If V, then W. If W and X, then Y or Z. Not Y. Not Z. How to deduce the conclusion?

Question

Prescript: I quoted the standard form of argument because its use of colours improves readability.

Source: 6 minutes 01 seconds juncture; Lecture 13, Video 5 (transcription inexistent);
MITx: 24.00x Introduction to Philosophy; by MIT Associate Prof Caspar Hare PhD (Princeton)

Hereafter, imagine and pretend that the conclusion were unseen and unstated. To wit, pretend that you were given only P1 to P4, from which you yourself must discover and derive the conclusion.

Because the antecedent of P2 is positive, but P3 and P4 are negations, the key idea is seemingly to apply modus tollens to P2:

5. ¬(Y or Z) ⟹ ¬(W and X).

Then apply DeMorgan's Law to distribute the negator (ie ¬):

6. ¬Y and ¬Z ⟹ ¬W or ¬X.

Now what must be done? Please explain and show all steps and thought processes.

Answer 1

So far you have argued :

We are given not(Y) and not(Z) as P3 and P4.

As you note, applying this and modus tollens to P2 gives us not(W and X), or equivalently not(W) or not(X).

[note : there may be some confusion here since you are writing an implication symbol rather than a symbol for logical equivalence. Formally, you should write each statement as two consecutive statements on separate lines, noting modus tollens and De Morgan in the margin.]

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Now lets consider what we wish to prove.

We wish to show : if V then not(X).

It is enough to show that if V is true, then not(X) must be true.

Assume V. Then by P1 we have W.

By your own reasoning restated above, we know not(W) or not(X) is true.

But since W is true, this means that not(X) must be true.

Therefore, if V then not(X).

It should be easy for you to formalise this argument.

Answer 2

Your approach (as completed by Nick) does work, but personally I would have just first assumed V (because we're trying to derive an if-then statement) and then further assumed X (because it's the negative of the desired consequent).

V gives you W. You now have W and X (because you assumed X) which gives you Y or Z. Either arm of Y or Z yields a contradiction (depending on what approach you use, it doesn't necessarily need to be the same contradiction). Therefore Not X. Therefore If V then Not X.

It really only differs in the order of the steps, but to me this is both more intuitive and more in line with a standard general approach (to establish an if-then, assume the antecedent, and then further assume the negative of the consequent).

Answer 3

Personally, I would apply propositional resolution, because I recognise this kind of problem. However, this is arguably a less thoughtful and educational route than constructing a linear proof. I'll include it here anyway, in case you were interested in how you might tell a computer to solve this problem.

The steps are:

1. Convert sentences to conjunctive normal form (CNF)
2. Apply the resolution rule

Step 1: Convert P1, P2 to CNF (P3, P4 are already in CNF).

Step 2: Apply the Resolution Rule

Complementary literals (e.g. W, ~W, Y, ~Y, Z, ~Z) simply cancel out. The remaining literals ~V, ~X are ORed together. We are left with ~V OR X.

The line above the conclusion means "logically entails".

I personally enjoy this method because it is simple, structured and requires few words.