A substitution instance of an argument occurs when a uniform substitution of statements for the variables results in an argument.
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.
Truth function:
The truth value of a compound proposition is a function of the truth values of its component statements and the logical operators
Statement variable:
A statement variable p, q, r, s …. can stand for any statement, simple or complex
Statement form:
A pattern of statement variables and logical operators
~ ( p v q )
Substitution Instance of a Statement:
Occurs when a uniform substitution of statements for the variables results in a statement.
Substitution Instance of an Argument.
Occurs when a uniform substitution of statements for the variables results in an argument.
Negation:
The truth table definition for negation shows for any statement p, ~ p will have the opposite truth value:
Conjunction:
The truth table definition for conjunction shows for any truth values for p, q, “p . q”
Disjunction
The truth table definition for (inclusive) disjunction shows for any truth values for p, q, “p v q” has the following truth values:
Inclusive disjunction:
at least one disjunct is true and possibly both. (We are using this one)
Exclusive disjunction:
at least one disjunct is true but not both.
Conditional
The truth table definition for the conditional shows for any truth values for p, q, “p ⊃ q”
Biconditional
The truth table definition for the biconditional shows for any truth values for p, q, “p ≡ q”
Truth-functional interpretations can obscure:
- Implied differences in ordinary language:
- Conditionals without inferential connection
- Counterfactuals with antecedents contrary to facts
Truth values assigned to every proposition:
Let R = true, S = false, and P = true
R ⊃ ( S · P )
Truth values not assigned to every proposition:
Let P = true, Q = unassigned
Order of operations:
The order of handling the logical operators within a truth-functional proposition; it is a step-by-step method of generating a complete truth table.
Contingent statements:
Statements that are neither necessarily true nor necessarily false (they are sometimes true, sometimes false).
Noncontingent statements:
Statements such that the truth values in the main operator column do not depend on the truth values of the component parts.
Tautology:
A statement that is necessarily true.
Self-contradiction:
A statement that is necessarily false.
Logically equivalent statements:
Two truth-functional statements that have identical truth tables under the main operator.
Contradictory statements:
Two statements that have opposite truth values under the main operator on every line of their respective truth tables.
Consistent statements:
Two (or more) statements that have at least one line on their respective truth tables where the main operators are true.
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
Contradictory Statements
Two statements that have opposite truth values on every line of their respective truth values
Consistent Statements
Statements that have at least one line on their respective truth tables where the main operators are both true.
Inconsistent Statements
Statements that do not have even one line on their respective truth tables where the main operators are both true.
Modus ponens:
A valid argument form (also referred to as affirming the antecedent).
Fallacy of affirming the consequent:
An invalid argument form; it is a formal fallacy.
Modus tollens:
A valid argument form (also referred to as denying the consequent).
Fallacy of denying the antecedent:
An invalid argument form; it is a formal fallacy.
