oldfag here - what the hell is that symbol before the Y in the first part?

>>13975459
it means the negation of y (NOT y)

I assume you've already learned some of the inference rules used in logic. From the four premises, here is the standard logical derivation of the conclusion D:

1. ¬Y
2. Z→Y
3. X∨Z
4. X→D
5. ¬Z (1,2 Modus Tollens)
6. X (3,5 Disjunctive Syllogism)
7. D (4,6 Modus Ponens)

If you got stuck working that out, then either you don't know the inference rules well enough or got stuck trying to figure out how to get started. This example involves simple enough applications of the rules that knowledge of the rules alone should make it pretty clear, but a general strategy that often works for logical proofs is to work backwards, since we know our goal (to derive D). We can reason as follows:

In premise 4 we can see our goal D. In this premise it's the consequent of an implication: X→D. If we could derive X, then modus ponens would immediately allow us to conclude D.

So now we try to derive X, since we know how to get from there to D. Premise 3 tells us something about X: either X or Z (X∨Z). From that we now know that if Z is false, then X must be true (in order for X∨Z to still be true). This reasoning is the basis of the rule of disjunctive syllogism. But can we indeed derive ¬Z?

Yes, from premises 1 and 2 we can conclude ¬Z immediately by modus tollens.

This reasoning must now be assembled in the proper order, which is what the question in your image is trying to ensure you understand.

Ask if any further clarifications are needed, or if you don't understand one of the derivation rules.

>>13975476
shit. when did they get rid of the "!" for logical negation?

! makes sense

>>13975677
depends on literature i think. our books were old and they only used ¬ as negation, so may be an old used symbol