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.
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.
Certainly, one may set out a variety of logical systems, picking out some logical vocabulary items as primitive, others as derived, which would be equivalent for all formal intents and purposes. However, they diversify on assessments from other aspects (philosophical, heuristic, operational, etc.).
In order to present one line of thought in response to the question, we shall turn to conceptual roots and consider the kernel ideas of two fundamental oppositions, contradiction and contrariety (as adapted to our present context). But first, two remarks:
- Logical treatment of oppositions and negations has been a historical challenge and an ongoing research topic.
- The primary touchstone of logic is the demands of mathematical discourse on reasons that we cannot go into, and it is not unexpected that miscellaneous problems arise when it is extended to other fields of discourse.
Contradiction is the familiar relation; two contradictory propositions cannot have the same truth-value simultaneously; one holds if only if the other does not. Contrariety, though important in (Aristotelian) term-based logic, has been pushed to the background in proposition-based logic, unfortunately. Two contrary propositions cannot be simultaneously true, but they can be false; thus, they allow a possibility for a "middle".
The Law of Non-Contradiction is regarded as the central principle. It is a characterising principle involving both contradictions and contrarieties, while the Law of Excluded Middle serves as a delimiting principle involving only contradictions.
Suppose we have two propositions, P and Q. Let us consider what their combinations R state. Referring to their truth-tables, we can observe the following (R assigned to false, F, indicates an inadmissible case):
R is ¬(P ∧ Q) which has a line of F in case that P and Q hold. R states that P and Q are mutually exclusive propositions, we cannot have both (⟹ non-contradiction).
R is P ∨ Q which has a line of F in case that neither P and nor Q holds. R states that P and Q are jointly exhaustive propositions, we can have either one or both, but not any other (⟹ tertium non datur).
R is P ⊕ Q which has two lines F in cases that P and Q simultaneously hold or simultaneously do not hold. R states that not only P and Q are mutually exhaustive, but also jointly exhaustive (⟹ a composite principle, while we prefer to proceed from the simple to the complex here).
Subsequently, we apply the preceding considerations to the regulating case for a proposition φ and its negation:
Law of Non-Contradiction: ¬(φ ∧ ¬φ)
φ and ¬φ are mutually exclusive; we cannot have both.
Law of Excluded Middle: φ ∨ ¬φ
φ and ¬φ are are jointly exhaustive; we cannot have any other.
The case for XOR: φ ⊕ ¬φ
This case does not state more than any tautology does.