Jan 22, 2022 08:02 PM
https://aeon.co/essays/paraconsistent-lo...tent-world
EXCERPTS: . . . By 1931, a young logician named Kurt Gödel had leveraged a similar paradox out of Russell’s own system. Gödel found a statement that, if provable true or false – that is, decidable – would be inconsistent. Gödel’s incompleteness theorems show that there cannot be a complete, consistent and computable theory of the world – or even just of numbers! Any complete and computable theory will be inconsistent. And so, the Enlightenment rationalist project, from Leibniz to Hilbert’s programme, has been shown impossible.
Or so goes the standard story. But the lesson that we must give up on a full understanding of the world in which we live is an enormous pill to swallow. It has been almost a century or more since these events, filled with new and novel advances in logic, and some philosophers and logicians think it is time for a reappraisal.
If the world were a perfect place, we would not need logic. Logic tells us what follows from things we already believe, things we are already committed to. Logic helps us work around our fallible and finite limitations. In a perfect world, the infinite consequences of our beliefs would lie transparently before us. ‘God has no need of any arguments, even good ones,’ said the logician Robert Meyer in 1976: all the truths are apparent before God, and He does not need to deduce one from another. But we are not gods and our world is not perfect. We need logic because we can go wrong, because things do go wrong, and we need guidance. Logic is most important for making sense of the world when the world appears to be senseless.
The story just told ends in failure in part because the logic that Frege, Russell and Hilbert were using was classical logic. Frege assumed something obvious and got a contradiction, but classical logic makes no allowance for contradiction. Because of the classical rule of ex contradictione quodlibet (‘from a contradiction everything follows’), any single contradiction renders the entire system useless.
But logic is a theory of validity: an attempt to account for what conclusions really do follow from given premises. [...] theories of logic are like everything else in science and philosophy. They are developed and debated by people, and all along there have been disagreements about what the correct theory of logic is.
Through that ongoing debate, many have suggested that a single contradiction leading to arbitrary nonsense seems incorrect. Perhaps, then, the rule of ex contradictione itself is wrong, and should not be part of our theory of logic. If so, then perhaps Frege didn’t fail after all.
Over the past decades, logicians have developed mathematically rigorous systems that can handle inconsistency not by eradicating or ‘solving’ it, but by accepting it. Paraconsistent logics create a new opportunity for theories that, on the one hand, seem almost inalienably true (like Frege’s Basic Law V) but, on the other, are known to contain some inconsistencies...
[...] there is a hard choice: give up any inconsistent theory as irrational, or else devolve into apparent mysticism. With these new advances in formal logic, there may be a middle way, whereby sometimes an inconsistency can be retained, not as some mysterious riddle, but rather as a stone-cold rational view of our contradictory world.
Paraconsistent logics have been most famously promoted by Newton da Costa since the 1960s, and Graham Priest since the 1970s. Though viewed initially (and still) with some scepticism, ‘paraconsistent logics’ now have an official mathematics classification code [...] These logics are now studied by researchers across the globe, and hold out the prospect of accomplishing the impossible: recasting the very laws of logic itself to make sense of our sometimes seemingly senseless situation. If it works, it could ground a new sort of Enlightenment project, a rationalism that rationally accommodates some apparent irrationality. On this sort of approach, truth is beholden to rationality; but rationality is also ultimately beholden to truth.
That might sound a little perplexing, so let’s start with a very ordinary example. Suppose you are waiting for a friend. They said they would meet you around 5pm. Now it is 5:07. Your friend is late. But then again, it is still only a few minutes after 5pm, so really, your friend is not late yet. Should you call them? It is a little too soon, but maybe it isn’t too soon … because your friend is both late and not late. (What they’re not is neither late nor not late, because you are clearly standing there and they clearly haven’t arrived.) Whatever you think of this, paraconsistent logic simply counsels that, at this point, you should not conclude, however provisionally, that the Moon is made of green cheese, or 2+2=5, or that maybe aliens did build the pyramids after all. That would be just bad reasoning.
Now, such situations are so commonplace that perhaps it seems implausible that some fancy system of non-classical logic is needed to explain what is going on. But maybe we are so enmeshed in contradictions in our day-to-day lives, so constantly pulled in multiple conflicting directions at once, that we don’t even notice, except when the inconsistency becomes so insistent that it can’t be ignored.
Paraconsistent logics help us find structure in the noise. [...] An immediate concern about a paraconsistent approach is that it looks like a kind of cheating. It seems to sidestep the hard work of philosophical theorising or scientific theory-building. The worry, articulated recently by the philosopher of science Alan Musgrave in ‘Against Paraconsistentism’ (2020), is that:
"It can plausibly be maintained that the growth of human knowledge has been and is driven by contradictions. More precisely, that it has been and is driven by the desire to remove contradictions in various systems of belief."
