A problem that we did in class last week left me with an
interesting question. There seemed to be
two ways of translating an argument into symbolic form, and I was wondering if
they were logically equivalent. It seemed
to me that, according to language of the argument, there was an overall
compound conditional, the antecedent of which was a conjunction of two
statements, and the consequent was itself a conditional statement:
(A&B)è(CèD)
But some people wanted to read it as:
[(A&B)&C]èD
It seems to me that the two formulations are actually logically
equivalent, but I was wondering if anyone could give a reason why they would
not be. I suppose I could easily look
this up, but I thought it would be an interesting blog topic.
Well, in the second one, A, B, and C are all necessary to imply D; if you don't have C for example you won't have D.
ReplyDeleteA: Rain requires water.
B: Requires cold front
C; Requires clouds.
D: Rain.
If there is not C, it is difficult to get D. And same goes for the first one, A and B are both necessary to imply the following. I believe that's correct.
I don't believe the second formulation leads to logical equivalence - it is true that both will lead to D - if you have A and B and C, then you will have D in either instance. But that is not the only aspect we are considering. The first considers is the relationship between (A&B) and (if C then D), whereas the consequent of the second formulation is merely D. Again, same final product, but you can for instance achieve the consequent of the second formulation by a conjunction of A&B&C, which is not possible with the first formulation.
ReplyDeleteIf A & B, infer that C implies D
ReplyDeleteIf A & B & C , infer D
Are you being careful to distinguish inference from implication?
ReplyDelete