Logic Midterm 3

Negatation

Unary, truth functional connective, tilde, means not, opposite truth value, t then f on true table column

Conjunction

Dot, means and, means but, means although, means despite, means even though, means moreover, means nevertheless, means still, true when both conjuncts are true

Disjunction

Means or, vee, means unless, means either…or…, means unless, means alternatively, false when both disjuncts are false

Conditional

Means if then, arrow, means it has an antecedent and consequent, false when antecedent is true and consequent is false

Biconditional

Means if and only if, double arrow, means just in case, means equivalent to, means both a sufficient and necessary condition, true when both bijuncts are the same true value

WFF Rules

Capital letters, correct # of (), correctly placed connectives, main logical operator

2^n

n represents # of atomic statements aka capital letters

2 atomic statement truth table

TT, TF, FT, FF

3 atomic statement truth table

Column 1: TTTT, FFFF Column 2: TT,FF,TT,FF Column 3: TFTFTFTF

Invalid truth tables

all true premises w/ false conclusion

Tautology

true on every row

Contradiction

false on every row

Contingent

true on some rows and false on others

Logically equivalent

all premises have same truth value on every row

Logically contradictory

all premises have diff truth values on every row, valid truth table

Logically consistent

all premises have same truth value on 1 or 1+ rows

Logically inconsistent

all premises are not all true on even a single row, valid but unsound

A is sufficient for B

A → B

A is necessary for B

B → A

A is necessary but not sufficient for B

(B → A) * ~ (A → B)

A is sufficient but not necessary for B

(A → B) * ~ (B → A)

Modus Ponens (valid)

Premises: A, A → B Conclusion: B

Affirming the consequent (invalid)

Premise: A→B, B Conclusion: A

Denying the antecedent (invalid)

Premises: A → B, ~ A Conclusion: ~B

Categories

Tautology, contradictions, and contingent

Relationships:

Equivalence, contradictoriness, consistency, and inconsistency