If paraconsistent logics allow us to rest easy with an inconsistent theory, then there would be no impetus to improve. Another way to put the objection is that paraconsistency seems to offer an easy way out of difficult problems, a way to shrug off any objection or counter-evidence, to maintain flawed or failed theories long after they’ve been discredited. Does archaeological evidence contradict ancient-alien theory? No worries! This is just a contradiction, no threat to the theory. Rational debate seems stymied, if not destroyed.
[...] the simpler theory is only better to the extent that the world itself is simple. If not, then not. So too with consistency. The virtue of any given theory then will be a matter of its match with the world. But if the world itself is inconsistent, then consistency is no virtue at all. If the world is inconsistent – if there is a contradiction at the bottom of logic, or at the bottom of a bowl of cereal – a consistent theory is guaranteed to leave something out.
[...] A more on-the-ground answer to Musgrave’s methodological worries – and a warning to anyone tempted by paraconsistency as some kind of free pass – is that working within paraconsistent logics makes things more difficult, not less. The paraconsistent idea is that classical logic makes too many arguments valid, too many proofs go through when they shouldn’t, and so these validities and proofs are removed from the logical machinery. That makes drawing conclusions in a paraconsistent framework much harder because there are fewer inferential paths available. Someone who attempts a paraconsistent ancient-aliens theory may find that constructing valid arguments in their new ‘more permissive’ system is too challenging to be worth the effort.
[...] The deep unease about paraconsistency, beyond methodological issues about scientific progress or practical problems about devising proofs, is what it would philosophically mean to accept a worldview that includes some falsity (where ‘false’ means having a true negation). How can a false theory be acceptable? ...
[...] The explanations an inconsistent theory provides, it must be admitted, may not look like what traditional philosophers have been expecting. But the expectations of traditional philosophers have not come to pass; indeed, Gödel gave us a mathematical proof that they will never come to pass. In the meantime, there are other kinds of valuable explanation right in front of us. As Schrödinger put it: ‘The task is not so much to see what no one has yet seen, but to think what no one has yet thought, about that which everybody sees.’
In 1921, a young Ludwig Wittgenstein’s Tractatus Logico-Philosophicus was published [...] Wittgenstein announces in the book’s forward that he has solved all the problems of philosophy. He marches to this conclusion through a relentless sequence of numbered propositions, appearing to lay out the nature of logic, what it can accomplish and, most crucially, what it cannot. ... According to Wittgenstein, the limits of logic mean that what is truly important in life can be shown but not said. He writes: ‘The solution of the riddle of life in space and time lies outside space and time.’ But then, he adds: ‘The riddle does not exist.’
And so, Wittgenstein is forced to conclude that all of his talk about showing and saying and riddles has itself been illegitimate. Yes, ‘there are, indeed, things that cannot be put into words … They are what is mystical’ – but the mystical is itself logically impossible, a senseless contradiction. Wittgenstein must admit that his whole beautiful book has been, by his own lights, nonsense, and the problems of philosophy are not so much solved as passed over in silence.
Wittgenstein was looking, like many, for an Archimedean point, a place ‘outside’ the world. [...] Only from there, he thought, could the world be explained. In seeking even to articulate that, to draw the limits of what we can understand, Wittgenstein contradicts himself, and inevitably so. He finds a contradiction, just as Frege and Russell and Gödel did when they attempted a complete theory.
Wittgenstein took this as a kind of failure. But what if he had found what he was looking for and just didn’t recognise it? Perhaps Wittgenstein, like many others, felt pushed to make a false choice between a mysticism that provides some all-encompassing but inarticulate sense of the world, and a rational theory that is rigorous and precise but must be forever incomplete, inadequate.
This is a false choice if there can be a theory of the world that does both. Paraconsistency today does not have such a theory ready, but it holds out the (im)possibility of one, someday. It has recently been applied to religious worldviews (to Buddhism by Priest, to Christianity by Jc Beall). [...] ‘The proposition that contradicts itself,’ wrote the later Wittgenstein, ‘would stand like a monument (with a Janus head) over the propositions of logic.’
