Help with a proof of validity?
I have a simple question.
In constructing a proof of validity for a symbolic argument, and employing modus ponens/modus tollens, I'm not sure whether I'm allowed to assume the truth of the antecedent/falsity of the consequent in order to further the proof, potentially isolate a term for later use, etc. If not, I presume I would need to derive from other lines the conditional's antecedent or consequent, and then proceed with either M.P. or M.T.
Which is correct?
Here is an example argument (one of the exercises in Kalish's Techniques of Formal Reasoning).
P1: (D -> A) -> T
P2: F -> ~A
P3: W
P4: ~F -> (W -> D)
C: T
My proof of validity for that argument runs as follows...
1. D -> (A -> T) [Exposition, P1]
2. F -> ~A [P2]
3. ~~A / A [Assumption of falsity of consequent in 2]
4. ~F [M.T. 2,3]
5. ~F -> (W -> D) [P4)
6. W -> D [M.P. 4,5]
7. W [P3]
8. D [M.P. 6,7]
9. A -> T [M.P. 1,8]
10. T [M.P. 3,9]
I greatly appreciate any help, thank you.
In constructing a proof of validity for a symbolic argument, and employing modus ponens/modus tollens, I'm not sure whether I'm allowed to assume the truth of the antecedent/falsity of the consequent in order to further the proof, potentially isolate a term for later use, etc. If not, I presume I would need to derive from other lines the conditional's antecedent or consequent, and then proceed with either M.P. or M.T.
Which is correct?
Here is an example argument (one of the exercises in Kalish's Techniques of Formal Reasoning).
P1: (D -> A) -> T
P2: F -> ~A
P3: W
P4: ~F -> (W -> D)
C: T
My proof of validity for that argument runs as follows...
1. D -> (A -> T) [Exposition, P1]
2. F -> ~A [P2]
3. ~~A / A [Assumption of falsity of consequent in 2]
4. ~F [M.T. 2,3]
5. ~F -> (W -> D) [P4)
6. W -> D [M.P. 4,5]
7. W [P3]
8. D [M.P. 6,7]
9. A -> T [M.P. 1,8]
10. T [M.P. 3,9]
I greatly appreciate any help, thank you.
Comments (2)
About your proof, you can't just assume that a variable if true or false if you don't have this data. Actually, you can do this when both true and false values for the variable in question leads you to the same answer, meaning that the answer does not depend on this variable, like the following exemple:
In this exemple, both P and ¬P implies in Q, which mean that anything implies in Q. This is equivalent to (P^¬P) ? Q, which is always true. But this is not the case, because if A if true, it's conclusion isn't the same as it would be if A was false. If you choose values for a variable, you will only know what would be the result of that if this were to be really the truth value, but you don't know what it is.
I don't know if it answers your question. In any case, one thing I like to do is always begin by checking if the argument is invalid. If I can't make the premises true and the conclusion false, then I start using inference rules to prove the argument validity.
*Sorry for any grammar error, i haven't been practicing writing in english.
Consider it on a more intuitive level: "If it is the case that if the moon is blue, then cats bark, then roger is happy" creates much different conditions from "if the moon is blue, then it is the case that if cats bark, roger is happy".
More formally you can see the transformation you use does not preserve truth value under the following value assignments:
A D T | (A?D)?T | A?(D?T)
F F F F T
A rule of inference must preserve truth value regardless of truth assignment, so "exposition" can not be a rule of inference.