Exam 2 Flashcards
(28 cards)
Antecedent
The proposition to the left of the horseshoe is the antecedent
Argument Form
An arrangement of logical operators and statements and statement variables in which a consistent replacement of the statement variables by statements results in an argument.
Biconditional
A biconditional is only true when the component propositions share a truth value (both true or both false). Otherwise it is false.
Symbolized by the tribal =. This is generally used to capture the phrase “if and only if”.
Compound Statement
A statement that has at least one simple statement as a component. E.g. “It is hot and humid today.”
Conditional
A conditional is only false when the antecedent is true and the consequent is false. Otherwise, it is true.
Our fourth symbol is the hook or horseshoe ⊃ which is used to translate conditionals. The exception is the phrase “only if” which signals the consequent.
Conjuct
Each of the statements in a conjunction is called a conjunct. Many English words are symbolized using the dot ● including: and, while, but, yet, moreover, also, still, although, etc.
Conjunctions
A conjunction is only true when both conjuncts are true. Otherwise, it is false.
Our second symbol is the dot ● which is used to symbolize conjunctions
Consequent
The proposition to the right of the horseshoe is the consequent
Consistent Statements
Two or more statements that have at least one line on their respective truth tables where the main operators are true. Two or more statements that can both be true at the same time.
R ⋁ B and R ⋁ ~B are consistent
Contingent Statements
A statement that is sometimes true and sometimes false
E.g. A ⦁ B
Contradictory Statements
Two statements that have opposite truth values on every line of their respective truth tables
E.g. ~F ⋁ S and F ⦁ ~S are contradictory statements
Disjunct
Each of the propositions in a disjunction is called a disjunct.
Disjunction
A disjunction is only false when both disjuncts are false. Otherwise, it is true.
The third logical operator is the wedge ⋁ which is used to symbolize disjunctions. We use this to translate words and phrases like “or”, “unless”, and the phrase “either… or…”
Exclusive Disjunction
Both disjuncts cannot be true at the same time. “You will pass or fail this course.”
Inclusive Disjunction
Both disjuncts can be true at the same time. “Premiums will be waived in the event of injury or unemployment.”
- What we use
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). Two or more statements that can’t all be true at the same time (though they can all be false).
C ⦁ ~M and C ≣ M are inconsistent
Logical Equivalence
Two statements that generate identical truth tables.
E.g. ~(A ⦁ B) is logically equivalent to ~A ⋁ ~B
Logical Operator
Special symbols that are used to modify or combine simple statements.
Main Operator
The logical operator that has in its range the largest component or components in a compound statement.
Negation
Negation switches the truth value of the proposition it is applied to.
Symbolized by the tilde ~. We use this symbol to translate phrases like “not” or “it is not the case that…”
Self-Contradiction
A statement that is always false
E.g. P ⦁ ~P
Simple Statement
Statements that do not have any other statement as a component. E.g. “It is hot today.”
Tautology
A statement that is always true
E.g. P ⋁ ~P
Well-Formed Formula
In our system, we will call “grammatically” correct statements Well-Formed Formulas (WFF)
