# Is this proof using only modus ponens correct?

The mouse is either quick or slow. If it is quick, it will escape the cat. If it is slow, it will take the cheese. If it takes the cheese, it will not escape the cat.

A = “the mouse is slow”
B = “the mouse is quick”
C = “the mouse will escape the cat”
D = “the mouse will take the cheese”

In the language of propositional logic, and in terms of the atomic propositions A to D, write down an encoding for the compound propositions given in the text passage

From this My encoding is

A OR B
B IMPLIES C
A IMPLIES D
D IMPLIES NOT C

I then have to prove that the mouse will not escape the cat and I am given A as a fact. The only inference rule I can use is modus ponens

A is given as a fact so

A, A IMPLIES D INFERS D (MP)

So I now have D as a Fact and from this I can do the following

D,D IMPLIES NOT C INFERS NOT C (MP) Proving NOT C as a fact

What I would like to know is if I am calculating this correctly or if this is completely wrong?

Thanks

So you have $A\implies D$ and $A$ is true

$D$ is true (this is the modus ponens)

$D$ is true and $D\implies\neg C$

Therefore $\neg C$ is true (again using modus ponens)

QED.

Remember, modus ponens is simply the following $(P$ is true and $P\implies Q)\implies Q$ is true

• I have seen QED at the end of statements before but it has never been explained. Is it just used to state that the end of proof has been completed? – Robert199110 Aug 7 '15 at 22:06
• It is just latin for "thus it is proven" theres no real reason to use it. its mostly for if you're too lazy to write out "thus [insert statement] is true" – Elliot G Aug 7 '15 at 22:08