Truth-functional Equivalencies Flashcards
What are DeMorgan’s laws? (Which disjunctions are equivalent to which conjunctions?)
- (p.q) = -pV-q
- (pVq) = -p.-q
p. q = -(-pV-q)
pVq = -(-p.-q)
What are the laws of the material conditional? (Which conjunctions and disjunctions are equivalent to which conditionals?)
p->q = -(p.-q)
p->q = -pVq
pVq = -p->q
-(p.q) = p->-q
What is the export-import law?
p->(q->r) = p.q->r
pV(q.r) = (p->q).(q->p)
pVq->r = (p->r).(q->r)
What are the laws of contraposition? (Switching the consequent and antecedent)
p->q = -q -> -p p->-q = q -> -p -p->q = -q -> p
What is the law of simplification? (Elimination)
p.q implies p and it implies q
What is the law of the assertion of the consequent?
p implies q -> p
-p implies p -> q
What is the principle of modus ponendo ponens?
(p->q).p implies q
What is the principle of modus tollendo tollens?
(p->q).-q implies -p
What is the principle of modus ponendo tollens?
- (p.q).p implies -q
- (p.q).q implies -p
What is the principle of modus tollendo ponens (also called the principle of the disjunctive syllogism)?
(pVq).-p implies q
(pVq).-q implies p
What is the principle of indirect proof?
-p->p implies p
What is the principle of reductio ad absurdum?
p->-p implies -p
What is the principle of strengthening the antecedent?
p->q implies p.r -> q
What is the principle of weakening the consequent?
p->q implies p -> qVr
p->q implies p->(r->q)
What are the principles of constructive dilemma?
(pVq).(p->r).(q->s) implies rVs (pVq).(p->r).(q->r) implies r (pVq).(p->r) implies rVq (pVq).(q->r) implies pVr (p->q).(-p->r).(q->r) implies r
What are the principles of destructive dilemma?
- (p.q).(r->p).(s->q) implies -(r.s)
- (p.q).(r->p).(r->q) implies -r
- (p.q).(r->p) implies -(r.q)
- (p.q).(r->q) implies -(p.r)
What conjunctions are the material conditional equivalent to?
p->q = -(p.-q) p->-q = -(p.q)
What disjunctions are the material conditional equivalent to?
p->q = -pVq -p->q = pVq
When is a conditional false?
A conditional is only false when p is true and q is false.
When is a disjunction false?
A disjunction is only false when p and q are false.
When is a conjunction false?
A conjunction is always false unless p and q are true.
What is completeness?
If a system is complete, everything implied is deducible.
What is soundness?
If a system is sound, everything deducible is implied.
What is the law of addition?
p implies pVq and q implies pVq
What is consistency?
A system is consistent if you cannot deduce a contradiction.
What are the laws of the biconditional?
piffq = (p->q).(q->p)
piffq = -piff-q
piffq = -(p.-q).-(q.-p)
piffq = p.qV-p.-q
piff-q = -piffq
piff-q = -(piffq)
Which disjunctions and conjunctions are the negation of a conditional equivalent to?
- (p->q) = p.-q
- (p->q) = -(-pVq)
What is the law of Clayvius?
-p->q implies p
What are some things that the material conditional is equivalent to?
p->q = p->-q
p->q = q->-p
-p->q = -q->p
p->q = -(p.-q)
p->q = -pVq
What is the converse of a conditional?
The converse of a conditional switches the consequence and antecedent.
So, the converse of p->q is q->p.
What is the inverse of a conditional?
The inverse of a conditional negates the antecedent and consequent.
So, the inverse of p->q is -p->-q.
What is introduction?
Introduction allows me to put any schema into a disjunction. p implies pVq.
What is elimination?
Elimination allows me to separate conjuncts. p.q implies p. p.q also implies q.
What is transitivity?
An example of transitivity is, p->q and if q->r then p->r.
If something is implied by the consequent of a conditional, the antecedent of that conditional also implies it. Only conditionals are transitive.
What is associativity?
Something is associative when moving the parentheses does not change the truth value of the schema. Only conjunctions and disjunctions are associative.
An example of associativity is (p.q).r = p.(q.r)
What is commutativity?
A schema is commutative when the order of its elements does not matter. Only conjunctions, disjunctions, and biconditionals are commutative.
For example, p.q = q.p
How is a biconditional related to a conjunction?
A biconditional is the conjunction of two conditionals.
The schema is p->q.q->p.
How is a biconditional related to the converse of a conditional?
A biconditional is a conjunction of a conditional and its converse.
For example, p->q.q->p.
How is a biconditional related to the inverse of a conditional?
A biconditional is a conjunction of a conditional and its inverse.
For example, p->q.-p->-q
