Thread: True or false: If someone in the library is wearing a hat, then I can steal it.

Assuming I am in the library and I am capable of stealing hats.

p: someone in the library is wearing a hat
q: I can steal it

Are you looking for a logical implication. Like ? In which case that would be right. P does imply Q.

However the converse, that Q implies P, is also true and so it is a logical equivalence

But does "am capable of stealing hats" mean "can steal any hat"?

For statement Q: "I can steal it" , 'it' doesn't necessarily have to refer to a hat. 'it' could be a book in the library, for example

if you are capable of stealing any hat from anyone in the library, and you cannot steal a hat from someone in the library (that is, you are not capable of stealing their hat), it must be that they are not wearing one, eh?

(contrapositive is true, thus the implication is true).

the real trouble here is that "natural languange" (such as English) doesn't "fit exactly" to "logical language". statements in English, for example, can be ambiguous (they're "maybe-true/maybe-false"), whereas statements within logic must be one OR the other (and certainly not BOTH)!

so trying to "illustrate" logical reasoning with examples in "everyday language" is a path fraught with peril and pit-falls. one must choose their words VERY carefully, and the clever ones among us will rush to point out the loop-holes.

Just bumping this over the spam

Any statement can never be proven true beyond a shadow of a doubt by examples that follow the rule of inference. However, it only takes one counterexample to prove a statement false.

For example: Suppose you want to know the truth value of 'If A then B'. In general for any math or nonmath statements A, you can find as many examples of 'A' and each time and conclude 'B' but you can never be entirely sure that you can conclude B. In your if-then statement, as long as you can show that you *can* steal the hat if a person in the library wears it, your if-then statement is true. Otherwise, if you can't show that you can steal the hat, your argument is weak. Also, if someone in the library is wearing a hat and you can't steal it, your statement would be false (For a given statement If P then Q, if it is known that P results in not Q, then 'P but not Q' is an example of a counterexample).