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:


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


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.

Peirce's Law, although a true logical truth, seems essentially nonsensical, but maybe someone knows how to bring zombies back to life?

  • 2
    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 '20 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 '20 at 20:30
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    "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 '20 at 4:17
  • If it is raining when if it is raining, then it is cloudy, then it is raining. – Mauro ALLEGRANZA Jul 7 '20 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 '20 at 6:40

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.

  • Thank you for your answer. The method to arrive at your example seems good to me, but the example itself seems wrong. The semantics of "to have the right to that land" clearly suggests that the right of someone is exclusive of other people's rights, which makes α → β false in your example. Is that a deliberate choice on your part? Can you explain? – Speakpigeon Mar 5 at 11:04
  • @Speakpigeon Of course α → β can be false and Peirce's law still holds. The problem is that in natural language unrelated statements seem false (or at least weird) when tied together by logical connectives. As I said in a comment (assuming deduction theorem and contraction) Pierce's law is equivalent to stating α |- α, β, i.e. "If we have the right to land then we have that right or others do" seems "relevant-istic" but "If we have the right to land then we either have that right or the moon is made of blue cheese" is just as true in LK but sounds more weird. [continues] – Fizz Mar 5 at 22:55
  • If you want the "least implausible" example from a natural language perspective, take α = β which results in sentence(s) that might make good stand-up comedy due to their repetition and obviousness. "If I have a dog then I have a dog or I surely have one". (You can obviously chain that one ad nauseam.) – Fizz Mar 5 at 22:58
  • @Fizz 1. "The problem is that in natural language unrelated statements seem false" Sure, but this shouldn't stop anyone who understand that Pierce's Law is true from providing a concrete example of it. The question of irrelevance is irrelevant here. - 2. "If I have a dog then I have a dog or I surely have one" This would be an example for ((p → p) → p) → p, not for ((p → q) → p) → p. – Speakpigeon Mar 6 at 11:14
  • @ It's an instance of both obviously. It's not a proper (non-degenerate) instance on the latter. – Fizz Mar 6 at 15:32

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.

  • "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" No. Pierce's Law is a true implication. Pierce himself proved that. 2. "The only way for it to be false" There is no way for a true implication to be false. – Speakpigeon Mar 6 at 17:43
  • @Speakpigeon: the phrase 'true implication' seems to have a specific meaning for you. Could you explain? – Ted Wrigley Mar 6 at 17:58
  • Nothing special. An implication which is true, in the ordinary sense of these words. Any proposition which is true is not false. – Speakpigeon Mar 6 at 18:22
  • @Speakpigeon: Then I'm not clear on why you're objecting to my analysis. Do you see something wrong in those first four bullet points? – Ted Wrigley Mar 6 at 18:27
  • I didn't object to that but if you want to know, point 2 and 4 is mathematical logic nonsense. Sorry you learned that stuff. However, this is independent of my initial comment, which seems clear enough and in no need of elaboration. – Speakpigeon Mar 6 at 18:53

One problem with translating material conditionals into english is that they often mislead since if-then statements in English normally have a range of meanings that go beyond the material conditional (see the indicative condition). To avoid these kinds of confusions, it helps to keep in mind that in logic the material condition "A -> B" is completely equivalent in meaning to the statement "~A or B" (this is one of the rules of 'conditional exchange', see p. 302 of this chapter on truth-functional logic). So by the same rule "(p -> q) -> p" must be equivalent to "~(p -> q) or p", which is equivalent to "~(~p or q) or p", and then you can use one of de Morgan's laws (see p. 300 of the book chapter above) to show this is equivalent to "(p and ~q) or p". And regardless of whether one or both sides of that last "or" statement is true, p would have to be true either way, so it should be clear enough conceptually why "(p and ~q) or p" -> p is tautologically true, since it's logically impossible you could have a situation where "(p and ~q) or p" is true but p is false.

Since it's tautologically true, you can put any claim you want in for p and q to give a concrete example of why it's true. For example, say p = "the sky is blue" and q = "rabbits are herbivores" so that ~q = "rabbits are not herbivores". Then you could write it like this:

(the sky is blue AND rabbits are not herbivores) OR (the sky is blue) logically implies (the sky is blue).

  • The question is "What would be a real life, concrete example of Peirce's Law?". Your answer completely fails to address this question. – Speakpigeon Mar 6 at 17:34
  • @Speakpigeon - How are "the moon is made of green cheese" and "rabbits are carnivorous" not concrete statements? Are you bothered by the fact that they are not true in real life? But it's a tautology, so its validity doesn't depend on whether the concrete statements are true or false. As an analogy, if there was a question asking for a "real life, concrete example" of the tautology "A -> A", would you accept "(the sky is blue) -> (the sky is blue)" but reject "(the sky is green) -> (the sky is green)"? Or would you accept both, or reject both? – Hypnosifl Mar 6 at 20:17
  • We seem to disagree about what is real life. – Speakpigeon Mar 7 at 16:41
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    But the "logic of human deductive reasoning" tends to involve different notions of "implication" than material implication, as discussed in the SEP indicative conditionals article--for example, many if-then statements in ordinary speech are assertions about what conclusions would follow in a possible world where the antecedent was true, more like a statement from modal logic than the material conditional. So with some other definition of "→" there is no reason to think "((p → q) → p) → p" would be true for all p's and q's--do you disagree? – Hypnosifl Mar 8 at 19:22
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    Also there is the point that Peirce himself, and all the logicians who cite "Peice's law", are assuming "→" represents the material conditional--if you want to ask about the statement "((p implies q) implies p) implies p" using a different definition of "implies", it's misleading to call this Peirce's law, just like one couldn't call F=ma "Newton's second law" if one redefined the meaning of the symbol "a". – Hypnosifl Mar 8 at 19:25

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