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What would be a real life, concrete example of Peirce's Law?

((p → q) → p) → p

There is a Wikipedia article on it, if you are unfamiliar with it:

https://en.wikipedia.org/wiki/Peirce's_law

There is also on Wikipedia of a discussion about giving a concrete example, which went nowhere:

https://en.wikipedia.org/wiki/Talk:Peirce's_law

Peirce gave a short explanation of the formula:

((x → y) → x) → x. This is hardly axiomatical. That it is true appears as follows. It can only be false by the final consequent x being false while its antecedent (x → y) → x is true. If this is true, either its consequent, x, is true, when the whole formula would be true, or its antecedent x → y is false. But in the last case the antecedent of x → y, that is x, must be true. https://www.jstor.org/stable/2369451?seq=10#metadata_info_tab_contents

Peirce's Law is a rather simple formula and it is proved valid in mathematical logic--through a Natural Deduction proof or using a truth table--so it should be easy to a give concrete, real life example of it.

  • For (P →Q) → P:"If us working on improving yields implied that our product would be better in 6 months, then we would work on improving yields". John Baez recommends getting from this to ((P →Q) → P) → P by cases with P true and P false considered separately. The latter case appeals to ex falso quodlibet:"We're not doing it, which means the implication is technically true, which means we are doing it". – Conifold Jul 6 at 19:29
  • @Conifold If p is false, then p is true doesn't make for a real life example.Technically true in mathematical logic if p is true, but not true in real life. – Speakpigeon Jul 6 at 20:30
  • "If p is false, then p is true" is not valid in mathematical logic either. The form used is "if p false implies any q then p is true", i.e. a version of ex falso quodlibet. Few find EFQ intuitive so one should not expect more from Peirce's law. Not to mention that people are generally bad at intuitively interpreting even two nested implications, let alone three, regardless of logic used or Peirce's law. – Conifold Jul 7 at 4:17
  • If it is raining when if it is raining, then it is cloudy, then it is raining. – Mauro ALLEGRANZA Jul 7 at 6:13
  • Peirce's law is "similar" to Modus Ponens. MP "defines" the condition for asserting the consequent: we can assert Q when "if P, then Q and Q" hold. The same for Peirce's law regarding the antecedent: we can assert P when "P if, if P, then Q" hold. – Mauro ALLEGRANZA Jul 7 at 6:40
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A misleading feature I observe in many textbooks on logic is that they restrict translations to and from natural language to a few strict patterns, upshot of which is a habit of interpretations bereft of the huge power of informal semantics, which, however, mathematicians freely enjoy (I have always thought that it would be very nice if the English language had a verb "enmean," in analogy to "enable," beside "interpret", for those cases we bring the phenomena naturally that have no meaning to being meaningful). Hence, one has actually the freedom, for example, to translate 'α → β' as "whenever α obtains, β obtains as well," or "on the condition that α, β occurs," etc. according to the context. Also, it is not required to form one long sentence as a translation of a compound proposition.

Returning to Peirce's law ((α → β) → α) → α with these remarks in mind, we may take α → β as "whenever we have α, we have β" and '(α → β) → α' as "we have α on the condition that whenever we have α, we have β." Thus, we can compose such a sentence:

If we have the right to that land on the condition that whenever we have the right [to that land], others [also] have the right to it, then we [do] have that right.

Notice that, though we have admitted others also have the right whenever we have, we are disinterested in the others' right (sure, we could exchange us and them). In any way, the resultant sentence would necessarily pick out α, and β would be excluded.

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Pierce's law works on the peculiarity of the truth table of implication, in which an implication is only false when the antecedent is true and the consequent is false. Implication is always true otherwise. We can show Pierce's law through reductio ad absurdum:

  1. For Pierce's law to be false, ((P → Q) → P) must be true and P false
  2. If P is false (p1), (P → Q) must be true in all cases.
  3. If (P → Q) is necessarily true (p2) and P is false (p1), then ((P → Q) → P) is false
  4. However, if ((P → Q) → P) is necessarily false (p3), and P is false (p1), then (((P → Q) → P) → P) is true, contradicting the premise we made in p1

P must be true, because asserting that it is false leads to contradiction.

What Pierce's law says in (somewhat) plain English is that if we ask whether "(B is contingent on A) is contingent on the existence of A", then that question itself implies the (logical) existence of A. For a real world example, let's consider lambs and sheep:

  1. The existence of sheep (P) implies the existence of lambs (Q).
    • (P → Q): This could be true or false: true in the 'natural' mode of ovine reproduction; false if we are (say) cloning lambs out of base genetic material or some such.
  2. The idea that the existence of sheep implies the existence of lambs implies sheep exist.
    • ((P → Q) → P): This too could be true or false. In the case that sheep exist this is always true (existing sheep may or may not imply lambs, but they still exist). In the case that sheep do not exist, this is always false (the terms always evaluate to T → F, whether or not lambs exist). In essence, this assertion says that if sheep do (or do not) exist, they do (or do not) exist whether or not they have a relationship to lambs. Lambs become irrelevant to the discussion.
  3. Asking whether point #2 implies the existence of sheep de-evolves to an implicative tautology: effectively it says that the existence of sheep implies the existence of sheep.
    • ((P → Q) → P) → P: This can only be true, though that is hardly obvious from the formulation. The only way for it to be false is to construct a scenario in which the existence of sheep implies their non-existence (a kind of Schrödinger's sheep, if you will...).

I hope that clarifies things, instead of confusing them further.

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