Talk:Peirce's law

message: *if P and Q are boolean, we can just consider the four possible combinations of P and Q and show that Pierce's Law is true for all four:
**P=0,Q=0: ((0→0)→0)→0 = (1→0)→0 = 0→0 = 1.
**P=0,Q=1: ((0→1)→0)→0 = (1→0)→0 = 0→0 = 1.
**P=1,Q=0: ((1→0)→1)→1 = (0→1)→1 = 1→1 = 1.
**P=1,Q=1: ((1→1)→1)→1 = (1→1)→1 = 1→1 = 1.
*just remember that the implication operator (→) only takes on the value of 0 when 1→0, in all other cases, even 0→0, it's true. Digfarenough 18:10, 2 January 2006 (UTC)
replies: text: The → as in ((P→Q)→P)→P does not stand for the implication operator, so it is not obvious why this proof should work, with my limited mathematical knowledge. However, I think I can still point out that, this proof assumes "the law of the excluded middle", while Peirce's Law in the context implies "the law of the excluded middle". The two are different, because the proof did not show that Peirce's Law would be false without the law of the excluded middle.218.250.191.237 (talk) 15:14, 28 March 2008 (UTC)
replies: text: What do you mean by "... → ... does not stand for the implication operator ..."? Of course, it does. What else could it mean here? JRSpriggs (talk) 01:42, 30 March 2008 (UTC)
replies: text: Material conditional, surely. "Implication" ain't conditional. 96.245.85.214 (talk) 05:39, 18 April 2018 (UTC)
thread_title:  A better explanation? 

message: Bah, I forget about my pet article for a bit and someone comes through and removes a vital section while someone else shows up with a question.
(duplicated at User talk:68.49.135.47)
Basically, what's going on is that we need to show a contradiction, e.g today is Tuesday and today is Wednesday.  It's a bit bizarre.  Let's start with these assumptions:
1. Today is Tuesday.
2. We never eat meat on Tuesday.
3. We ate meat today.
Notice that 2 and 3 together mean that today is not Tuesday:
1. Today is Tuesday.
2+3. Today is not Tuesday.
This is a contradiction, so something was wrong.  In this case, one of our three assumptions is wrong, but we don't know which.  Perhaps today is Wednesday.  Perhaps we do sometimes eat meat on Tuesday.  Perhaps we ate meat yesterday and not today.  All of those are valid ways to fix our assumptions.
Now, in the case of our Pierce's Law example, we have two assumptions and one hypothesis:
1. A line is a glue line if every pear is glued to a kiwi.
2. A glue line has pears.
H. Today, we saw a line without pears.
Through our little mental experiment, that becomes:
H'. Today, we saw a line with every pear glued to a kiwi, but no pears.
So, we combine 1 and H'.  Now we know this:
1+H'. Today we saw a glue line, but it had no pears.
2. A glue line has pears.
That gives us our contradiction, and also your confusion.  You jumped here to start, and that didn't make any sense.  Instead, our contradiction means that something was wrong.  We trust our two assumptions, but the hypothesis could be wrong.  It was wrong, meaning that the line we saw had pears.  So we alter our hypothesis:
1. A line is a glue line if every pear is glued to a kiwi.
2. A glue line has pears.
H. Today, we saw a line with pears.
This is consistent, meaning that there is no contradiction.  Notice that we can't combine 2+H:
2+H. Today, we saw a glue line.
is wrong.
Hope that helps! -FunnyMan 21:01, 2 January 2006 (UTC)
thread_title:  Article fixed 

message: Dear FunnyMan, Thanks for your interest.  Please be more specific as to what you find confusing and we can all discuss it here on the talk page, as that is likely to be the best policy for improving the article.  In the meantime, though, I think that pinning a "scarlet letter" on the main article page is likely to be counter-productive, leading the casual reader of Wikipedia to lose confidence in the enterprise as a whole. So now that you've drawn initial attention to your concerns, maybe you would consider removing it.  Many regards, Jon Awbrey 16:00, 7 January 2006 (UTC)
thread_title:  Confusion 

message: I am not an expert on Peirce's logic but I know that (at least in one point of his career) he was hesitant to all axioms, owing to his empiricism/fallibilism/ pragmaticism.
In the second sentence of the introduction, the wiki states:
 "It was taken as an axiom in his first axiomatisation of propositional logic." 
Whereas, in the excerpt in his own description Peirce writes: 
 "This is hardly axiomatical." 
Are there two notions of "axiom" being used? Or is there an error? 
In any case the current explanation  is rather confusing.  — Preceding unsigned comment added by Quaquaquaqua (talk • contribs) 17:46, 30 May 2013 (UTC)
replies: text: Remember this is not an article about Peirce or his system of logic. It is merely an article about one formula in the language of propositional logic. JRSpriggs (talk) 06:19, 31 May 2013 (UTC)
thread_title:  Question concerning Axioms 

message: Since this expression is true for all values of P and Q, it might benefit from being written using the double arrow of material implication ((P→Q)→P)⇒P. How about it? Is there a reason this isn't used?
KazKylheku (talk) 21:50, 16 June 2016 (UTC)
replies: text: No. Is there some reason why we should confuse the readers by using two different symbols (arrows) for material implication? JRSpriggs (talk) 14:18, 17 June 2016 (UTC)
thread_title:  Arrow?