A is a sufficient condition for B. B is a necessary condition for A. • A is true only if B is true. Only if B is true is A true.

The key is to remember that it is both statements “A” and “A implies B” that together allow us to infer statement B. So if statement B is not true it is either because A isn’t true or because “A implies B” isn’t true. The possibility of the implication being untrue is often overlooked.

A powerful aspect of abstraction is that many different situations become the same when you forget some details.

It is important to note that this is logically different from concluding that the hypothesis is false, which would be the opposite rather than the negation. If you have insufficient evidence to support a hypothesis, it means that the truth value is still unknown. Maybe you need more data. Maybe you need a better experiment.

It doesn’t mean that you won’t ever try, and that nobody else will, it’s just that right now you’ve decided this is your starting point and is the basis of your logical system, or your belief system.

encapsulate the most important building block of logical arguments: logical implication.

It doesn’t make more things true, it just uncovers more true things than we saw before.

many arguments in real life go wrong because of problems with the groundwork rather than with the argument per se.

The primary aim of normal language is communication, whereas the primary aim of logical language is to eliminate ambiguity.