## Inconsistent Mathematics

• 27
Well, here's an odd thing:

Inconsistent Mathematics

Apparently they are constructed by removing the Principle of Explosion - which I find has the less recent name ex contradictione quodlibet: (A & ~A)⊃B.

It is proposed as a way of getting around Russel's Paradox. Hence it might allow mathematics to be deduced from mere logic.

Interestingly, given recent conversations, Modus Ponens is invalid in certain paraconsistent logics... multivalued logic for instance.
• 1
Inconsistent mathematics would be useless and boring. Imagine applying a mathematical concept derived from inconsistent logical system to an engineering project, it would be disastrous.

Inconsistent logic may find some application in artificial intelligence though.
• 27
Inconsistent mathematics would be useless and boring.

• 27
Too obscure, it seems, for folk to comment on. At stake is the nature of language, or at least the best way to analysis it.

Over a hundred years ago - it doesn't seem that long - the idea was that we could build a coherent axiomatic system from which the whole of arithmetic, and hence ultimately mathematics, could be derived. Frege, Russell, Whitehead, (the early) Wittgenstein and others thought that language would submit to a similar treatment.

But this was found to lead quite rapidly to inconsistency. Russell rediscovered that sets that are members fo themselves cause all sorts of problems, and when Gödel used numbers to count theorems, he found that the whole enterprise was either inconsistent or incomplete.

When this sunk in philosophers stated to ignore formal treatments of language, leading to some excellent work on natural languages. It seemed that perhaps one might build a consistent, incomplete account of what might be said. This lead to the preeminence of natural deduction in logic, to the later Wittgenstein and to Davidson's project. Formal logic went off into possible worlds.

But now it seems that there might be an alternative. Rather than an incomplete yet consistent account of mathematics and language, we might construct an inconsistent yet complete account...

If we get rid of Ex Falso Quadlibet and Modus Ponens, what insight do we gain into logic, maths and language?

Perhaps this rough mud map of the significance of the issue will annoy some folk enough to respond.
• 13
My only, somewhat tangential, thought to contribute here is that in my own proposed extension to logic, you incidentally get something kind of like a paraconsistent logic "for free", yet without technically violating the principle of bivalence.

In that proposed extension, we abstract out the propositional force of the usual indicative propositions logic normally deals with, so instead of propositions like "x is F" we have gerund incomplete sentences like "x being F", to which we can then re-apply that propositional force a la "this state of affairs is the case: x being F". All of our usual logic still applies with just those gerunds even before we re-apply the propositional force, e.g. all F being G, and x being F, entails x being G; we don't have to actually propose that any of those states of affairs are the case to discuss the logical relationships between them.

My initial motive for abstracting out that propositional force was so that we could then apply different kinds of propositional force to the same gerunds without impacting their logical relations to each other: specifically, instead of proposing that some state of affairs is the case, we could rather propose that it be the case: a prescriptive or imperative proposition rather than a descriptive or indicative one.

But a side-effect of that, that makes it relevant to this thread, is that with the usual indicative propositions reconstructed as an indicative function wrapped around a gerund state of affairs, you can do things very much like paraconsistent logics, without actually violating the principle of bivalence.

In other words, if instead of saying "x is F" or "x is not-F", we say "there-is(x being F)" and "there-is(x being not-F)", we open up the possibility to say both of those things at the same time without any strictly formal contradiction. We would require a special rule that says "there-is(x being not-F)" entails "not(there-is(x being F)" if we wanted to enforce the usual kind of substantive consistency, and we don't have to introduce such a rule if we want to allow for paraconsistency. Just applying classical logic to this kind of construction automatically gives you something about tantamount to paraconsistency.
• 11
If we get rid of Ex Falso Quadlibet and Modus Ponens, what insight do we gain into logic, maths and language?

That truth means something very different to logicians than it means to philosophers in general. That's without speaking of the general public.

At the very base of it then, truth and falsehood are labels you apply to words by combining them with some stipulated object. "2+2=4" is true, why, because the objects which sit on the end of those symbols' interpretations all satisfy the equation. Change the meaning of the symbols and "2+2=4" is false. With the freedom to vary, and even explicitly construct much of, the frame of interpretation; the use case of a logic; all bets are off regarding a global interpretation of truth and falsity that works over all logics.

In that regard, what truth really means or how one ought to think about it won't be settled in formal logic alone - there can be no theorem of a logic that that logic is "right" or "apt" or "fit for purpose" as those terms are evaluations of it as a whole system. Generalities over systems; like languages/logics capable of arithmetic not being able to contain their own truth predicate on pain of inconsistency; hold over a broad swathe of logical systems. At its base, then, insofar as truth is a notion in formal logic; a notion of truth's aptness in any use case emerges as a frothing sea - of norms and heuristic - may shape and wash up smoothed wood - chunks of formalism-.
• 27
Two things there that look problematic.
"...the objects which sit on the end of those symbols' interpretations..." It seems to me erroneous to think of numbers as things; rather, they are ways of doing things with words. A number is not an object so much as what we do when we count; a way of using words.

