Inconsistent Mathematics
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.
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.
Comments (48)
Inconsistent logic may find some application in artificial intelligence though.
But read the article.
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.
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.
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-.
"...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.
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.
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.
Quoting Banno
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.
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.
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.
Mathematics earns meaning through use.
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.
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.
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:
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.
"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?
It would be better called "Deconstructed Math" or "Existential Math", or how about Possible World Math :D
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...
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...
I don't agree. There's something important going on here.
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.
Please don't clutter up this thread with your non sequiturs.
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...
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.
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.
This is different.
(My bolding).
See Paraconsistent Logic
Fishfry and Tones have far better grasps than me of the logic and set theoretic aspects of this subject. I just like the observation that naive set theory has some value. :smile:
What do you think it is? Continuing with my example contradiction x is 1 AND x is not 1, what is the value of x?
Try this one:
Australasian Journal of Logic Paraconsistent Measurement of the Circle
Speaking for myself, paraconsistent logic is a "higher form" of logic and contains, as a proper subset classical logic (sentential, predicate, categorical) barring reductio ad absurdum. So, I'm not as I impressed as I would've liked to be by the fact that the area of a circle is "indifferent to changes in logic" - it must be so.
Secondly, I'm not so sure that we can be as mechanical in our thinking in paraconsistent logic as we can be in classical logic. In the former, we just plug in the propositions, apply natural deduction rules, and out at the other end we get a true proposition. In the latter, I surmise, we have to be constantly alert to the possibility that we aren't making some kind of mistake like how mathematicians have to take precautions that they aren't dividing by zero when tackling algebraic problems.
By the way, you haven't really addressed the issue I raised. A contradiction simply doesn't make any kind of sense at all. Banno is a member of TPF AND Banno is not a member of TPF. Ergo, Banno is...??? ( :chin: )
That would not be correct. The logic being proposed is as formal as any.
Quoting TheMadFool
Indeed; and yet here we have a paraconsistent logic that begins to make sense.
:ok: My bad.
Quoting Banno
Ok, so for what it's worth, here's what I think:
Nicolai A. Vasiliev (father of paraconsistent logic) is said to have admitted that he was influenced by Nikolai Lobachevsky (father of hyperbolic geometry)
[quote=Wikipedia]Reasoning by analogy with the "imaginary" geometry of Lobachevsky, Vasiliev called his novel logic "imaginary", for he assumed it was valid for the worlds where the above-mentioned laws (law of the excluded middle and the law of noncontradiction) did not hold, worlds with beings having other types of sensations. [/quote]
Since Nicolia A. Vasiliev is borrowing a page from Nikolai Lobachevsky's hyperbolic geometry, specifically the rejection of a postulate (Euclid's controversial (?? :chin: ??) fifth postulate), we need to look into the so-called Three Laws Of Thought:
1. Law of identity. A = A
2. Law of the excluded middle. p v ~p
3. Law of noncontradiction. ~(p & ~p)
At this point I'm out of depth but in my humble opinion these laws of thought are, at the end of the day, postulates i.e. they're assumptions. Given this, we're free to deny any or all of them and investigate what doing so might lead to. That's exactly what Nicolai A. Vasiliev, inspired by Nikolai Lobachevsky, did and we have logic that tolerate contradictions (paraconsistent logic).
Truth is there doesn't seem to be any real difference between paraconsistent logic and madness/stupidity as inconsistencies are the hallmark of all three. In that sense, Nicolai A. Vasiliev's paraconsistent logic is a study of insanity/inanity. (Mentally ill & mentally retarded) people since time immemorial have been using paraconsistent logic. :chin:.
[quote=Oscar Levant]There's a fine line between genius and insanity. I have erased this line.[/quote]
The Wisdom Of The Fool
Hey! It'about me! The Mad Fool! :scream:
Also, paraconsistent logic may prove its worth in matters inherently subjective:
X: The movie was phenomenal!
Y: No, it was not!
Z: It was both phenomenal and also not phenomenal!
De gustibus non est disputandum
Last but not the least, The Paradox Of Paraconsistent Logic:
Paraconsistent logic makes sense AND paraconsistent logic doesn't make sense!
...what happens if we do not assume these? Can we find a way to do that, which still maintains a capacity to construct arguments?
That's what is being proposed.
What this thread is about is that this fringe approach to logic has recently shown some interesting aspects of mathematical proof - the example above begins work towards a proof of integral calculus in a paraconsistent logic.
Quoting TheMadFool
But here we have a way to perhaps understand these inconsistencies in a coherent way. Madness and stupidity is perhaps to do with incoherence rather than inconsistency.
Hence the somewhat surprising break between consistency and coherence.
The world would grind to a halt and from then on chaos.
X: The UN will convene at 10:00 AM and the UN will not convene at 10:00 AM
Y: WTF?
From my own painful personal experience, one goes through both physical and mental paralysis when face to face with a contradiction!
That this discussion between the two of us is even possible depends on the laws of thought.
Quoting Banno
I know next to nothing about calculus so I'm not going to be able to make a comment that would further the discussion. I'll say this though, Godel's incompleteness theorems seem mighty relevant.
Quoting Banno
Coherence & Consistency? :chin:
I don't think these two are different. Coherence is only possible if there are no inconsistencies. In other words, coherent iff consistent.
Math had started out of practical uses in ancient Egypt for building the pyramids etc.
Would it be able to cover the area of the Traditional and Modal Logic, where they cannot cope with some of the real world cases in the arguments?
Quoting Banno
Existentialism as opposed to Rationalism, and denoting absurdity, irrationality and unpredictability?
Even if it is Inconsistent Math, if it is a Math, it would have some consistency, one would imagine.
Quoting TheMadFool
Quoting Banno
Yes, you don't. There's one issue.
Indeed; and cats can be black.
If you don't mind me butting in, I feel non-euclidean geometry is important to non-classical logic. N. A. Vasiliev (father of paraconsistent logic) drew an analogy between his paraconsistent logic and N. Lobachevsky's (father of non-euclidean geometry) and all we need to do is take the analogy just one step further and say that in higher-dimensional analogs of classical logic, "contradictions are true" (more on this below).
A simple example of this is Lobachevsky's take on Euclid's fifth postulate itself. In 2 dimensions, true that only one line can be drawn through a point that's parallel to another line but in higher dimensions (I hope I got this right), an infinite number of lines can go through a point such that they're all parallel to another line.
However, note that we have to switch between euclidean space and noneuclidean space for there to be a contradiction i.e. each of the two spaces above are self-consistent; it's only when the two are juxtaposed that a contradiction rears its ugly head so to speak.
Thus, as I mentioned earlier, whenever a contradiction arises, it should send alarm bells ringing - we're looking at the same thing in different contexts (analogously different dimensions). For this reason, I recommend paraconsistent logic for human experiences that are inherently subjective, each point of view providing its own "dimension" and assigning its own unique truth value to a proposition.
Is perception paraconsistent?
Cool video.
But no, perception is not paraconsistent, although there might be interesting paraconsistnt descriptions of perception.
Quoting Banno
I agree with @Banno also on the paraconsistency. At least I can see both ways. It's just a thing were you focus. Similar to those pictures where you can see two different pictures. A little bit of training your eyes and it's easy to notice. Of course that's a subjective opinion.
Yet really hard to think that something in nature would be paraconsistent. It may just be a useful model to depict things when we don't know something.
The classic picture: