Reading formulas

Here $\leftrightarrow$ is read 'means exactly the same as'. For 'can be inferred', 'is implied' and 'must be true', an option passes only if it holds in every situation in which all the passage's sentences hold. 'Probably true' and 'true in real life' do not pass. Ordinary word meanings may be used in applying the test; facts about the topic that the passage does not give may not.

Of the three standard rearrangements of an if-then statement (converse, inverse, contrapositive), only the contrapositive keeps the meaning. The converse (if B then A) and the inverse (if not A then not B) do NOT follow; they are equivalent to each other, not to the original.

The part that follows 'if' is sufficient: it guarantees the other part. 'Only if', below, points the other way. 'A if and only if B' asserts both directions: if B then A, and if A then B.

B is necessary for A. Knowing that B holds tells you nothing about A; knowing that B fails tells you that A fails.

The group named after 'only' becomes the one that every member of the other group belongs to. It does not say that every X is a Y.

Replace 'unless' with 'if not'. Its contrapositive, 'if not A then B', is the same statement. On its own, the sentence says nothing about the case where B occurs.

'Some' means at least one and possibly all. It does not by itself mean 'not all'. 'At least' and 'at most' are read inclusively.

The opposite of 'all' is 'not all', not 'none'. You need this when you apply the negation test to an option that contains 'all' or 'every'.

The opposite of 'at least one' is 'none'. Read the other way, the opposite of 'no X is a Y' is 'at least one X is a Y'.

Denying 'both' says only that at least one of the two fails. It does not say that both fail.

Denying 'either' (inclusive or) says that both fail. This is what 'neither A nor B' means.

Here $\to$ is read 'leads to'. State the mildest opposite of the option (the opposite of 'all' is 'not all'). If the conclusion then no longer follows from the evidence, the option is an assumption the argument needs. If the argument survives, it is not an assumption.