Modus Ponens 2: More Translation Practice

Hot dog hot dog

Lesson:
1. Review validity.
2. Review translation.
3. Simple proofs.

Translation
Here's how we do a basic conditional translation:

E.g. 1, If you're happy and you know it then clap your hands. = H>C

First write down the first letter of a key word in the antecedent (e.g., 'H' from happy) then write down the symbol '>' and finally write down the first letter of a key word in the antecedent (e.g., 'C' from clap).

Order doesn't matter!
When we translate, the symbol for antecedent always goes first followed by '>', followed by the symbol the consequent.

E.g. 2, Clap your hands if you're happy and you know it. = H>C.

Notice that even though the words of the sentence are ordered differently from e.g., 1, we still symbolize it the same as e.g., 2. Doing so preserves the logical structure. That is, "If you're happy and you know it then clap your hand" and "Clap your hand if you're happy and you know it" both have the same logical structure. To see why, let's look at a similar example but with a different logical structure.

E.g. 3, If you clap your hands then you're happy and you know it. = C>H.
or
E.g. 4, You're happy and you know it if you clap your hands. = C>H.

Notice that e.g. 1 and 2 mean something different than e.g. 3 and 4. If we had translated e.g., 2 as C>H it would mean the same thing as e.g., 3.

Translation Practice
A.

1. If I were a Boy just for a day I'd Roll out of bed in the morning.
2. I'd be Coming right back home to you if I could Fly.
3. If I could Find a way I'd take back those Words that hurt you and you'd stay.
4. You shoulda put a Ring on it if you Liked it.
5. If I Eat too much I feel Tired. 
6. If you Lose yourself in the music, the moment then you Own it, you better never let it go. 
7. You only get one Shot, do not miss your chance to blow if this opportunity Comes once in a lifetime.

Modus Ponens and Modus Tollens

Modus Ponens
1. Modus ponens refers to particular underlying structure of an argument that uses a conditional (if-then statement) as one premise and the antecedent (the part following the 'if) of the conditional as the other premise. The conclusion is the consequent (the part following the 'then') of the conditional.

Otis Ponens

Here's an example:
P1. If [I put Money in the machine] then [I'll get a Snickers bar].
P2. [I put Money in the machine].
C.  [I got a Snickers bar]. 

We can symbolize the underlying structure using a capital letter to represent each clause.
1.  M>S ('>' means 'if___then___'
2. M
3. S

Modens Ponens is a valid argument form. This means that no matter what you substitute for the variables (even if they are false in the 'real' world) the argument is valid. Validity refers to the structure of an argument NOT its truth (i.e., soundness). We can define validity two different ways:

(a) An argument is valid if and only if if all its premises are true and it's impossible for its conclusion to be false.

(b) An argument is valid if and only if if all its premises are true then its conclusion must also be true. 

Modus Tollens
Modus tollens is like a modus ponens in reverse. It has the following structure: one premise is a condidtional and the other premise is the negation of the consequent. The conclusion is the negation of the antecedent. 

Here's an example:
P1. If [I put Money in the machine] then [I'll get a Snickers bar].
P2. [I don't have a Snickers bar].
C.  [I didn't put Money in the machine].

Symbolized, modus tollens looks like this:
1. M>S
2. ~S ('~' means 'not')
3. ~M

Proofs :
(Modus Ponens Only)

A. 

1. A>(B>C)
2. A
3. B    /C

B. 

1. (A>B)>(C>D)
2. A>B
3. C   /D

C. 

1. A>(B>(C>(D>(E>F))))
2. A
3. B
4. C
5. D
6. E   /F

D. 

1. (A>B)>(C>(D>E))
2. A>(A>B)
3. A
4. A>C   /D>E

E.

1. (A>(B>C))>(A>B)
2. A>(B>C)
3. A   /C

F. 

1. (P>Q)>(R>((R>Q)>P))
2. R
3. P>Q
4. R>Q
5. P>S   /S