I read Kelley's book (the art of reasoning: An Introduction to Logic and Critical Thinking 4th edition). On page 150, I found this statement: "A special problem arises with statements that have the form "All S are not P". Despite its appearance, this is not standard form. It is ambiguous. Consider the statement "All politicians are not crooks". Does this mean that no politicians are crooks (an E proposition) or that some politicians are not crooks (an O proposition)? It could mean either."

I have two questions. If I have a statement " Not all S are P", is it ambiguous too? Can it be translated to be an I proposition or an O proposition?

  • 1
    Not every S is P is not considered ambiguous, it is translated as Some S is not P, see Square of Opposition.
    – Conifold
    Oct 31, 2021 at 8:52
  • Thank you for your explanation. Oct 31, 2021 at 8:59
  • I think what the book wants to say is that from that no total statement follows, ie All S are not P, or all S are P. Only some S are P, some S are not P. So ambiguity is used in that sense
    – Nikos M.
    Oct 31, 2021 at 9:54
  • The book is obviously correct. The same wording could mean either no s is p or some s is not p. Consider the following: Not all women are reptiles. Would you take that to mean Some women are not reptiles? That would be foolish. The intent is NO women are reptiles. Playing with the at least one woman is not a reptile definition would be petty. We know NO is the proper quantifier there. We also know when we are NOT SURE we will appropriately use SOME s is not p. Translating NOT ALL as SOME s are not p is super high percentage.If you are not sure then go that route.Be aware of the other though.
    – Logikal
    Oct 31, 2021 at 16:34

1 Answer 1


"All S are not P". Despite its appearance, this is not standard form. It is ambiguous.

"All S are not P" is readily understood in natural languages as "Not all S are P".

For example:

Wilfried: Politicians are liars.

Wilbur: Well, all politician are not liars.

Wilfried: Yes, all politicians are liars.

However, "All S are not P" can be mistaken to be "All S are not-P", i.e., "No S is P", which is of course very different from "Not all S are P".

However, if we want to mean that all S are not-P, we don't say "All S are not P", but "No S is P". Thus, there is normally no ambiguity. However, some people mistake one for the other.

"Not all S are P" does not seem to be ambiguous. The three following ways of parsing it yield the same semantics:

Not(all S are P)

Not(all S) are P

Not(all) S are P

This is equivalent to "Some S are not P"

  • What is the difference between "All S are not P" and "All S are not-P"? Oct 31, 2021 at 13:03
  • @MartinFirdaus are not P = (are not) P = not(are) P. That is, the negation applies to the verb. While: are not-P = are(not(P)). Here, the negation applies to the predicate P. Oct 31, 2021 at 17:00
  • 1
    Oh I see, thank you for your explanation. Nov 1, 2021 at 0:15

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