Can Peirce's Law be proven without contradiction?

Question

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)

Answer 1

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".

Answer 2

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.

Answer 3

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.