Speakpigeon wrote: ↑Wed May 01, 2019 9:02 pm
Scott Mayers wrote: ↑Wed May 01, 2019 5:47 pm
SOME call it a paradox for it being unresolved in their opinion. This is noted in most texts to be fair about the controversies of others but not usually of those writing the texts themselves.
I'm not sure what concern you might have for this thread and so would need a better example that you could express of what is contradicting here.
If you already agree but pointing out that it isn't an actual contradiction, I agree and my post is in support of that point.
The inference A ∧ ¬A ⊢ B isn't a contradiction but it is seen as a paradox. It is seen as a paradox in the sense that it contradicts our intuition, which is that B has nothing to do with A ∧ ¬A and so cannot be implicated by it.
It seems clear that A ∧ ¬A ⊢ B is in contradiction with our logical intuition and Aristotelian logic. And I don't know of any justification that A ∧ ¬A ⊢ B should be valid. It may follow from the definition of the material implication but there is no justification of that definition.
Further, A ∧ ¬A (in A ∧ ¬A ⊢ B) is an actual contradiction. It doesn't seem to make any sense to infer anything from a contradiction. The fact that none of the systems of axioms in use in mathematics today include a contradiction doesn't change the fact that most mathematicians accept the inference A ∧ ¬A ⊢ B as valid even though it is nonsense.
However, this isn't the topic of this thread. The topic is whether falsifying A ∧ ¬A ⊢ B would affect any mathematical result.
EB
I'm not understanding the concern at all here except that you maybe assume contradiction cannot be useful in a system. ?
In my own physics theory I utilize "contradiction" as the linchpin of causation or 'force'. Multi-value logic does this by having at least a 'third' value and where the term "con-tra-dict-ion" means
with third spoken/commanded part/factor.
The particular Propositional Calculus I used above that uses binary values takes the contradiction as being unallowed for closure but even with the point I make about a 'force' there is a "logical force" that compels us to avoid situations that are contradictory. But that doesn't HAVE to be the end but a 'trigger' to do something else.
For example, lets say you set up a three valued system such as {0, 1, P}. Here '0' can represent "false", '1' as "true", and "P" can stand for "Next". When you have anything that might normally contradict in binary, you might have a rule such as
(0 & 1) ⭢ P
...which when P is triggered, it can have a rule that does anything. We DO have the rule, "STOP", when we run into this case and so can be interpreted as P == STOP, by an assigned semantic constant.
All the Principle of Explosion does is DEFINE those systems that use the minimal binary level logic of truth values as 'invalid' for the reason that anything 'false' is infinitely inclusive. So it is defined by assigning the MEANING of 'false' as equivalent to being both (false & true).
Here is the 'explosion' expressed:
Let something be given or determined as 'false'. With the defining of something false as meaning 'false & true' ,
False
= (False & True)
= (False & True) & True
= ((False & True) & True) & True
....etc
So anything false leads to an infinity of truths.
The 'principle' is just an agreement in this sense. But we don't HAVE to require a system of reasoning that is limited to binary values or to a single conclusion.
Logic is just like any rule book to some game. You can have a game that
allows this 'explosion'. Games that use 'cycles' that can potentially go on forever is like this.