If we are living in an inconsistent world, in a world with contradictions from the foundations of mathematics to the triviality of dinner appointments, then its logic will leave room for falsity, doubts and disagreements. That is something many philosophers outside analytic and logical traditions have been urging for some time. As Simone de Beauvoir wrote: ‘Let us try to assume our fundamental ambiguity … One does not offer an ethics to a god.’ God would have no need of a logic, not even a paraconsistent one. But maybe we do... (MORE - missing details)
EXCERPTS: . . . By 1931, a young logician named Kurt Gödel had leveraged a similar paradox out of Russell’s own system. Gödel found a statement that, if provable true or false – that is, decidable – would be inconsistent. Gödel’s incompleteness theorems show that there cannot be a complete, consistent and computable theory of the world – or even just of numbers! Any complete and computable theory will be inconsistent. And so, the Enlightenment rationalist project, from Leibniz to Hilbert’s programme, has been shown impossible.
Or so goes the standard story. But the lesson that we must give up on a full understanding of the world in which we live is an enormous pill to swallow. It has been almost a century or more since these events, filled with new and novel advances in logic, and some philosophers and logicians think it is time for a reappraisal.
If the world were a perfect place, we would not need logic. Logic tells us what follows from things we already believe, things we are already committed to. Logic helps us work around our fallible and finite limitations. In a perfect world, the infinite consequences of our beliefs would lie transparently before us. ‘God has no need of any arguments, even good ones,’ said the logician Robert Meyer in 1976: all the truths are apparent before God, and He does not need to deduce one from another. But we are not gods and our world is not perfect. We need logic because we can go wrong, because things do go wrong, and we need guidance. Logic is most important for making sense of the world when the world appears to be senseless.
The story just told ends in failure in part because the logic that Frege, Russell and Hilbert were using was classical logic. Frege assumed something obvious and got a contradiction, but classical logic makes no allowance for contradiction. Because of the classical rule of ex contradictione quodlibet (‘from a contradiction everything follows’), any single contradiction renders the entire system useless.
But logic is a theory of validity: an attempt to account for what conclusions really do follow from given premises. [...] theories of logic are like everything else in science and philosophy. They are developed and debated by people, and all along there have been disagreements about what the correct theory of logic is.
Through that ongoing debate, many have suggested that a single contradiction leading to arbitrary nonsense seems incorrect. Perhaps, then, the rule of ex contradictione itself is wrong, and should not be part of our theory of logic. If so, then perhaps Frege didn’t fail after all.
Over the past decades, logicians have developed mathematically rigorous systems that can handle inconsistency not by eradicating or ‘solving’ it, but by accepting it. Paraconsistent logics create a new opportunity for theories that, on the one hand, seem almost inalienably true (like Frege’s Basic Law V) but, on the other, are known to contain some inconsistencies...
[...] there is a hard choice: give up any inconsistent theory as irrational, or else devolve into apparent mysticism. With these new advances in formal logic, there may be a middle way, whereby sometimes an inconsistency can be retained, not as some mysterious riddle, but rather as a stone-cold rational view of our contradictory world.
Paraconsistent logics have been most famously promoted by Newton da Costa since the 1960s, and Graham Priest since the 1970s. Though viewed initially (and still) with some scepticism, ‘paraconsistent logics’ now have an official mathematics classification code [...] These logics are now studied by researchers across the globe, and hold out the prospect of accomplishing the impossible: recasting the very laws of logic itself to make sense of our sometimes seemingly senseless situation. If it works, it could ground a new sort of Enlightenment project, a rationalism that rationally accommodates some apparent irrationality. On this sort of approach, truth is beholden to rationality; but rationality is also ultimately beholden to truth.
That might sound a little perplexing, so let’s start with a very ordinary example. Suppose you are waiting for a friend. They said they would meet you around 5pm. Now it is 5:07. Your friend is late. But then again, it is still only a few minutes after 5pm, so really, your friend is not late yet. Should you call them? It is a little too soon, but maybe it isn’t too soon … because your friend is both late and not late. (What they’re not is neither late nor not late, because you are clearly standing there and they clearly haven’t arrived.) Whatever you think of this, paraconsistent logic simply counsels that, at this point, you should not conclude, however provisionally, that the Moon is made of green cheese, or 2+2=5, or that maybe aliens did build the pyramids after all. That would be just bad reasoning.
Now, such situations are so commonplace that perhaps it seems implausible that some fancy system of non-classical logic is needed to explain what is going on. But maybe we are so enmeshed in contradictions in our day-to-day lives, so constantly pulled in multiple conflicting directions at once, that we don’t even notice, except when the inconsistency becomes so insistent that it can’t be ignored.
Paraconsistent logics help us find structure in the noise. [...] An immediate concern about a paraconsistent approach is that it looks like a kind of cheating. It seems to sidestep the hard work of philosophical theorising or scientific theory-building. The worry, articulated recently by the philosopher of science Alan Musgrave in ‘Against Paraconsistentism’ (2020), is that:
"It can plausibly be maintained that the growth of human knowledge has been and is driven by contradictions. More precisely, that it has been and is driven by the desire to remove contradictions in various systems of belief."
