Post-analytic Philosophy?

i) Introduction

ii) Objective Truth?

iii) Philosophy Must Be Political?

iv) Richard Rorty

The term 'post-analytic philosophy' was first used in the mid-1980s. At that time it referred to those philosophers who were indebted to analytic philosophy, but who, nonetheless, believed that they'd moved on from it (for whatever reasons).

The term seems, prima facie, odd. After all, how can philosophers be 'post'- or anti-analysis? Surely even most examples of post-analytic philosophy will contain analyses of sorts. (This isn't necessarily to say that philosophy must consist entirely of analysis.)

Thus, the term must instead refer to the tradition (in a broad sense) of analytic philosophy. But which aspects of that tradition? Which particular philosophers? Did all analytic philosophers have an philosophical essence in common? And let's not forget that philosophical analysis occurred well before the analytic tradition got under way. (What is it that Hume, Hobbes, Aquinas, etc. did if it wasn't - at least in part - analysis?)

The above are all problems which, to some extent, subside once the history and use of the term 'post-analytic philosophy' is studied.

However, it is indeed analysis that some philosophers seem to have a problem with. Or, rather, perhaps it's more accurate to say 'philosophical analysis' rather than the simple 'analysis'. This is obviously the case because the words 'philosophical analysis' are more particular than 'analysis' and it may/will contain assumptions as to what philosophical analysis actually is.

Objective Truth?

If we want to put meat on what post-analytic philosophers see to be the problem (or simply a problem) with analytic philosophy, it's best to consult late-20th century and contemporary American pragmatism. This school is itself seen as being part of the post-analytic movement (i.e., which isn't a determinate or real school).

Many would say that such American pragmatists have a problem with the very notion of objective truth, realism and representationalism. They are things they see as being an idée fixe throughout the history of philosophy. And this, indeed, is no less the case when it came to 20th-century analytic philosophy.

A personal objection to this is that I've hardly read a single analytic philosopher mention - or use - the words “objective truth”. (I have read, however, Peter van Inwagen's 'Objectivity'.) Then again, it can easily be countered that a philosopher needn't use the actual words “objective truth” in order for him to be committed to the notion of objective truth. In other words, perhaps he simply calls it by another name.¹

In any case, the position that objective truth doesn't exist (or that it's not a worthy aim in philosophy) goes alongside a stress on the contingency of cognitive activity, the importance of convention and utility, and, indeed, the idea that human (or social) progress can never be ignored – not even in philosophy. Nonetheless, here again I don't see how there's an automatic (or prior) problem with accepting all this and still engaging in analytic philosophy (or in philosophical analysis).

For example and in very basic terms, one could offer a philosophical analysis of philosophical analysis (or some part thereof). And then, as a result, see philosophical problems with such philosophical analysis. Despite that, such a philosopher would still be in the domain of analytic philosophy (or of philosophical analysis).

Strangely enough, Richard Rorty seems to agree with this position. Or, at the least, he says something similar. In an interview conducted by Wayne Hudson and Win van Reijen, Rorty states:

"I think that analytic philosophy can keep its highly professional methods, the insistence on detail and mechanics, and just drop its transcendental project. I'm not out to criticize analytic philosophy as a style. It's a good style. I think the years of superprofessionalism were beneficial." 

I said the position is “similar” to the one advanced by Rorty. It's similar in the sense that an analytic philosopher needn't “drop [his] transcendental project”. That is, an analytic philosopher may be fully aware of Rorty's positions/arguments (or the general positions of post-analytic philosophers) and still be committed to the transcendental project. (Of course we'd need to know what Rorty means by the words “transcendental project”.)

Philosophy Must be Political?

It seems that the position of many post-analytic philosophers is primarily political - or at least primarily social – in nature. Hilary Putnam (1985), for example, has said that analytic philosophy has “come to the end of its own project—the dead end”. That can be taken to mean that philosophy should connect itself more thoroughly with other academic disciplines. Or, more broadly, that analytic philosophy should connect itself with culture or society as a whole.

The problem is that, on and off, analytic philosophy has already connected itself to many other disciplines. (Admittedly, that's been more the case since the 1980s and the rise of cognitive science.) To give just a couple of examples: the logical positivists connected themselves to science (or at least to physics). And, to give another example, philosophers in the 19th century connected themselves to logic, mathematics and, again, to science. This non-ostentatious “interdisciplinary” nature of philosophy has been the case, in fact, throughout the history of philosophy.

One can also say that philosophy can connect itself to other disciplines - and even culture as a whole - and still remain analytic philosophy. Philosophers can still practice philosophical analysis. (This, again, raises the question as to what analytic philosophy - or philosophical analysis - actually is.)

A philosopher may also ask why he should connect himself to other disciplines - never mind to something as vague (or as broad) as culture. In other words, a philosopher must have philosophical reasons as to why this would be a good thing, just as a philosopher must have philosophical reasons as to why it's a bad thing. That means that there'll be philosophical angles to this very debate. However, it can be added, those angles needn't always be philosophical in nature.

Another slant on this philosophy-society "binary opposition" is that it is argued that analytic philosophy is too professional and therefore too narrow. In other words, analytic philosophers are over-concerned with very tiny, narrow and specialised problems which have almost zero connection to society as a whole or indeed to anything else.

More technically and philosophically, it can also be argued that certain central commitments and assumptions of analytic philosophy have been shown to be indefensible. (Hence Putnam's own words quoted earlier.)

Yet all disciplines can be said to concerned with narrow or specialised issues or concerns. Yet this is an accusation more often aimed at analytic philosophy than at any other academic subject.

Richard Rorty

Richard Rorty appears to be talking about analytic philosophy as it was in the past (say, the 1950s to the 1970s), not as it is today or as it has been since, say, the 1980s.

Take the view that analytic philosophy has as its primary aim a form of knowledge which grounds all other forms of knowledge. This is odd. It's true that much traditional philosophy has placed various philosophical domains in the position of what used to be called First Philosophy. (It was once metaphysics, then epistemology, then philosophy of language, then philosophy of mind...) However, in the 20th century this has been far from the case. Indeed philosophers - throughout the 20th century - have argued against the nature of a first philosophy.

Take naturalists (e.g., the logical positivists, then Quine): arguably, they placed science (or simply physics) in the role of first philosophy. (Although such naturalists saw physics as being primary, that isn't in itself a commitment to also seeing it as some kind of first philosophy.)

It must be said that just as Rorty's post-philosophy is a philosophical position, so too is the Wittgensteinian attempt to “dissolve” and then disregard philosophical problems (if not philosophy itself). This position can be said to be held by Putnam and John McDowell, as well as by Rorty. (A more specific example of this would be the “problem” of how mind and language are connected to the world.)²

In any case, there's just as strong case for arguing that Rorty's later position was more a case of post-philosophy than post-analytic philosophy. In other words, like Heidegger and Derrida, Rorty had a problem with the whole damn show that is Western philosophy. And, here again, it can be argued that Rorty's position was more political (or social) than strictly philosophical. That said, a position that rejects philosophy in toto can't help being philosophical – in some or many ways – itself, as Rorty would have no doubt happily admitted. (Jacques Derrida did admit this.)

Notes

1) If you say that an argument (or a single statement) is “warranted and therefore assertible”, then is that a case of being wedded to the notion of objective truth? Or is the notion of warranted assertibility a different species entirely?

2) I put the word “problem” in scare quotes because the very stance of seeing such problems as problems means - according to Rorty, Derrida, Heidegger, etc. - that we've fallen prey to particular philosophical “style of thinking”. However, that position too would be a philosophical position.

Is Constructive Empiricism's Use of Counterfactuals Illicit?

Bas van Fraassen

Possible Worlds

At first glance it may seem odd that James Ladyman (2000) argues that constructive empiricist talk of observability may well require a commitment to possible worlds. Thus it would also seem - again on the surface - that any kind of empiricist must shun possible worlds. 

Possible worlds - to state the obvious – are neither empirical entities and nor are they observable (not even in principle). Thus, as a consequence, constructive empiricists would require unobservable entities in order to legitimise or justify their talk of observable entities.

Let's put the observability-in-principle position thus:

x is observable iff observers were in a suitable place, then they could observe x.

Now that's a bone fide counterfactual. Thus an old-style empiricist may now ask:

What are the truth-conditions of the statement “x is observable” (i.e., the statement above)?

Perhaps the constructive empiricist would say:

x is observable-in-principle because it has truth-conditions(-in-principle).

Why should the old-style empiricist accept either observables-in-principle or indeed truth-conditions-in principle? The whole point of observables is that they can be seen, smelt, heard, touched, etc. at the present moment in time (or at least they can... in principle!).

The constructive empiricist, thus, may require his truth-conditions to exist at possible worlds. Nonetheless, the truth-conditions can't exist at this moment in time because possible worlds don't exist at this or at any moment in time – at least not according to the empiricist.

Context-dependence

I'm not sure if I understand Bas van Fraassen's reply to such points. It involves a strong use of the notion of “context-dependence”. In basic terms, when we “fix the context” of a counterfactual claim about observability, that somehow stops the claim from being modal in nature.

What is fixed is the epistemic community – all the “suitably constituted observers” who're relevant to the counterfactual claim of observability. This, itself, is supposed to make the counterfactual conditional non-modal in nature. That is, once we explicate the nature of (ideal) observers in (ideal) situations, it is these things which we can empirically investigate (Monton and van Fraassen 2003, 413-414). Nonetheless, we still have the non-Humean move from what's empirically observable at this present moment in time and in this place at this time, to that which is not empirically observable at this moment in time or in any place at this moment in time.

Thus statements about what's true at this moment in time - and at this place at this time - slide into what would be the case at other times and at other places. There's still a non-Humean jump that's not eased with these technical additions.

Van Fraassen adds extra detail to this.

He argues that the principle of observability can be cashed out in terms of the objective properties of the world. Moreover, we can use our best scientific theories in order to determine the truth (or content) of “x is observable” (Monton and van Fraassen 2003, 415-416). But, again, there are hidden modal assumptions in all this technical detail and even, as James Ladyman argues, hidden commitments to possible worlds.

Take this statement:

If Bertrand Russell's teapot in Andromeda showed itself to observers (suitably prepared, etc.), then they could observe it.

It seems like a sleight of hand to say that to understand the statement above is - even though a counterfactual - entailed by the facts or the phenomena of the observable (empirical) world. Presumably that must be a reference to teapots and observers which and who exist at this moment in time and can be observed at this moment in time. However, wouldn't Hume have argued that a bone fide empiricist couldn't jump from teapots in our solar system to teapots in Andromeda without begging a few questions or assuming a few facts?

I mentioned facts in the last sentence. Ladyman (2004, 762) talks, instead, about “laws”. That is, he says that

“unless we take it that the specification by science of some regularities among the actual facts as laws … is latching onto objective features of the world”.

Wouldn't that mean, in our case, that the laws and objective features of Andromeda are assumed to be like the laws and objective features of the earth and our solar system? Yes, the laws of Andromeda are the same as the laws of the earth and our solar system. And, I assume, Russell's teapot would behave in a pretty similar way to how it would behave if it were floating near the moon or even in the sky above us.

Nonetheless, Ladyman does go on to say that only objectively-existing laws (rather than “pragmatically selected empirical regularities”) can justify (or warrant) our claims about the nature of Russell's teapot or any other phenomena of Andromeda. So it's not the constructive empiricist's claims about the nature and behaviour of Russell's teapot in Andromeda that are problematic per se. It's that in order to justify (or warrant) those claims the constructive empiricist would need to commit himself to entities which aren't kosher from an empiricist's point of view: viz., objective laws. Again, the only thing that a constructive empiricist can rely on are pragmatically-selected empirical regularities, not objective laws. Indeed the acceptance of objective laws commit one to a metaphysics that's not empiricist in nature.

References

Monton, B., and van Fraassen, B., (2003) “Constructive Empiricism and Modal Nominalism”, British Journal for the Philosophy of Science, 54: 405–422.

Ladyman, James. (2000) “What's Really Wrong With Constructive Empiricism? Van Fraassen and the Metaphysics of Modality”, British Journal for the Philosophy of Science, 51: 837–856.

- (2004) “Constructive Empiricism and Modal Metaphysics: A Reply to Monton and van Fraassen”, British Journal for the Philosophy of Science, 55: 755–765.

Realism, Anti-realism, and Evidence-transcendent Statements

This piece deals with the nature of truth-valued statements which have semantic contents which are said to be “evidence-transcendent”. In less technical terms, the nature of unobservability and observability-in-principle are tackled within an anti-realist versus realist context.

The classic cases are covered: including the doubling in size of the universe, Bertrand Russell's flying teapot, Michael Dummett's organisms in Andromeda, past and future events, electrons, other minds and what it is to be bald.

Within these contexts, we'll also try to clarify what it is to understand statements which have evidence-transcendent content.

Realist Truth

The realist position on truth can appear strange, at least prima facie. Take this statement:

“In 607 AD there were precisely one million people with ginger hair in Europe.”

According to the realist, that's either true or false. He may also say that it's determinately true or false (i.e., it's truth is fixed in time).

Similarly for this statement:

“Is is true [false] that Theresa May, at this precise moment, is dreaming about flowers.”

If Theresa May isn't asleep, it's false. That would be easy – in principle – to determine. Though what about if she is asleep at this precise moment? Is it still determinedly true or false that she's dreaming about flowers?

Despite that, it may well be the case that although one takes a realist position on this, one needn't take a similarly realist position on all other domains of discourse. (This is often said of anti-realism, not realism.)  More specifically, statements about Bertrand Russell's flying teapot or Dummett's organisms in Andromeda (both covered later) may well be determinately true. Nonetheless, is it automatically the case that a realist should also have exactly the same position, for example, on statements about the future? Perhaps a realist believes that statements about the future throw up problems which aren't encountered in these other domains.

Understanding Statements

Following on from that, an anti-realist can ask a realist two questions:

i) If you understand the statement “It is true [or false] that that the universe sprang into existence just five minutes ago, replete with traces of a long complex past” [worded by Bob Hale], then how do you understand that sentence?

ii) What gives you the warrant to say that it's either true or that it's false?

The realist may now reply:

What do you mean by the word 'understand' [as in “understand that sentence”]?

A standard picture is that in order to understand p, one needs to understand both p's truth-conditions and then somehow decide whether or not those truth-conditions obtain. So, in the case of the statement about the universe doubling in size, how would the truth-conditions for the universe being the same size differ from the truth-conditions of a universe which has doubled in size? (I'm assuming here that the philosophical puzzle of a doubled universe works. As it is, there are arguments against it.) Secondly, how would someone be warranted (or justified) in saying that the universe has or hasn't doubled in size?

The argument is that if the realist can't answer these questions, then his position is untenable. That is, he doesn't know what he's talking about. Or, less judgementally, he doesn't understand what he's talking about. That means that we have no means of understanding what a realist position on truth (at least as regards the doubled-universe scenario) amounts to.

The Doubled Universe

Since we've just mentioned the doubled-universe scenario, Bob Hale talks in terms of what he calls “chronically e-transcendent statements” [1999]. (The 'e' is short for 'evidence'.) He cites the doubled-universe case:

“Everything in the universe has doubled in size.”

As well as:

“The entire universe sprang into existence just five minutes ago, replete with traces of a long and complex past.”

(These statements have been much discussed in philosophy; though not always in the context of the realism vs. anti-realism debate.)

If the universe had doubled in size (so the argument goes), then there'd be no way of telling that it had actually done so. Thus we couldn't say that it has or that it hasn't doubled in size. Nonetheless, isn't it the case that it either has or it hasn't doubled in size? And if that's the case according to the realist, the statement is indeed determinately true or determinately false.

Unobservable Electrons

There are many problems for the anti-realist position too; especially if anti-realism is closely tied to acts of verification (or to verificationism).

Take the many unobservable phenomena of science (specifically of physics). Can it be said that statements about, say, electrons are similar in kind to statements about our doubled universe or flying teapots in distant galaxies? Certain anti-realists would say that even though electrons aren't observable, we're nonetheless led to posit their existence because of the evidence supplied by phenomena which are indeed observable. Thus, although electrons are too small to be observed, we're led to them by observable phenomena - plus, of course, lots of theory. (Couldn't the realist argue that he's led to his statements about determinate truth about the unobservable-in-principle by what is actually observable?)

The idea that an electron is posited due to phenomena we can observe (along with theory) is parallel - or additional - to the idea of something's being observable-in-principle.

It could be said that something as tiny as the electron could be observable in principle; except for the large problem that it's deemed to be a “theoretical entity” anyway. That is, besides mathematical structure (as well as theory), there would be nothing to observe even if we could observe it. On the other hand, we can say that a distant something in our solar system could be observed in principle. That may mean that this something isn't a theoretical entity at all. Well, in a sense, it is a kind of theoretical entity in that it hasn't actually been observed. Though being, say, a teapot, it could be observed if we were able to travel to the distant place it inhabits. (Let's forget the science here!)

There's one clear problem for this observable/unobservable opposition. This is that there isn't always (or never) a clear dividing line between observation-statements and theoretical statements. That can be because observation-statements involve theory and theoretical statements involve (elements derivable from) observation. Still, whatever problems there are here, they're not as problematic as those statements about unprovable mathematical statements; and certainly not as problematic as our doubled-universe scenario.

Other Minds

A similar problem arises for anti-realism when it comes to other minds. We can't observe the goings-on in other people's minds. Nonetheless, like electrons, we're led to acknowledge other minds because of the things we can indeed observe. However, in this case we still need to accept that behaviour (including speech and writing) isn't conclusive evidence for other minds.

There are many problems thrown up by other minds. Behaviourism, for one, was/is one response to these philosophical problems. And that's why certain types of behaviourist relied exclusively on behaviour (whether physical or verbal behaviour) in their experiments and musings. That meant that other minds ceased being a problem for behaviourists because minds in effect didn't (really) exist. Or, at the least, behaviourists believed - at one time - that the mind wasn't a fit subject for science.

Is John Bald?

There's also the problem of statements which involve vague concepts or references to vague states-of-affairs (if there can be such a thing!). Take the well-known case of whether a certain person is bald.

To clarify with a statement: “John is bald.” This can certainly be said to have truth-conditions (which certain earlier examples didn't have). Nonetheless, in a certain sense, truth-conditions don't really help here. That is, we have access to John and to John's head. What we don't have access to is whether or not it's true or false that he's bald. (I'm taking it here that someone can be bald even if they have a few hairs left.) Since it's already been said that truth-conditions aren't the problem, then perhaps we do have a problem with the “vague predicate” that is “bald”.

Here we encounter problems covered by a sorties paradox. Can we ignore them for now? Perhaps we can. It can be said, for example, that we can make a stipulation as to what makes someone bald. (This is deemed to be problematic if taken as a sorties paradox.) We can say that anything less than 100 hairs constitutes baldness in a given male. Consequently it can be said that it's determinately true that John is bald or not bald (i.e., post-stipulation).

What if we accept the sorties paradox? Then we'd be unable to decide (care of truth-conditions or anything else) whether or not John is bald. Nonetheless, the realist, yet again, would argue that it's a determinate fact which makes it the case that either John is bald or John isn't bald. The problem is that if we accept the paradox, we can't know either way.

Michael Dummett, for one, had a problem with this realist conclusion.

In terms of the word “bald”, that would mean that our use of words like that would have “confer[red] on them meanings which determine precise applications for them that we ourselves do not know”. Basically, that would mean that the world tells us if John is bald or not. Or, at the least, the world (including John's head) determines the truth or falsity of the statement “John is bald”. In addition, the world determines the truth regardless of whether or not we can ever determine it to be true or false. Yet surely whether or not someone is bald is something to do with what we decide. The world has no opinion on this or on anything else.

Still, this sorties paradox has an impact on the nature/reality of baldness even if we accept a conventional stipulation about baldness. That is, the logical process which leads from having, say, 1000 hairs to having a single hair is still ultimately paradoxical. That is, step by step we can move from the statement “A man has a thousand hairs is not bald” to the statement “A man with three hairs is not bald” without a hiccup.

Another way of looking at this is to say that if the realist is correct, then any indeterminacy there is has to do with our vague predicates or vague statements, not the world itself (or with John's baldness).

The Teapots/Organisms of Andromeda

Michael Dummett offers us this statement:

“'There are living organisms on some planet in the Andromeda galaxy.'”

That statement, according to Dummett's realist, is “determinately true or false” [1982].

In response, the anti-realist adds an extra dimension to this case in terms of the aforementioned idea of observability-in-principle. Dummett expresses the anti-realist's (as well as, I suppose, the realist's) position in this way:

“'If we were to travel to the Andromeda galaxy and inspect all the planets in it, we should observe at least one on which there were living organisms.'”

Basically, because the science and the practicalities are so far-fetched in this case, we can't do anything else but forget them. In other words, we need to give the anti-realist the scientific benefit of the doubt. The problem here, though, is that if we give the anti-realist the benefit of the doubt about this currently unobservable situation (which is nonetheless supposedly observable-in-principle), then we can - or must – do the same in the countless other cases of unobservable phenomena in science (particularly in physics). Having said all that, these provisos may not be to the point here.

In any case, if the aforementioned organisms are observable-in-principle, then perhaps they can't be (fully) theoretical entities. Or, less strongly, if the Andromeda organisms are theoretical entities at the present moment, then they needn't remain theoretical entities simply because they can be observed in principle. (Though, again, perhaps the atomic and subatomic world may one day be observed; though not if the entities concerned are simply “theoretical posits” and/or mathematical structures.)

Statements About the Future

Dummett also brings up another example of something that's unobservable-in-principle: a future event. How can we deal with truth-valued statements about future events?

My prima facie position is twofold. 

i) Such statements are neither true nor false. ii) If such statements are neither true nor false, then they serve little purpose.

The realist believes that statements about future events are determinately true or false. According to Dummett, the realist believes that 

“there is [ ] a definite future course of events which renders every statement in the future tense determinedly either true or false” [1982].

I find realism towards statements about the future even more difficult to accept than realist claims about other domains. I would agree with Dummett when he says that the only way that a future-tensed statement can be true or false at this moment in time would “only [be] in virtue of something that lies in the present”. This is surely Dummett hinting at some form of determinism in that what is the case at this moment in time will have a determinate causal affect on what will be the case in the future. (Try to forget arguments against determinism here; as well as references to quantum mechanics, backwards causation, action-at-a-distance, etc.)

Let's take that deterministic position to be the case. That is, a future-tensed statement is true or false at the present moment in time because of what is the case at the present moment in time. That's the case even though the event referred to is in the future. That's fair enough; though it's clear that the realist would have no way of knowing whether or not it's true or false at the present moment. Nonetheless, we've already seen that the realist happily and willingly accepts his position of epistemic deficiency.

Instead of using the word “determinism”, Dummett talks about “physical necessity” instead.

Dummett picks up on an interesting consequence of what was said in the previous paragraph. What the realist must do, Dummett argues, is tell us what are the truths about the statement (or situation) at the present moment in time and how these truths bring about the truth of a statement about a future event. That means that only known truths at the present moment can contribute to truths about future events – at least within this context of “physical necessity” or determinism.

Dummett spots a double problem with the realist's position here. The realist can't determine the present-time truths which would bring about truths about the future. And, by definition, he can't determine - as a consequence of this - a statement about the future that's true at the present moment. That means that although the realist acknowledges his lack of a means to determine the truth of a statement about a future event, he hasn't even got a way of determining the present-time truths which will determine - or cause by virtue of physical necessity – the truth of statements about future events. Thus, in order to make sense of his realism, the least we should expect from the realist is the truths of statements about current situations which would cause - or determine - the truths of statements about future events. Without all that, realism towards statements about the future make little sense.

References

Dummett, Michael. (1982) 'Realism'.

Hale, Bob. (1999) 'Realism and its Oppositions'.

Paraconsistent Logic: Inconsistency, Explosion and Relevance

(1) Do paraconsistentists really accept the conjunction P & ¬P? (2) Does that conjunction really “generate every theorem in the language”?

The following essay will question the paraconsistent acceptance of inconsistencies. It will also question the related acceptance of logical explosion and logical triviality (which paraconsistent logicians also reject) by classical logicians.

The main theme of this piece (if sometimes implicit) is that both logical explosion and logical triviality result from taking logical statements, premises or propositions as being empty logical strings (or syntactic strings) — i.e., notations without semantic content. Indeed the position advanced here can be deemed to be (if loosely and in a limited sense) against logical formalism, in which logical strings are treated as being autonomous of — and independent from — semantics.

(The positions expressed above amount to the same argument I’ve previously provided for the purely logical renditions of the Liar Paradox and even for Gödel sentences — see here and here. It’s of course the case that all such logical strings are provided with what is called a “semantics”. Yet that is a semantics purely in the limited sense that these strings are somewhat arbitrarily classified as being “true”, “false”, as having an “extensional domain”, etc.)

So the following is an essay in the philosophy of logic (which will explain the dearth of logical notation). In other words, this essay is not a work in logic itself.

Introduction

As the American philosopher C.I. Lewis once claimed (as quoted by Bryson Brown) that no one “really accepts contradictions”. From that it can be said that the prime motivation for paraconsistency (as can sometimes be gleaned from what various paraconsistent logicians themselves say — at least implicitly) is mainly epistemological. Sometimes it’s also inspired by theories, experiments and findings within quantum physics.

The following is also at one with the position of the American philosopher David Lewis (1942–2001) who argued (see here) that it’s impossible for a statement and its negation to be true at one and the same time. (Lewis believed in the “reality” of possible worlds. He also believed that in none of these possible worlds is the conjunction P &¬P true.) Having said that, all this depends on what exactly is said about the embracing of both P and ¬P; as well as on how that embracing is defended.

A related objection is that negation in paraconsistent logic isn’t (really) negation: it’s merely, according to B.H. Slater, a “subcontrary-forming operator”. Indeed the dialetheic philosopher Graham Priest (1948-) explicitly states that paraconsistent negation isn’t Boolean negation. Thus Priest also uses the (epistemic and psychological) word “denial” when referring to negation.

Thus if the acceptance of inconsistencies is largely an epistemological move (as shall be argued), then that move isn’t really (or isn’t actually) an acceptance of both P and its negation (i.e., at one and the same time) at all.

The Acceptance of P∧ ¬P

The American philosopher Bryson Brown says that

“a defender of [C.I.] Lewis’s position might argue that we never really accept inconsistent premises”.

Yet Brown immediately follows that statement with a defence of inconsistency which doesn’t seem to work.

Brown continues:

“After all, we are finite thinkers who do not always see the consequences of everything we accept.”

Perhaps C.I. Lewis’s reply to those words might have been that we don’t “accept inconsistent premises” that we know — or that we think we know — to be inconsistent. Of course it’s the case that having finite minds is a limit on what we can know. Nonetheless, we still don’t accept the conjunction P ∧ ¬P. (It needn’t always be entirely a case of symbolic autonyms.) Paraconsistent logicians don’t accept the statement “1 = 0” either; and virtually no one would accept the conjunctive statement “John is dead and John is alive”.

As for not seeing the consequences of our premises.

No, we don’t see all the consequences of all the premises we accept. However, we do know the consequences of some of the premises we accept. So the finiteness of human minds doesn’t stop us accepting certain premises — or even entire arguments — either. Still, Brown may only be talking about inconsistent premises which reasoners simply aren’t sure about. In such cases, then, the limitations of our minds is salient: we can’t know all the consequences of all the premises we accept. In addition, we can’t know if the all the premises and conclusions we accept are mutually consistent.

Similarly, do we (or do quantum physicists, scientists, theorists, paraconsistentists, etc.), as Bryson suggests, accept inconsistent premises for (to use Brown’s word) “pragmatic” reasons? Would C.I. Lewis (again) have also said that even in this case “we never really accept inconsistent premises”?

Brown goes on to say that

“[i]nference is a highly pragmatic process involving both logical considerations and practical constraints of salience”.

This talk of a “pragmatic process” and “salience” is surely bound to make us less likely to accept inconsistent premises, rather than the opposite.

Take salience.

Not only will inconsistent premises throw up problems of salience (or relevance): such problems will also (partly) determine our choice between two contradictory — or simply rival — premises. What’s more, further talk of (to use Brown’s words) “how best to respond to our observations and to the consequences of what we have already accepted” will, again, make it less likely that we would accept inconsistent premises, not more likely.

In other words, P may have observational consequences radically at odds with the observational consequences of ¬P. So why would we accept both — even provisionally?…

… Unless, that is, accepting both P and ¬P is simply an (epistemic) way of hedging one’s bets! So is that really all that (philosophical) paraconsistency amounts to?

Logical Explosion

The American philosopher Dale Jacquette (1953–2016) put the paraconsistent position when he said that

“logical inconsistencies need not explosively entail any and every proposition”.

What’s more, “contradictions can be tolerated without trivialising all inferences”. Here we have the twin problems (for paraconsistent logic) of logical explosion and logical triviality.

[Ex contradictione sequitur quodlibet = (one of a few translations) “from contradiction, anything follows”.]

To be honest, I never really understood the logical rule (as Brown puts it) that

“if someone grants you (or anyone) [inconsistent] premises, they should be prepared to grant you anything at all (how could they object to B, having already accepted A and ¬A?)” .

How does this work? What is the logic — or the philosophy — behind it?

In other words, how does anything follow from an inconsistent pair of premises (or propositions) being (taken to be) true, let alone everything?

An inconsistent pair of premises (when taken together) surely can’t have any consequences — at least not any obvious ones. (You can derive, it can be supposed, logical strings such as ¬¬A ∧ ¬A and similar trivialities.) In terms of truth conditions (if we take our symbols — or logical arguments — to have semantic interpretations and even truth conditions), how could we derive anything from the premises “John is a murderer” and “John is not a murderer” if both are taken to be true? We can, of course, treat both premises only as-if-they-were-true — but surely that’s not paraconsistency.

In terms of the technical logic of explosion.

Let’s take explosion step by step so it can be shown where the problems are.

One symbolisation can begin in the following way:

i) If P and its negation ¬P are both [assumed to be] true,

ii) then P is [assumed to be] true.

So far, so good (at least in part).

If the conjunction P ∧ ¬P is (assumed to be) true, then of course P (on its own) must also be true. Here, the inference itself is classical; even though the original conjunction P ∧ ¬P isn’t.

Following on from that, we have the following:

iii) From i) and ii) above, it follows that at least one other (arbitrary) claim (symbolised A) is true.

This is where the first problem (apart from the conjunction of contradictories) is found. It can be said that some proposition or other must be the consequence of P; though how can — or why — is that consequence (A) arbitrary? An arbitrary A doesn’t follow from P. Or, more correctly, some A may well follow; though not any arbitrary A. (This is regardless of whether or not A, like P, is actually true.)

So perhaps all this isn’t actually about consequence.

“Consequent” A, instead, may just sit (or be consistent) with P without being a consequence of — or following from — P. Thus if A isn’t a consequence of P (or it doesn’t follow from P), then the only factor of similarity it must have with P is that both are (taken to be) true. However, if that’s the case, then why put A together with P at all? Why not say that P is arbitrary too? If there’s no propositional parameter between P and A, and if A doesn’t actually follow as a consequence of P, then why state (or mention) A at all?

Then comes the next bit of the argument for explosion. Thus:

iv) If we know that either P or A is true, and also that P is not true (or ¬P), then we can conclude that A (which can have any — or no — content) is true.

This is where the inconsistent conjunction is found again. Here there’s a (part) repeat of i) and ii) above. That is, P is both true and also not true; and again we conclude A. In other words, A follows the conjunction P∧ ¬P. This can also be seen as A following P and also A following ¬P (i.e. separately).

Again, why an arbitrary A? Instead of any A following from an inconsistent conjunction, why not say that A can’t’ follow from an inconsistent conjunction? Yet (as is now clear), the broad gist is that because we have both P and ¬P together, then it’s necessarily (or automatically) the case that any arbitrary A must follow from such an inconsistent conjunction.

We now encounter logical triviality; which is very similar to logical explosion.

Logical Triviality

Basic translation: For every p (i.e., for every proposition or statement), every p is true.

Instead of dealing with any (arbitrary) proposition (or theorem within a system/theory) following from an inconsistent conjunction, we now have every proposition (or theorem) doing so. It goes as follows:

If a theory contains a single inconsistency, then it must be trivial. That is, it must have every sentence as a theorem.

There are two problems here, both related to the points already made about logical explosion.

Why does an inconsistency have “every sentence as a theorem”? Sure, if this is indeed the case, then one can see the triviality of the situation. Nonetheless, how does the conjunction P ∧ ¬P generate every sentence as a theorem? Indeed, how does P ∧ ¬P generate even a single sentence? Surely the conjunction P ∧ ¬P generates nothing!

This isn’t to say that inconsistencies aren’t a problem for theories. Of course they are. However, arguing that the conjunction P ∧ ¬P itself generates every sentence as a theorem is another thing entirely…

… Or is it?

At the beginning of the last paragraph it was stated that I’ve rarely seen a defence of logical explosion — only bald statements of it. However, Bryson Brown does present C.I. Lewis’s “proof” of logical triviality (the bedfellow of explosion). Nonetheless, before that Brown does argue that “this defence [of Triv] is just a rhetorical dodge”. And, indeed, that’s how it can be seen. That is, it seems that the logical rule that “from any inconsistent premise set, every sentence of the language follows” is indeed rhetorical in nature. This logical rule is “rhetorical” because it simply can’t be taken literally. That is, it can’t literally be the case that the conjunction P & ¬P can generate every sentence of the language.

So perhaps the proof (or rule) actually amounts to stating (or even shouting) the following:

If a person accepts (or doesn’t even note) an inconsistency (such as the conjunction P & ¬P), then he or she may as well accept any statement!

In terms of the logical notion of the unsatisfiable nature of such premise sets, things seem to be much more acceptable. This is Brown’s formulation of that situation:

“A set Γ is inconsistent iff its closure under deduction includes both α and ¬α for some sentence α; it is unsatisfiable if there is no admissible valuation that satisfies all member of Γ.”

Unlike Triv, this seems perfectly acceptable. Of course there’s “no admissible valuation” of α & ¬α!… At least not in my own (non-formal and philosophical) book.

Logical Relevance

If relevance logic is a type of paraconsistent logic (see Graham Priest here), then that may well be relevant to some — or many — of the points raised above about explosion and triviality.

The main point is that if relevance is a logical stance, then nothing explodes from accepting both P and ¬P. That’s because it’s not the case than an arbitrary A can follow from a conjunctive inconsistency. Nor does it follow that if both P and ¬P are part of a theory (which, for example, arguably occurs in some formulations of quantum mechanics — see my ‘Is Graham Priest’s Dialetheism a Logic of Quantum Mechanics?’), then they trivially bring about every sentence as a theorem.

On the other hand, if we accept the relevance of relevance, then the very acceptance of a conjunctive contradiction (or inconsistency) may also be problematic. If both P and ¬P are accepted, it’s hard to see relevant derivations (or consequences) which follow from contradictory propositions. Of course we can accept that P (on its own) has relevant derivations and(!) that ¬P (on its own) has relevant derivations. But does the actual conjunction P & ¬P have relevant — or any — derivations?

For example, what follows from the propositions “The earth is in the solar system” and “It is not the case that the earth is in the solar system”? Taken individually, of course, much follows from both P and ¬P. But what is the case when P and ¬P are taken together as being jointly true (i.e., as a conjunctive truth)?

In symbols, the semantic heart of the argument above can be expressed in the following way:

If

A → B

is a theorem, then

A and B must share a non-logical constant (sometimes called a propositional parameter).

On the other hand, that (if indirectly) means (if jumping to propositions rather than the symbols A and B) that

If we have the following:

i) (P ∧ ¬P) → Q, Y, Z…

then we must have this consequent too:

ii) then Q, Y, Z…

Yet i) and ii) can’t be a argument in relevance logic.

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Note

(1) To show how radically non-relevant the principle of explosion is, let’s deal with an everyday statement rather than with — possibly misleading — symbolic letters. Thus:

i) Jesus H. Corbett is dead.

ii) Jesus H. Corbett is not dead.

iii) Therefore Geezer Butler is a Brummie.

This isn’t the classical-logic point that two true premises necessarily engender a true conclusion regardless of the propositional parameters of the premises and conclusion. In the classical case, then, all the premises can be genuinely true, along with the conclusion, even if they share no semantic content.

Now take logical triviality.

In this case, the premises above are supposed to generate all statements (or theorems) precisely because i) and ii) are mutually contradictory. This means that the propositional parameters of these premises are irrelevant: only their truth values matter. Not only that: we have now “proved” that Geezer Butler is a Brummie from the premises “Jesus H. Corbett is dead” and “Jesus H. Corbett is not dead”.

[I can be found on Twitter here.]