Implication & Equivalence Rules

Implication Rules I

100

p > q / q > r

Using the hypothetical syllogism (HS) I'd get "p > r"

200

p > q / ~q

I'd use Modus Tollens (MT) to get ~p.

300

p

I'd use addition (Add) to get "p v q"

400

p * q

I'd use Simplification (Simp) to get p

500

(p > q) * (r > s) / p v r

I would use the constructive dilemma to get "q v s"

Equivalence Rules I

100

~~p

I would use Double Negation (DN) to get "p"

200

p v p

I would use Tautology (Taut) to get "p"

300

p * (q v r)

I would use Distribution (Dist) to get (p * q) v (p * r)

400

~p v q

I would use Material Implication (Impl) to get "p > q"

500

(p * q) v (~q * ~p)

I would use Material Equivalence (Equiv) to get p <-> q

Implication Rules II

100

(b v r) v q / ~q

I would use Disjunctive Syllogism (DS) to get "(b v r)"

200

(n * (p > r)) > (w <--> r) / ~(w <--> r)

I would use Modus Tollens (MT) to get ~(n * (p > r))

300

((t v p) > (n * r)) * ((q v b) > (p > m)) / (t v p) v (q v b)

I would use Constructive Dilemma (CD) to get (n * r) v (p > m)

400

p > q / p > r (Put them together)

I would use Conjunction (Conj) to get (p > q) * (p > r)

500

(p v (t > (q * (s v h)))) > r / q > (p v (t > (q * (s v h))))

I would use Hypothetical Syllogism (HS) to get (q > r)

Equivalence Rules II

100

((a * b) > (c v r)) * ((c v r) > (a * b))

We could use Material Equivalence (Equiv) to get (a * b) <--> (c v r)

200

~((v > (r v b)) * s)

We could use DeMorgan's Rule (DM) to get ~(v > (r v b)) v ~s

300

(h * j) > ((p v r) > (s <--> t))

Use Exportation (Exp) to get ((h * j) * (p v r)) > (s <--> t)

400

2 steps--get me to "p"; 3 steps if you use double-negation rule: (~p > p)

Use Material Implication (Impl) to get (p v p), followed by Tautology (Taut) to get p.

500

3 steps: Get to (p <--> q); use conjunction for the middle step. p > q / ~p > ~q

use Transposition (Trans) to get (q > p), and use Conjunction to get [(p > q) * (q > p)]. Material Equivalence (Equiv) will get you (p <--> q).

English Please

100

It's not the case that I'm both teaching you and playing with my son.

Using DeMorgan's (DM), I am either not teaching or not playing with my son.

200

You are either texting in class or scratching your private area. You are not scratching your private area. What then are you doing?

Using Disjunctive Syllogism (DS), you are texting in class.

300

If we get a drastic weather change, I get a migraine. If I get a migraine, I get overly spacey.

Using Hypothetical Syllogism (HS), we'd get: "If we get a drastic weather change, I get overly spacey."

400

If we have a son, we'll name him Arthur. If we have a daughter, we'll name her Lucy. We will have a son or daughter. Therefore...

Using the Constructive Dilemma, we'd get "We'll name the child Arthur or Lucy."

500

I'm teaching logic, and I'm either helping or further confusing my students.

Using Distribution (Dist), we'd get "I'm teaching logic and helping my students, or I'm teaching logic or further confusing my students."