If paraconsistent logics allow us to rest easy with an inconsistent theory, then there would be no impetus to improve. Another way to put the objection is that paraconsistency seems to offer an easy way out of difficult problems, a way to shrug off any objection or counter-evidence, to maintain flawed or failed theories long after they’ve been discredited. Does archaeological evidence contradict ancient-alien theory? No worries! This is just a contradiction, no threat to the theory. Rational debate seems stymied, if not destroyed.
[...] the simpler theory is only better to the extent that the world itself is simple. If not, then not. So too with consistency. The virtue of any given theory then will be a matter of its match with the world. But if the world itself is inconsistent, then consistency is no virtue at all. If the world is inconsistent – if there is a contradiction at the bottom of logic, or at the bottom of a bowl of cereal – a consistent theory is guaranteed to leave something out.
[...] A more on-the-ground answer to Musgrave’s methodological worries – and a warning to anyone tempted by paraconsistency as some kind of free pass – is that working within paraconsistent logics makes things more difficult, not less. The paraconsistent idea is that classical logic makes too many arguments valid, too many proofs go through when they shouldn’t, and so these validities and proofs are removed from the logical machinery. That makes drawing conclusions in a paraconsistent framework much harder because there are fewer inferential paths available. Someone who attempts a paraconsistent ancient-aliens theory may find that constructing valid arguments in their new ‘more permissive’ system is too challenging to be worth the effort.
[...] The deep unease about paraconsistency, beyond methodological issues about scientific progress or practical problems about devising proofs, is what it would philosophically mean to accept a worldview that includes some falsity (where ‘false’ means having a true negation). How can a false theory be acceptable? ...
[...] The explanations an inconsistent theory provides, it must be admitted, may not look like what traditional philosophers have been expecting. But the expectations of traditional philosophers have not come to pass; indeed, Gödel gave us a mathematical proof that they will never come to pass. In the meantime, there are other kinds of valuable explanation right in front of us. As Schrödinger put it: ‘The task is not so much to see what no one has yet seen, but to think what no one has yet thought, about that which everybody sees.’
In 1921, a young Ludwig Wittgenstein’s Tractatus Logico-Philosophicus was published [...] Wittgenstein announces in the book’s forward that he has solved all the problems of philosophy. He marches to this conclusion through a relentless sequence of numbered propositions, appearing to lay out the nature of logic, what it can accomplish and, most crucially, what it cannot. ... According to Wittgenstein, the limits of logic mean that what is truly important in life can be shown but not said. He writes: ‘The solution of the riddle of life in space and time lies outside space and time.’ But then, he adds: ‘The riddle does not exist.’
And so, Wittgenstein is forced to conclude that all of his talk about showing and saying and riddles has itself been illegitimate. Yes, ‘there are, indeed, things that cannot be put into words … They are what is mystical’ – but the mystical is itself logically impossible, a senseless contradiction. Wittgenstein must admit that his whole beautiful book has been, by his own lights, nonsense, and the problems of philosophy are not so much solved as passed over in silence.
Wittgenstein was looking, like many, for an Archimedean point, a place ‘outside’ the world. [...] Only from there, he thought, could the world be explained. In seeking even to articulate that, to draw the limits of what we can understand, Wittgenstein contradicts himself, and inevitably so. He finds a contradiction, just as Frege and Russell and Gödel did when they attempted a complete theory.
Wittgenstein took this as a kind of failure. But what if he had found what he was looking for and just didn’t recognise it? Perhaps Wittgenstein, like many others, felt pushed to make a false choice between a mysticism that provides some all-encompassing but inarticulate sense of the world, and a rational theory that is rigorous and precise but must be forever incomplete, inadequate.
This is a false choice if there can be a theory of the world that does both. Paraconsistency today does not have such a theory ready, but it holds out the (im)possibility of one, someday. It has recently been applied to religious worldviews (to Buddhism by Priest, to Christianity by Jc Beall). [...] ‘The proposition that contradicts itself,’ wrote the later Wittgenstein, ‘would stand like a monument (with a Janus head) over the propositions of logic.’
If we are living in an inconsistent world, in a world with contradictions from the foundations of mathematics to the triviality of dinner appointments, then its logic will leave room for falsity, doubts and disagreements. That is something many philosophers outside analytic and logical traditions have been urging for some time. As Simone de Beauvoir wrote: ‘Let us try to assume our fundamental ambiguity … One does not offer an ethics to a god.’ God would have no need of a logic, not even a paraconsistent one. But maybe we do... (MORE - missing details)