For my logic final we were given the following argument in which we needed to show the proof for using only the rules of inference and replacement. I’ve worked on this proof for days and I can’t get it and it is driving me nuts. I was wondering if anyone could figure it out for me.

Here’s the argument that we were given:

- XZ
- YZ // (XY)Z.

And as a reminder, here are the rules of inference:

Modus Ponens (MP)

pq

p____

q

Modus Tollens (MT)

pq

~q____

~p

Pure Hypothetical Syllogism (HS)

pq

qr__

pr

Disjunctive Syllogism (DS)

pq

~p___

q

Constructive Dilemma (CD)

(pq)*(rs)

pr___

qs

Conjunction (Conj

p

q__

p*q

Simplification (Simp)

p*q

p

Addition (Add)

p___

pq

And here are the rules of replacement:

De Morgan’s Rule (DM

~(p*q) :: (~q~q)
~(pq) :: (~p*~q)

Commutativity (Com)

(pq) :: (qp)

(p*q) :: (q*q)

Associativity (Assoc)

[(pq)r] :: [p(qr)]

[p*(q*r)] :: [(p*q)*r]

Distribution (Dist)

[p*(qr)] :: [(p*q)(p*r)]

[p(q*r)] :: [(pq)*(pr)]

Double Negation (DN)

p :: ~~p

Transposition (Trans)

(pq) :: (~q~p)

Material Implication (Impl)

(pq) :: (~p~q)

Material Equivalence (Equiv)

(pq) :: [(pq)*(qp)]
(pq) :: [(p*q)(~p*~q)]

Exportation (Exp)

[(p*q)r] :: [p(qr)]

Tautology (Taut)

p :: (pp)

p :: (p*p)

Of course I found a link to this as I was near done writing out all of the stoopid code that would’ve saved me a whole lot of time

http://www.mathpath.org/proof/proof.inference.htm