HULK have trouble with math exercises, so HULK smash? This cheesegrater is rather amusing in homework help threads. (I know, I know, it's self-assigned homework.)
Ax (monkey(x) => banana(x))
I assume this axiom is meant to reflect the statement, "all monkeys like bananas", so I infer that banana(x) means that x likes bananas.
Ex (monkey(x) ^ button(x) => banana(x))
I assume this axiom is meant to reflect the statement, "monkeys who press the button get a banana". The Ex should be an Ax, because every
monkey who presses the button gets a banana, and not just, say, Lucy. Also, from this axiom it appears that banana(x) means that x gets/has a banana, but you're already using banana(x) to mean x likes bananas, so you need to get rid of banana(x) and replace it with likesBananas(x) and hasBanana(x), or something.
Ex (monkey(x) ^ banana(x) => happy(x))
Similarly, this Ex should also be Ax, because every
monkey who gets what they like is happy, not just Lucy.
Ax,y (light(x) => button(y)) ------ I wanted to provide x as will be either 'on' or 'off', but is this the right way to do?
I would treat light as a boolean constant, so "light" means that the light is on, and "¬light" means the light is off. (This is the same as the 0-ary predicate approach "light()" that Desiato suggested.) That way the tautology "the light is on or off" will be represented by the tautology "light v ¬light" in your logic, instead of needing to put it in manually as the axiom "light(off) <=> ¬light(on)" or something. Your second version "Ax (light() => button(x))" is better, but doesn't exactly represent the situation, "Basil will press the button if the light is on," because it says that everyone
will press the button if the light is on. You want the axiom "light() => button(Basil)".
Axiom 3: Ex (chimpanzee(x) => (x=Basil)) ---this is the change. I came across a example in the book that does this. Unfortunately the book isn't comprehensive enough for a self-learner like me.
Your original "chimpanzee(Basil)" was better, as it says, exactly, "Basil is a chimpanzee". The new axiom says that is an entity which, if it is a chimpanzee, is Basil. I'm not quite sure what the example in the book was doing, but
P.S. Let me know if the word-filters are making anything hard to understand and I'll insert LRM characters to keep things from getting cheesegrated.
P.P.S. Chimpanzees are not monkeys!!!
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.
"With math, all things are possible." —Rebecca Watson