Have you got a definition of satisfiable and/or valid? What kind of structures do these objects have?
I am struggling to prove whether the following statements are true or false, and consequently to prove why that is.
Mainly because I don't understand the difference between satisfiable and valid, so if someone could explain that, I'd highly appreciate it!
(i) For all F, F is satisfiable or ~F is satisfiable.
(ii) For all F, F is valid or ~F is valid.
F is satisfiable if F is true in at least one interpretation. For example, 1 + 1 = 0 is satisfiable because it is true in . F is valid if it is true in all interpetations. For example, 1 + 1 = 0 and 1 + 1 = 2 are not valid, but 1 = 1 is. Obviously, if a formula is valid, it is satisfiable. (This assumes that there is at least one interpretation.)
Concerning the first claim, a stronger one is true: F is satisfiable or ~F is valid for all F.
First, I have to ask. Have you learned the definitions of formulas, interpretations and when a formula is true in an interpretation? Without it, it's meaningless to go forward.
(for all i, P(i) holds) or (for all i, P(i) does not hold)?
Now, let i range over interpretations instead of numbers and let P(i) say that the given formula F is true in an interpretation i.
(1) F is valid ⇔ F is true in all interpretations i ⇔ for all i, P(i) holds .
(2) ~F is valid ⇔ ~F is true in all interpretations i ⇔ F is false in all i ⇔ for all i, P(i) does not hold.
Does it have to be that (1) or (2) holds?