"...what truth really means..." Why should we assume that there is only one true way to talk about truth - what it really means?

Not that I think you don't have a point; but your point might be made clearer.
• 29
As they saying goes, the most powerful mathematical language is where 0=1. How awesome to use it to prove anything!

It used to be a joke, but I think it goes in with the age old line of thinking, that simply assumes having a black box that takes care of a certain (here paradoxical) problem and we contemplate how the World would be then.

Similar like using super(hyper)tasks or something similar.
• 11
"...the objects which sit on the end of those symbols' interpretations..." It seems to me erroneous to think of numbers as things; rather, they are ways of doing things with words. A number is not an object so much as what we do when we count; a way of using words.

Thing is about as clear as you can get for a generic end point that a string gets mapped to. You put the objects in the background then map the symbols to them (see definition of interpretation here). I know you know this, if you could suggest a better word for the background object that an element of a theory is mapped to, I'm all ears.

Might be a mathematician's bias, I'm quite happy referring to abstracta as thingies. Infinity is a thingy.

Why should we assume that there is only one true way to talk about truth - what it really means?

That's rather the point I'm trying to make: if logics aren't designed for the same use case, if they have different actionable concepts of truth, believing any formal notion is unique or basic or fundamental doesn't seem to reflect the variation between concepts of truth in use and how competing logics might be compared.

I believed it would be easier to connote the idea that the heuristic concept of truth varies between logics with the "really means" phrase, the alternative seemed to me to engender a discussion about the individuation/contextual genesis of heuristic truth concepts.
• 27
...a generic end point that a string gets mapped to

Perhaps in the final analysis "1" doesn't map on to anything; https://thephilosophyforum.com/discussion/8110/1-does-not-refer-to-anything.

"Truth" has the same root in as "tree": PIE drew-o-, suffixed variant form of root deru- "be firm, solid, steadfast," and indeed the uses to which mathematicians and logicians and politicians put the word differ markedly, but there is the underpinning notion of reliability. A heuristic mathematics would be constructed even as the proto-indo-European language lead to the construction of English and Hindi.

Perhaps mathematical Platonism will give way to a recognition of maths as another language in which we should look not to meaning but to use.
• 4
Perhaps mathematical Platonism will give way to a recognition of maths as another language in which we should look not to meaning but to use.

What do you mean by that? Seems interesting to say that mathematics could get along well without certain principles governing how we operate in it.
• 7
Perhaps mathematical Platonism will give way to a recognition of maths as another language in which we should look not to meaning but to use.

Mathematics earns meaning through use.
• 7
The point of Mathematics is the universal consistency, accuracy and infallible knowledge it gives. If it is inconsistent, then it loses its' point.

We can introduce the concept of inconsistent maths in some possible worlds for some metaphysical debates, but that world would be a world of chaos and confusion.
• 27
The point of Mathematics is the universal consistency, accuracy and infallible knowledge it gives. If it is inconsistent, then it loses its' point.

Unless in can be pointedly inconsistent. Take a look at the articles cited.
• 7
Unless in can be pointedly inconsistent. Take a look at the articles cited

Sure. That sounds like again the different interpretations stemmed from different definitions of "inconsistent". Still wondering if it would be much practical in the real world.
• 58
This is really cool. I like the way the SEP article is written too. It's got a bit of sass to it.
• 7
If the Russell Contradiction does not spread, then there is no obvious reason why one should not take the view that naive set theory provides an adequate foundation for mathematics, and that naive set theory is deducible from logic via the naive comprehension schema.

At first I thought these ideas nonsense, but upon further reading, noting that nothing done here likely disturbs the course of mathematics as it is normally practiced - outside of foundations - I find these notions appealing. Of course, having been exposed to naive set theory many years ago and thinking it the real deal, only to be shocked by what has overtaken it, I am a tad biased. :cool:
• 26

First, a confession: I loooove paradoxes a term in logic reserved for contradictions but in the vernacular also applicable to the counterintuitive, the ironic, the strange, basically all manners of WTFery. I don't know why? Frankly, I can't make heads or tails of some of them, Wikipedia has a list, but I'm like a moth to a flame when it comes to paradoxes. This itself is a paradox because what happens usually is one first masters logic and only then investigates its limitations but look at me - all I know are the basics but I'm already delving into advanced topics.

