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My guess is that they say "iff", but I wanted to make sure as math books have never made it explicit.

To take the field axioms for example, is it fair to say that it says both:

  1. $F_x\rightarrow A_x$ (i.e. "If some object $x$ is a field, the it also satisfies the field axioms"), and,

  2. $A_x\rightarrow F_x$ (i.e. "If some object $x$ satisfies the field axiom, then it is also a field").


marked as duplicate by Stefan Perko, joriki, Community Jun 28 '16 at 11:12

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  • $\begingroup$ What makes you think that all axioms use the same type of implication? BTW, every "if" statement has an equivalent "only if" statement. For example, "if A then B" is equivalent to "A only if B". $\endgroup$ – barak manos Jun 28 '16 at 10:28
  • $\begingroup$ Axioms are self-contained statements, they may involve if statements or iff statements depending on what they want to say. You are confusing it with a definition, which always implies iff (a structure is defined to be a field iff it satisfies the field axioms). $\endgroup$ – Morgan Rodgers Jun 28 '16 at 10:32
  • $\begingroup$ First of all, all types of axioms don't use the same thing, If there are two ways a thing can be dealt with, owning to different circumstances they particularly use 'if' in order to write the constraints on to the things while if something satisfies a relation only and only if some particular values or sets of values are true, the axioms use' iff' to convey this meaning. You can also check out some more details here en.wikipedia.org/wiki/If_and_only_if . $\endgroup$ – Harsh Sharma Jun 28 '16 at 10:33
  • $\begingroup$ You example is more a definition: "a structure $\mathcal F$ is a Field iff it satisfy the field axioms". $\endgroup$ – Mauro ALLEGRANZA Jun 28 '16 at 10:35