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Good evening, I heard the proof by contradiction is required for Peirce's law. AFAIK, truth tables are not related directly to proofs by contradiction, and if of an operation $\text {op}$ we have a truth table 

P(p) P(q)   P(p op q)
  0    0        1
  0    1        1
  1    0        1
  1    1        1

such that $P$ is the predicate, or logical/truth value of a proposition, then

$$ \forall p\forall q, P(p\ \text{op}\ q)=1\Leftrightarrow\forall p\forall q, p\ \text{op}\ q $$

So isn't 

(p    q   if p, then q  if p, q; and then p    modus ponens           Peirce's Law)
P(p) P(q)    P(p → q)     P((p → q) → p)   P(((p → q) → p) → q)   P(((p → q) → p) → p)
  0    0         1              0                   1                      1       
  0    1         1              0                   1                      1       
  1    0         0              1                   1                      1       
  1    1         1              1                   1                      1

a proof of Peirce's law?
(you can make the maths yourself, noting this is classical logic and that $p \to q \Leftrightarrow \lnot p\lor q$; and all works without admiting any proof by contradiction at all)

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JMCF125
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    I think you heard something wrong. "cannot be proved without admitting proof by contradiction" is not the same (at all!) as "a proof by contradiction does not exist" – Hagen von Eitzen Nov 20 '13 at 21:59
  • No, that's not a proof, because it doesn't have the truth table for the material conditional. Also, truth tables don't satisfy the definition of a proof as used in most works on logic (they can often get converted to proofs in some logical systems with, but that doesn't make them into proofs). – Doug Spoonwood Nov 22 '13 at 15:40
  • @Doug, I wrote that before I knew a truth table is equivalent to a proof by contradiction, and can be accepted iff a proof by contradiction is acceptable. It's gone now. And AFAIK they are proofs, they show that for all possible values of $p$ and $q$, the operation works. How could they not be proofs whenever proof by contradiction is accepted. – JMCF125 Nov 22 '13 at 16:44
  • @JMCF125 A proof table isn't equivalent to a proof by contradiction. A proof table comes as a semantic way of analyzing things. A proof by contradiction, comes as syntactic. They only work out as "proofs" in a loose sense of the term if and only if you have completeness of the logical system at work. Also, truth tables can do more than proofs by contradiction. Additionally, truth tables can come as acceptable even for systems without proof by contradiction. – Doug Spoonwood Nov 22 '13 at 18:20
  • @DougSpoonwood, «Additionally, truth tables can come as acceptable even for systems without proof by contradiction»: that's not what Henning Makholm said, read his second paragraph please. – JMCF125 Nov 22 '13 at 22:44
  • @JMCF125 Henning Makholm says "Yes, that does sound like a stretch, but really the basis for accepting the truth table as complete is that we know that in every relevant world p will either be true or false (and likewise for q). And that is the same thing that lies at the root of proof by contradiction: If we can prove that something cannot be false, then it must be true because being true and being false are the only options." You can use truth tables in finite-valued logics with more than two truth values. We can have proof by contradiction in some systems with more than two truth values. – Doug Spoonwood Nov 26 '13 at 21:41

3 Answers3

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What I said in the linked answer was that Peirce's law requires proof by contradiction or something essentially equivalent to it. For this purpose reasoning by truth table counts as "essentially equivalent to proof by contradiction".

Yes, that does sound like a stretch, but really the basis for accepting the truth table as complete is that we know that in every relevant world $p$ will either be true or false (and likewise for $q$). And that is the same thing that lies at the root of proof by contradiction: If we can prove that something cannot be false, then it must be true because being true and being false are the only options.

Intuitionistic logic rejects proof by truth table as well as proof by contradiction and the principle that $P\lor \neg P$ is always true. I was using "essentially equivalent to proof by contradiction" as a (possibly too) fanciful shorthand for "a reasoning principle that, when added to intuitionistic logic, produces the usual classical logic".

hmakholm left over Monica
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  • I would never before have thought of proofs by contradiction as equivalent to truth tables. But you do make a good case. – JMCF125 Nov 22 '13 at 17:02
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    @JMCF125 I'm not sure it's such a good case. Henning Makholm's argument basically implies that sole axioms for the pure implicational calculus, as well as sole axioms for classical logic under the rules of detachment and substitution are "essentially equivalent" to proof by contradiction. But, those axioms also hold for some logical systems where the principle of bivalence fails. – Doug Spoonwood Nov 22 '13 at 18:13
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I use Polish notation. C stands for the material conditional and goes before arguments. Thus these formation rules will suffice here:

  1. All lower case letters of the Latin alphabet are well-formed formulas (wffs).
  2. If $\alpha$ and $\beta$ are wffs, then so is C$\alpha$$\beta$.

