Jack, logical conditioning is dealt in great detail in Future Logic,part III.

That said, the form If/not-then is nothing very new really. In ordinary discourse, we say commonly things like "if so and so, it does not follow that such and such". If X, not-then Y is stated in modern logic as simply the negation of if X then Y, i.e. as not(X->Y), i.e. X does not imply Y.

Keep in mind that for every form we need a counter form (though there are very rare exceptions, as I recall).

(Jack, please help Fred if you can. As I said, I am in the middle of a difficult bit of writing. It is difficult for me to turn my attention elsewhere. Many thanks.)

Fred, another word of advice if I may. It is better for you to present your proposed solution to the problem and then have that corrected by others, than to ask others to do the thinking for you. One learns best from one's errors (if any), than from ready-made work by others.

Thanks again for your message. I think we have been taught to use disjunction if a sentence is in the form of 'unless A, B'. But it is important to note that we can also offer alternative disambiguations if they exist. I've got this (O ∨ ¬(S ∧ Y) ∧ L) as what I think is right, because it can be paraphrased in English as 'James is outraged unless it is not the case that Daisy prefers salad to desert and Ed hates Yorkshire puddings, and that Ed laughs.

Thank you Avi. I've noted my idea of the solution below, but I'm not 100% sure about it. If you can think of any disambiguations for this statement that would be a real help. No worries if you don't have time, I really appreciate it nonetheless!

Jack, logical conditioning is dealt in great detail in Future Logic,part III.

That said, the form If/not-then is nothing very new really. In ordinary discourse, we say commonly things like "if so and so, it does not follow that such and such". If X, not-then Y is stated in modern logic as simply the negation of if X then Y, i.e. as not(X->Y), i.e. X does not imply Y.

Yes, I'm currently studying FL. I guess my point was that modern logic has no way of "symbolically" representing a nonsequitur ("it does not follow"). The result of negating a conditional is

¬(p => q) = p & ¬q

But this doesn't have the same meaning as "q does not follow from p", or "if p, not-then q". It's one of the problems with a purely "truth-functional" logic as you clearly explain in Chapter 24.3.

I just wanted to remind you of the first few questions I asked if thats ok. I noticed you used conditionals to represent the double-turnstile symbol in the truth table. In my textbook, the double turnstile is used to symbolise tautological validity between arbitrary sentences of TFL. It is not a symbol of TFL, but rather a symbol of augmented English used to describe arbitrary sentences of TFL. The conditional is a symbol of TFL, unlike the double-turnstile. Also, the letters in the questions are used to symbolise any arbitrary sentence of TFL, not specific ones.

Based on this information, would this change any of your answers?

I have put the questions I am struggling with below. Sorry to bring this up again, I just wanted to check.

Question: Are the following claims about tautological validity true or false?*
*If you could explain your position on each question that would be really useful.

1. If A ⊨ B, then B ⊨ A.
2. If A ⊨ C, then A ∧ ¬C ⊨ C.
3. If A ∨ B ⊨ C ∧ D, then A ⊨ C.

I also want to say how grateful I am for your help. I am working through this textbook by myself and your explanations have really helped me over the last week.