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So the law of the excluded middle, as I have read in every logic textbook that I have read, has been ( ϕ ∨ ¬ ϕ ) , but this seems somewhat unintuitive, since I was under the impression that the intuition was that only one could be true, i.e. ( ϕ ⊕ ¬ ϕ ) . Now I understand that the latter is derivable from the former using the law of non-contradiction, but my question is why we don't refer specifically to the latter form when we talk about the law of the excluded middle.

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    Because "standard" modern symbolic logic formalizes arguments using "inclusive or" as primitive and "exclusive or" as a derived one. Nov 30, 2021 at 10:38
  • I think that the "shift" from exclusive to inclusive or occurred in The Algebra of Logic Tradition: John Venn (1881) seems exclusive while Peirce (1885) is inclusive: the motivation seems to be to validate De Morgan's laws. Peano (1889) explicitly call it vel (Latin inclusive or, compared to aut). Nov 30, 2021 at 11:34
  • In parallel, there is Frege (1879)'s approach: "Of the two ways in which the expression "A or B" is used, the first, which does not exclude the coexistence of A and B, is the more important, and we shall use the word "or" in this sense." Nov 30, 2021 at 15:03
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    It is because it is helpful to distinguish the law of excluded middle from the law of non-contradiction. By using an exclusive or, you are effectively stating both at once. Some logics have one but not the other, so we need to be able to state them separately.
    – Bumble
    Nov 30, 2021 at 17:28
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    We can use XOR to express LEM, and Reichenbach even advocated such per wiki ref: It is correct, at least for bivalent logic—i.e. it can be seen with a Karnaugh map—that this law removes "the middle" of the inclusive-or used in his law. And this is the point of Reichenbach's demonstration that some believe the exclusive-or should take the place of the inclusive-or. In fact combining XOR with conjunction produce a field which can represent any logic obtainable with usual Boolean lattice formed by conjunction and disjunction... Dec 1, 2021 at 7:32

3 Answers 3

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It's one of TWO principles. One principle is that out of S and (not S), at least one must be true. Another, separate principle is that S and (not S) cannot both be true. The first principle gives us completeness, the second gives us freedom from contradictions.

Obviously if you combine both then exactly one of S and (not S) must be true.

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It depends on which language of logic is being used; there's one answer for classical disjunction, and another for linear disjunctions.

In classical fragments, there is only one disjunction. Note that "ϕ ⊕ Ψ" can be defined as (isomorphic to) the definition "(ϕ ∨ Ψ) ∧ ¬(ϕ ∧ Ψ)". Substituting for the special case of LEM, we are therefore asking for "(ϕ ∨ ¬ϕ) ∧ ¬(ϕ ∧ ¬ϕ)". But note that "ϕ ∧ ¬ϕ" is always false (via proof by truth tables), so our right-hand side of our topmost conjunction is always true, and may be removed; as a result, "ϕ ⊕ ¬ϕ" reduces to "ϕ ∨ ¬ϕ". This also holds intuitionistically. (This reduction is actually an equivalence!)

For linear logic, which has two disjunctions, the details are different. Quoting SEP on linear logic (and fixing their connectors for readability):

For example, the law of excluded middle (A ∨ ¬A) is considered valid in the classical world and absurd in the intuitionistic one. But in linear logic, this law has two readings: the additive version (A ⊕ ¬A) is not provable and corresponds to the intuitionistic reading of disjunction; the multiplicative version (A ⅋ ¬A) is trivially provable and simply corresponds to the tautology (A ⊸ A) that is perfectly acceptable in intuitionistic logic too.

Interpreted via the common vending-machine semantics for linear logic, "ϕ ⊕ ¬ϕ" asserts LEM for ϕ, or that ϕ is decidable; it says that we may either have ϕ or ¬ϕ, but we don't choose which.

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So the law of the excluded middle, as I have read in every logic textbook that I have read, has been ( ϕ ∨ ¬ ϕ ) , but this seems somewhat unintuitive, since I was under the impression that the intuition was that only one could be true, i.e. ( ϕ ⊕ ¬ ϕ ) . Now I understand that the latter is derivable from the former using the law of non-contradiction, but my question is why we don't refer specifically to the latter form when we talk about the law of the excluded middle.

Yes, you are absolutely correct. The standard formulation which says "ϕ ∨ ¬ϕ" does not preclude ϕ being both true and false, and this would be in effect a third alternative between true and false, which is precisely what the Law of Excluded Middle (LEM) wants to precludes.

The LEM is as the label says the law which excludes a "middle" alternative between ϕ and ¬ϕ. This just means that ϕ is either true or false, and so not possibly both, as ϕ ∨ ¬ϕ suggests.

The problem comes with mathematical logic. Philosophers and mathematicians working on a mathematical model of logic starting with George Boole in the 19th century have come to redact very nearly all the definitions of the key terminology of formal logic, including the word "logic" itself, which is routinely presented as nothing but a formal discipline.

Another example of this is the pseudo "law of non-contradiction", which in effect falsely says that there are no contradictions. The correct formulation is "Law of Contradiction", written ¬(ϕ ∧ ¬ϕ), which just says, correctly, that contradictions exist and that they are false.

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