Second, I want to discuss contradictions, the big daddy of inconsistencies. From the articles I read online, logicians are scared to death of contradictions because what they can lead to, their consequence - aptly named the principle of explosion (ex contradictione quodlibet). Even beginner's like me understand what it is.

However, for my money, a contradiction is, in and of itself, something nonsensical, it doesn't make sense, period! Why? Well, take a look at this contradiction: x is 1 and x is not 1. Clearly, x is not 1 is a denial of x is 1. In other words, the word "not" in x is not 1 asserts that x is 1 is false. Thus, x is 1 and x is not 1 states x is 1 is true and x is 1 is false. This issue with contradictions has nothing to do with ex contradictione quodlibet (I'm not saying contradictions cause undesirable effects).

Thus, even if one creates a logical system, like paraconsistent logic or dialetheism, by somehow blocking the principle of explosion, it doesn't change the fact that contradictions are inherently nonsensical as described above.

I guess what I'm really saying is that though paraconsistent logic and dialetheism tolerate contradictions, these systems drawing the line at ex contradictione quodlibet, the contradictions themselves as they appear in such systems are at the end of the day empty words.
• 6
But now it seems that there might be an alternative. Rather than an incomplete yet consistent account of mathematics and language, we might construct an inconsistent yet complete account...

"We might construct..." In other words, if we allow that anything goes, then we are able to do anything, so we might also be capable of doing everything. But of course, that's just the imagination running wild. That's why philosophy is considered to be a discipline. But pure mathematics, who knows what that is?
• 7
I feel Inconsistent Math is a misnomer.
It would be better called "Deconstructed Math" or "Existential Math", or how about Possible World Math :D
• 27
That doesn't strike me as a useful. If you want to make use of a many-valued logic, it is perhaps best to do it explicitly.
• 27
Still wondering if it would be much practical in the real world.

Flat Euclidean space can be seen as a special case in which parallel lines stay the same distance from each other. Consistent mathematics is a special case in which (A & ~A)⊃B.

Now, who'd a thunk non-euclidean space could be useful...
• 27
I find these notions appealing.

High praise, seeing as you would have a better grasp of what it is about than we commoners.

I find I had roughly assumed that incoherent and inconsistent were the same thing; but it seems we can have a coherent mathematics that is inconsistent...
• 27
the contradictions themselves as they appear in such systems are at the end of the day empty words.

I don't agree. There's something important going on here.
• 27
In other words, if we allow that anything goes, then we are able to do anything,

You are off on your usual pattern of commenting without grasping what is going on. What you wrote here is exactly wrong; it is what has explicitly been rejected by rejecting (A & ~A)⊃B.

• 27
Possible worlds is a quite different area. It explicitly assumes consistency. This does not. Of course, if this could be made coherent, then it might be applied to possible world semantics.

Nor does it have anything to do with existentialism. But it might be a sort of deconstruction, in which consistency is seen as a special case...
• 12
If we get rid of Ex Falso Quadlibet and Modus Ponens, what insight do we gain into logic, maths and language?
I suppose it means that descriptions of the world can not make sense, but that not making sense isn't the same thing as being impossible.
• 27
I suspect you have missed something. The supposition is that we might drop (A & ~A)⊃B and yet with suitable alterations retain the "making sense" part.
• 12
I suspect you have missed something. The supposition is that we might drop (A & ~A)⊃B and yet with suitable alterations retain the "making sense" part.
If the case of A and not A, then B. There is some way in which A and not A implies B. But, A and ~A is a contradiction. So, some As are contradictory. Why not? If language is objectively inspired there's no reason the world should have to follow our rules for it. On occasion the rules we have don't match the world.
• 27

This is different.

Contemporary logical orthodoxy has it that, from contradictory premises, anything follows. A logical consequence relation is explosive if according to it any arbitrary conclusion B is entailed by any arbitrary contradiction A, ¬A (ex contradictione quodlibet (ECQ)). Classical logic, and most standard ‘non-classical’ logics too such as intuitionist logic, are explosive. Inconsistency, according to received wisdom, cannot be coherently reasoned about.

Paraconsistent logic challenges this orthodoxy. A logical consequence relation is said to be paraconsistent if it is not explosive. Thus, if a consequence relation is paraconsistent, then even in circumstances where the available information is inconsistent, the consequence relation does not explode into triviality. Thus, paraconsistent logic accommodates inconsistency in a controlled way that treats inconsistent information as potentially informative.
(My bolding).

See Paraconsistent Logic
• 4
Black holes are an examples of inconsistency within pockets of our otherwise rational universe. Below the event horizon causality breaks down and all our laws of thought are violated. The law of explosion is not a law. It's an error, for when does a paradox become a contradiction? When we don't like it. Are minds are not made from infallible rules and although we must follow logic, logic doesn't have to respect us..
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