Truth-functional Equivalencies

Question 1

What are DeMorgan’s laws? (Which disjunctions are equivalent to which conjunctions?)

Answer 1

• (p.q) = -pV-q
• (pVq) = -p.-q
  p. q = -(-pV-q)

pVq = -(-p.-q)

Question 2

What are the laws of the material conditional? (Which conjunctions and disjunctions are equivalent to which conditionals?)

Answer 2

p->q = -(p.-q)

p->q = -pVq

pVq = -p->q

-(p.q) = p->-q

Question 3

What is the export-import law?

Answer 3

p->(q->r) = p.q->r

pV(q.r) = (p->q).(q->p)

pVq->r = (p->r).(q->r)

Question 4

What are the laws of contraposition? (Switching the consequent and antecedent)

Answer 4

p->q = -q -> -p
p->-q = q -> -p
-p->q = -q -> p

Question 5

What is the law of simplification? (Elimination)

Answer 5

p.q implies p and it implies q

Question 6

What is the law of the assertion of the consequent?

Answer 6

p implies q -> p

-p implies p -> q

Question 7

What is the principle of modus ponendo ponens?

Answer 7

(p->q).p implies q

Question 8

What is the principle of modus tollendo tollens?

Answer 8

(p->q).-q implies -p

Question 9

What is the principle of modus ponendo tollens?

Answer 9

• (p.q).p implies -q

- (p.q).q implies -p

Question 10

What is the principle of modus tollendo ponens (also called the principle of the disjunctive syllogism)?

Answer 10

(pVq).-p implies q

(pVq).-q implies p

Question 11

What is the principle of indirect proof?

Answer 11

-p->p implies p

Question 12

What is the principle of reductio ad absurdum?

Answer 12

p->-p implies -p

Question 13

What is the principle of strengthening the antecedent?

Answer 13

p->q implies p.r -> q

Question 14

What is the principle of weakening the consequent?

Answer 14

p->q implies p -> qVr

p->q implies p->(r->q)

Question 15

What are the principles of constructive dilemma?

Answer 15

(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

Question 16

What are the principles of destructive dilemma?

Answer 16

• (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)

Question 17

What conjunctions are the material conditional equivalent to?

Answer 17

p->q = -(p.-q)
p->-q = -(p.q)

Question 18

What disjunctions are the material conditional equivalent to?

Answer 18

p->q = -pVq
-p->q = pVq

Question 19

When is a conditional false?

Answer 19

A conditional is only false when p is true and q is false.

Question 20

When is a disjunction false?

Answer 20

A disjunction is only false when p and q are false.

Question 21

When is a conjunction false?

Answer 21

A conjunction is always false unless p and q are true.

Question 22

What is completeness?

Answer 22

If a system is complete, everything implied is deducible.

Question 23

What is soundness?

Answer 23

If a system is sound, everything deducible is implied.

Question 24

What is the law of addition?

Answer 24

p implies pVq and q implies pVq

Question 25

What is consistency?

Answer 25

A system is consistent if you cannot deduce a contradiction.

Question 26

What are the laws of the biconditional?

Answer 26

piffq = (p->q).(q->p)

piffq = -piff-q

piffq = -(p.-q).-(q.-p)

piffq = p.qV-p.-q

piff-q = -piffq

piff-q = -(piffq)

Question 27

Which disjunctions and conjunctions are the negation of a conditional equivalent to?

Answer 27

• (p->q) = p.-q

- (p->q) = -(-pVq)

Question 28

What is the law of Clayvius?

Answer 28

-p->q implies p

Question 29

What are some things that the material conditional is equivalent to?

Answer 29

p->q = p->-q

p->q = q->-p

-p->q = -q->p

p->q = -(p.-q)

p->q = -pVq

Question 30

What is the converse of a conditional?

Answer 30

The converse of a conditional switches the consequence and antecedent.
So, the converse of p->q is q->p.

Question 31

What is the inverse of a conditional?

Answer 31

The inverse of a conditional negates the antecedent and consequent.
So, the inverse of p->q is -p->-q.

Question 32

What is introduction?

Answer 32

Introduction allows me to put any schema into a disjunction. p implies pVq.

Question 33

What is elimination?

Answer 33

Elimination allows me to separate conjuncts. p.q implies p. p.q also implies q.

Question 34

What is transitivity?

Answer 34

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.

Question 35

What is associativity?

Answer 35

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)

Question 36

What is commutativity?

Answer 36

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

Question 37

How is a biconditional related to a conjunction?

Answer 37

A biconditional is the conjunction of two conditionals.

The schema is p->q.q->p.

Question 38

How is a biconditional related to the converse of a conditional?

Answer 38

A biconditional is a conjunction of a conditional and its converse.
For example, p->q.q->p.

Question 39

How is a biconditional related to the inverse of a conditional?

Answer 39

A biconditional is a conjunction of a conditional and its inverse.
For example, p->q.-p->-q