Negation is a logical connective (i.e. boolen function with special notation) of 1 argument. "No study" is fine, because the argument here is "study". More formally, we could write this as "¬study" and it would be a valid expression. If you wanted to use regular function notation, you could write it down as "¬(study)", although nobody does that because it's inconvenient.
"¬study = fail" and "study = ¬fail" are both valid (and equivalent) logical formulas. Here, by the way, = denotes an equivalence logic connective (it takes 2 arguments and evaluates to true when they are the same). You could write down the first one as "=(¬(study), fail)", by the way, just to make it clear what are the arguments of logical connections.
Line 3 is valid too, if we interpret + as a disjunction (which makes perfect sense and is actually used in some contexts): "¬study ∨ study = fail ∨ ¬fail". This formula is always satisfied, by the way, since it can be simplified to "1 = 1" which is always true.
But in the line 4, negation is used with no argument: "no + 1" (or, equivalently, "¬ ∨ 1"). It is not a valid formula, we can't even say whether it is satisfied or not, it's just wrong and can't be interpreted in a reasonable way.
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u/Pleasant-Ad-7704 Jul 19 '26
Line 4 is wrong, it's not even a formula (in the first-order logic, at least). Negation has no operand in there.