Thus, Peirce's Law becomes CCCpqpp. Arthur Prior's book Formal Logic indicates that a sole axiom for the pure implicational calculus of propositions (the only rules of inference are detachment and uniform substitution for variables) given by Lukasiewicz in 1936 is

3 CCCpqrCCrpCsp.

I used prover9 to find the following proof (even if I'm clever enough to find a proof, it's almost always faster then me... when I can get it to find me a proof). The notation 3 p/Crs indicates that p gets substituted with Crs in wff 3. The notation 3 p/Crt * C4-5 indicates that we'll substitute p with Crt in wff 4, it has the same form as the wff which starts with a C, then has wff 4, and ends with wff 5. 4 is already in our set of logical theorems or axioms, and consequently we'll detach 5 as a theoerem by the rule of detachment.

3 CCCpqrCCrpCsp.

 3 p/Cpq, q/r, r/CCrpCsp, s/t * C3-5

5 CCCCrpCspCpqCtCpq.

 5 r/Crp, p/Csp, s/p, q/CpCsp * C5 q/Csp, t/Csp-6

6 CtCCspCpCsp.

 3 p/CCrpCsp, q/Cpq, r/CtCpq, s/u * C5-7

7 CCCtCpqCCrpCspCuCCrpCsp.

 6 t/CtCCpqCqCpq, s/p, p/q * C6-8

8 CCpqCqCpq.

 3 r/CqCpq, s/r *C8-9

9 CCCqCpqpCrp.

 7 t/CqCCppq, q/p, s/p * C9 p/Cpp, r/Crp-10

10 CuCCrpCpp.

 10 u/CuCCpqCqq, r/p, p/q * C10 r/p, p/q-11

11 CCpqCqq.

 11 p/CCqCpqp,q/Crp * C9-12

12 CCrpCrp. 

 3 p/r, q/p, r/Crp * C12-13

13 CCCrprCsr.

 3 p/Crp, q/r, r/Csr, s/t * C13-14

14 CCCsrCrpCtCrp.

 3 p/Csr, q/Crp, r/CtCrp, s/u * C14-15

15 CCCtCrpCsrCuCsr.

 15 t/Cpq, s/CCCspqp, r/Csp, p/q * C3 r/CCspq -16

16 CuCCCCspqpCsp.

 16 u/CuCCCpqrqCpr, s/p, p/q, q/r * C16 s/p, p/q, q/r-17

17 CCCCpqrqCpq. 

 3 p/CCpqr, r/Cpq * C17-18

18 CCCpqCCpqrCsCCpqr. 

 18 p/Crp, q/r * C13 s/Cpq-19

19 CsCCCrprr.

 19 s/CsCCpqpp, r/p, p/q * C19 r/p, p/q-20

20 CCCpqpp.

There's plenty of other already known axiom sets for the pure implicational calculus of propositions... Prior's book lists a 3-axiom set, a 4-axiom set, 12 2-axiom sets, and 4 sole axiom sets. All of those systems only have detachment and substitution as rules of inference. None of them use or allow for proof by contradiction.

Doug Spoonwood
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I think that the statement "the proof by contradiction is required for Peirce's law" must be interpreted in a different way.

I refer to these previous post about Implicational calculus.

My starting point is the answer by @Doug Spoonwood : according to Lukasiewicz' axiomatisation, one axiom is enough to derive all the "implicational" tautologies.

The issue is that in this way there is place for negation ($\lnot$) in the system.

The easiest way is to introduce the falsum ($\bot$); with it we may use $\lnot A$ as an abbreviation of $A \rightarrow \bot$.

In the previous post, I argued that, in order to derive the classical laws of negation, like Double Negation, we need Peirce's law plus the Ex Falso Quodlibet axiom :

$\vdash \bot \rightarrow A$.

Also in order to prove the equivalence between Peirce's law and the Law of Excluded Middle, we need the symbol $\bot$ as primitive and the EFQ axiom.

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Mauro ALLEGRANZA
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  • Why do you end with a question? :) Anyway, wouldn't the law of the excluded middle be equivalent to proof by contradiction, inutilizing the efforts to prove Peirce's Law is can be derived without proof by contradiction? Or am I missing something. +1, thanks for the answer, I thought this was already a dead post. – JMCF125 Feb 18 '14 at 22:44