LOGIC SYMBOLISM HELP!

Question

Consider the following translation scheme:Use symbols to translate

A: The American team will win a gold medal.
B: The British team will win a gold medal.
C: The Canadian team will win a gold medal.


1. All of the teams will win a gold medal.
2. Not all of the teams will win a gold medal.
3. At least two of the teams will win a gold medal.
4. At most two of the teams will win a gold medal.
5. If any of the teams wins a gold medal, all of them will.

Answer 1

I know this is an old question, but it is an interesting one. The way it is phrased suggests that this is propositional (sentential) logic rather than predicate, so the answers I provide are using that framework.

 

When engaging in this process, we want to use symbolic logic to express the intended meaning of the statement (not necessarily the precise grammar). I'm going to use the following conventions:

 

∧ = and operator

∨  = or operator

~ = not operator

⊃ = conditional operator

≡ = biconditional operator (not needed here)

 

1. All of the teams will win a gold medal. This claim is satisfied if each of the propositions is true.

 

(A ∧ (B ∧ C))  

 

2. Not all of the teams will win a gold medal. This is simply the negation (not) of the above solution.

 

~(A ∧ (B ∧ C))

 

3. At least two of the teams will win a gold medal. This is a little more complicated to symbolise in propositional logic, but this is one way to do it. Basically, if any two teams win, then the following statement is true.

 

(A ∧ B) ∨ (A ∧ C) ∨ (B ∧ C)

 

4. At most two of the teams will win a gold medal. Frequently the answer is found in thinking about the mirror image of a statement. In this case, the "mirror" is to say that at least one team will lose.

 

(~A ∨ ~B) ∨ ~C

 

DeMorgan's rule of replacement also suggests this form:

 

~(A ∧ B) ∨ ~C

 

5. If any of the teams wins a gold medal, all of them will.

 

((A ∨ B) ∨ C) ⊃ (A ∧ (B ∧ C))