These exercises test your ability to apply the rules of inference to a set of premises. Here, we are given the following premises:
(1) C -> ~E (2) D -> M (3) ~D -> E
Let's start with identifying common variables. Premise (2) and (3) have D and ~D. We can use transposition on one of these premises to convert both into D or ~D. Let's say we use transposition on premise (2):
(4) ~M -> ~D (2) transposition
Now we can take the resulting premise (4) and combine it with premise (3) into the following chain:
~M ->~D -> E, which is equivalent to:
(5) ~M -> E
Now let's take the resulting premise (5) and see if we can make an inferences from premise (1). We can use transposition on premise (1) so that both premise (1) and (5) have a common element E:
(6) E -> ~C (1) transposition
Now we can combine the remaining premises (5) and (6) into the following chain:
~M -> E -> ~C , which is equivalent to:
(7) ~M -> ~C
We can again use transposition to the resulting premise to conclude that:
(8) Therefore, C -> M (7) transposition
mamieOctober 1, 2019
Hi Irina, that's super helpful! I am wondering why you chose premise (2) instead of premise (3)? Because it was more straightforward without the "not D" perhaps? Would you have ended up with the same thing if you had chosen premise (3)?
I am assuming that transposition is the same as taking the contrapositive, and I am wondering why it was necessary to take the contrapositive? Could you have not taken the common variable as
D -> M or would that not work?
When you combined (2) and (3) to make the chain not M -> not D -> E
How did you know that the E was as the end of the chain?
Finally, why did you take the contrapositive of premise (1)? because premise (5) says not m -> E and you need premise (1) to say E -> not C instead of C -> not E to match?
IrinaOctober 1, 2019
@mamie,
Great question. Transposition is the same as taking a contrapositive, transposition is just a formal name for it.
1. I randomly picked premise (2), you could do either one and end up with the same conclusion. If you take contrapositive of (3), we'll end up with this statement:
(4) ~E -> D
Then we would combine it with (2):
~E -> D -> M or ~E-> M
And finally combine it with premise (1) to conclude: C -> ~E -> M or C-> M
As it turns out, picking premise (3) first would lead us to the conclusion in fewer steps but starting with either one would lead to the same result.
2. The reason we have to take a contrapositive because one of the variables is negated, so in the initial premises we have D & ~D (2) (3) or E & ~E (1) (3). We can only make inferences if both variables are the same - DD or ~D~D in this specific scenario.
3. It is one of the rules of inference, known as hypothetical syllogism (HS). It tells us that when we have two or more statements such as: A-> B B-> C We can infer that A-> C .
In this case, we have two premises: ~M -> ~D ~D -> E
Since ~D is a common variable that allows us to connect these premises, we can infer that ~ M -> E.
4. That's correct, premise (1) says C -> ~E, I cannot make any inferences unless both premises have E E or ~E ~E in them.
I know formal logic rules are fairly abstract and can be tricky to apply, so please let me know if you need any further clarification. Also, it looks my original message got cut off, it is supposed to say "the resulting premise to conclude that:
