Conditional logic in abstract terms can be tricky. Hope this helps:
Premise 1: if S, ——> then T
Premise 2:
Conclusion: if /U/ ——> /S/
For the purposes of this explanation, the double forward slashes are another way of indicating “not.†Not just means the negation of something. Here is example in less abstract terms that might help:
Premise: if you’re a cat —-> then you’re a mammal
Contrapositive: if you’re / (not) a mammal/ ——> then you’re / (not) a cat/
That’s true in real life, right? If you’re not a mammal, then it must be true that you’re not a cat. You can be shark or a lizard, but if you’re not part of the mammal kingdom then I’m sorry to say, you’re certainly not part of any species of cat.
so:
Premise: cat —> mammal
Contrapositive: /mammal/ —> /cat/ (the slashes here mean “notâ€)
So, coming back up to our example written above, let’s consider the blank space one more time in this context.
Premise 1: if S, ——> then T
Premise 2:
Conclusion: if /U/ ——> /S/
Premise one tells us that if you’re an S, then it must be true you are a T. However, our conclusion starts with U. There’s no premise above with a U, so that’s the gap we have to bridge.
What if we add a new premise that connects T, the necessary condition of our first premise, to U? Then our conditional chain would look something like:
S—>T—>U
A conditional chain is just multiple individual conditional statements that all function together. Whether it’s written like this…
S—>T—>U
or like this…
S—>T T—>U
The logical force is the same. However, the first one is a little less cluttered and easier to read, when you get the hang of it!
If we do this, we end up with the below:
S—>T T—>U ——— /U/ —> /S/
You’re probably wondering where all the “nots†came from in that conclusion. Let’s break down the contrapositive to find out. Take a minute to think about what the contrapositive of S—>T—>U would be if we diagrammed it.
/U/—>/T/—>/S/
We get that from taking the contrapositive of each premise individually. Sooo…
Premise 1: S—> T Contrapositive of premise 1: /T/ —> /S/
Premise 2: T—>U Contrapositive of premise 2: /U/ —> /T/
The conclusion we are given is:
/U/ —> /S/
Now if you look at premise 2’s contrapositive, you’ll see it connects to premise 1’s contrapositive. /U/—>/T/—>/S/
Our right answer here, or missing premise, is the contrapositive of premise 2.
This is one way of getting there that I think helps for understanding how to bridge gaps and use contrapositives to do that. However, there is another way I’ll show you now.
Our original question is:
Premise 1: if S, ——> then T
Premise 2: _____?______
Conclusion: if /U/ ——> /S/
What if we tried taking the contrapositive of our conclusion to find the answer?
Conclusion: /U/—>/S/ Contrapositive: S—>U
S—>U, huh? Does that look familiar to something we did above?
Think back to:
S—>T—>U
If you were going to conclude that Premise 1 and Premise 2 lead to the conclusion:
Premise 1: S ——> T Premise 2: Conclusion: S—>U
You would need to add another premise that said T—>U. Because if we know that S —> T, and T—> U, then we know that S —> U, which is the contrapositive of our original conclusion /U/ - >/S/.
Because the contrapositive of a conditional statement is logically identical to the original conditional statement (Think if you’re a cat, then you’re a mammal. And if you’re not a mammal, then you must not be a cat) contrapositives are useful ways of figuring out conditional logic questions such as these.
