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Just a quick question I stumbled upon from my readings.

When some philosophers write A ↔ B and others write A =df B, is there supposed to be a difference?

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  • See also this post Nov 19, 2023 at 8:38
  • "Iff" can be formalized in different ways, according to the context. First of all, it is a relation between statements that can be either the bi-conditional connective of the formal language, or the semantical notion of logical equivalence, at the meta-level. The two relations are strictly related but different. Nov 19, 2023 at 11:20
  • What makes you think there isn't? Nov 21, 2023 at 7:45

3 Answers 3

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A ↔ B means that both propositions are equivalent: "x implies y" (A) is equivalent to "non-y implies non-x" (B).

Equivalences have to be demonstrated.

While A =df B means that the term A is defined by the term B: to be a bachelor (A) is defined to be an unmarried man (B).

Definitions are arbitrary, like giving a name to something.

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  • +1 For clarity. I'm not sure definitions are based on whim, impulse, or are without reason, though.
    – J D
    Nov 19, 2023 at 16:56
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    @JD Concerning definitions one is free like choosing a name for one's child. Nevertheless one should follow certain rules in choosing the name :-)
    – Jo Wehler
    Nov 19, 2023 at 17:23
  • Well spoken, sir!
    – J D
    Nov 19, 2023 at 19:27
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    @JD There's nothing technically wrong with defining "addition" to mean multiplication. It may be confusing to the readers, but as long as you use it consistently it's mathematically correct.
    – Barmar
    Nov 20, 2023 at 16:03
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'IFF' is a biconditional*. That means two propositions always have the same truth value since the LHS implies the RHS and vice versa. Thus, given P → Q and Q → P, then we can infer that Q ↔ P. These are types of propositional function (SEP).

'≝' is defined by. The latter indicates that the equivalence is the result of action by people to make things in some sense the same. For instance, we can stipulate the definition of a 'fizbee':

Fizbee ≝ the age of a hamster at 1,000 hours.

This example is a stipulative definition. But there are other acts of definition such as lexical definitions and operational definitions. The SEP has an article on definitions. See Robinson's work Definitions or Harris and Hutton's Definition in Theory and Practice for more information on what is entailed by the act of defining.

The important take away is that they are both are equivalence relations, but they each have different contexts. The first is a logical term and is generally used in sentential logic, and the second is a much deeper process that is related to semantics and natural language and is therefore much more linguistic and psychological.


  • Note Bumble's comment: Currently, Wikipedia is a complete mess when it comes to articles relating to biconditionals. There are articles on "Logical biconditional", "Logical equivalence", "Logical equality" and "If and only if". Also, they contradict each other. The one you linked (Logical biconditional) is perhaps the weakest, since it confuses logical equivalence with material equivalence.
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  • Currently, Wikipedia is a complete mess when it comes to articles relating to biconditionals. There are articles on "Logical biconditional", "Logical equivalence", "Logical equality" and "If and only if". Also, they contradict each other. The one you linked (Logical biconditional) is perhaps the weakest, since it confuses logical equivalence with material equivalence. It might be better to link to en.wikipedia.org/wiki/Logical_equivalence instead.
    – Bumble
    Nov 19, 2023 at 6:57
  • @Bumble The product of the glorious process of design by committee. Thanks.
    – J D
    Nov 19, 2023 at 16:52
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When some philosophers write A ↔ B and others write A =df B, is there supposed to be a difference?

Clearly, A ⇔ B and A ≝ B do not mean the same thing.

  1. A ⇔ B means that A and B are logically equivalent, i.e., that if one is true, the other is also true.
  2. A ≝ B means that A and B are used to mean the same thing, i.e., that if x is A, then x is also B, and vice versa.

A ⇔ B and A ≝ B cannot possibly mean the same thing because A ⇔ B may be true even if we don't use A and B to mean the same thing.

The logical equivalence A ⇔ B is a logical expression, while the definition A ≝ B is a linguistic one.

A ≝ B maybe implies A ⇔ B, but A ⇔ B certainly does not imply A ≝ B.

Presumably, A ⇔ B may imply A ≝ B depending on A and B.

Explain the logical relation between the two and collect your Nobel Prize.

EDIT -- An important point (see David Roberts' comment below), is that A ≝ B is oriented. That is, we use A ≝ B to define A in terms of B, not the reverse. And once we have posited A ≝ B, we are not going to posit B ≝ A.

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  • =df is not symmetric, in my experience; rather the left hand side is being just now defined to be exactly the right hand side, which is something that already has meaning. Nov 20, 2023 at 23:39
  • @DavidRoberts "=df is not symmetric" True, I agree, and it is an important point. I didn't want to go into what definitions are used for and how exactly they are used. See my edit. Nov 21, 2023 at 9:30

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