Thursday, 24 November 2016

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 all 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, of course, 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, of course, many problems for the anti-realist position too; especially if anti-realism is tied closely 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 nonethless 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). (Could 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. 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 ever) 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 a response to these philosophical problems. And that's why certain types of behaviourist relied exclusively on behaviour (whether physical or verbal behaviour) in his experiments and musings. That meant that other minds ceased being a problem for behaviourists because minds in effect didn't exist. Or, at the least, they believed - at that 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 his 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 and 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. One, such statements are neither true nor false. Two, if such statements are neither true nor false, then they serve little purpose.

The realist, of course, 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 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 (577).

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 neither 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'.




Wednesday, 16 November 2016

Anti-Realism, Intuitionism, and Mathematical Statements


Michael Dummett

The following piece is essentially about the anti-realist's problems with mathematical Platonism; as well as about how - and if - intuitionist mathematics ties in with anti-realism.

The focus will be on Michael Dummett's specific take on intuitionism; as well as his position on mathematical realism (i.e. Platonism). Understandably, at least within this limited context, the relation between truth and proof will also be discussed.

Dummett on the Platonic Realm

Michael Dummett asks a (fictional) Platonist the following question:

'What makes a mathematical statement true, when it is true?'”

The Platonist's answer is:

'The constitution of mathematical reality.'” [1982]

Of course if Dummett hadn't told us that he was talking to a (fictional) Platonist, the words “mathematical reality” could mean anything. After all, mathematical inscriptions on a page could be deemed to be, by some, mathematical reality. As it is, the Platonist's mathematical reality exists regardless of minds, before minds existed and will exist after the extinction of minds.

The anti-realist, along with many other philosophers of mathematics and mathematicians, will now ask:

How do we gain access to this mathematical reality?

More relevantly to the anti-realist's verificationism, the anti-realist will also ask:

How do we determine that we've accessed mathematical reality and how would we know that our mathematical statements match up with that reality?

Indeed wouldn't it need to be the case that even the mathematician's decision-procures and resultant proofs would also be required to match parts of this Platonic mathematical reality? No, not according to (most?) Platonists. The decision-procedures and resultant proofs are for us here in the non-platonic world. They're the way that mathematicians determine the truths of their mathematical statements. However, the results of these decision-procedures and proofs are indeed in the/a platonic realm; even though the procedures and proofs aren't.

Having said all that, and in the hope of capturing the Platonist position, I find it hard to know what all that means. That is, it's hard to understand or accept this Platonist separation of truth (or mathematical results) from decision-procedures and proofs. The intuitionist, of course, believes that there are no truths without decision-procedures (or “constructions”) and proofs because they quite literally bring about the truths and even numbers themselves. Mathematical truths wouldn't exist without them. (Despite that, one may also have problems with the Platonist construal of a mathematical reality which aren't particularly anti-realist or intuitionist is nature: e.g., what about our causal links to the platonic realm?)

What about other anti-realist problems with Platonism (i.e., realism)?

It can be said that a verifiable statement is also decidable statement. Thus it may/will be the case that unobservable (in principle) and observable states will be, respectively, unverifiable and verifiable. Does this apply to mathematics? Platonist mathematicians and philosophers say that mathematics doesn't concern the observable. We can, of course, observe mathematical equations on paper or on the blackboard. We can even introspect certain mathematical symbols. Despite these qualifications, in the Platonist view we're actually observing the symbolic representations of numbers and mathematics generally – not a number or mathematics itself. (In addition, a Platonist will happily accept – and even emphasise – the fact that mathematics and numbers can be applied to the world or used as the basis of structural descriptions of the world.)

Dummett on Truth and Proof

Michael Dummett goes into the technicalities of anti-realist truth without actually mentioning mathematics. In other words, what he says is the case about all/most domains of truth.

In terms of the lack of proof of a statement (although proofs aren't really applicable to verifiable non-mathematical statements), Dummett says that

we cannot assert, in advance of a proof or disproof of a statement, or an effective method of finding one, that it is either true or false”.  [1982]

There is a slight problem with that. The realist - against whom Dummett is arguing - doesn't say that he knows that a statement is true or false regardless of proof (or of “an effective method of finding one”). His position, in this respect, is effectively the same as the anti-realist's. However, the statement under consideration still has a determinate truth-value: regardless of proofs. The realist, sure, can't say that the statement is true. And he can't say that the statement is false. Though he can say that the statement is either determinately true or determinately false regardless of proofs. So, yes, even the realist, in Dummett's words, “cannot assert” (at this juncture) that “it is either true or false”. Instead he can simply say: It is either true or false.

Having said all that, I see very little point in saying that a mathematical statement is determinedly true or false regardless of decision-procedures and their resultant proofs. What does this claim amount to? Where does it get us? It's effectively equivalent to Bertrand Russell's teapot flying around somewhere in a distant galaxy. Yes, there could be such a thing. However, we can never establish that there is such a thing. Therefore what, exactly, is the point of saying that “there's a flying teapot somewhere in a distant galaxy”? 

Dummett (in his own way) puts the gist of the last paragraph in more circumspect and, indeed, Dummettian prose. On the proof-independent mathematical statement, Dummett says that

we shall be unable to conceive of a statement as being true although we shall never know it to be true, although we can suppose a true statement as yet unproved”.      [1982]

Yes, what's the point of “suppos[ing] a true statement as yet unproved”? As it is, however, there are indeed unprovable truths in mathematics. Dummett himself puts that in this way:

A Platonist will admit that, for a given statement, there may be neither a proof nor a disproof of it to be found.” [1982]

Despite that, what Dummett has just said is most certainly applicable to non-mathematical statements; whether about distant galaxies, the past, other minds and other problematic areas. This is even more the case when it comes to statements about the exact number of people in a given aeroplane at a given time; or, more mundanely, about whether or not Jesus H. Corbett is dead at time t.

Truth-conditions, Proof and Truth

Clearly it appears to be problematic to think in terms of truth-conditions when it comes to mathematical statements. (A Platonist, perhaps, could think in terms of such truth-conditions being in an abstract realm; which are accessed through “intuition”, “direct insight” or - metaphorical - “seeing”.) More concretely, it seems odd to demand truth-conditions for the statement 5 times 56 = 280. Strictly speaking, it has no truth-conditions. So what does it have? According to anti-realists or intuitionists, there is a procedure which can result in a determinate result. Is that decision-procedure also the proof of the mathematical statement? Is the way of determining the truth of a mathematical statement also a proof of the statement? Yes, but truth is not proof. The intuitionist believes that a proof leads to the truth of a mathematical statement. Without proof there is no truth. Nonetheless, a proof isn't the same thing as a truth.

Mathematical statements, when true, are decidable, not verifiable (because unobservable). Despite that, the decidability of mathematical statements (in a way) does the job that verifiability does when it comes to statements which can be observed (or only observed “in principle”). That is, there's a decision-procedure for deciding the truth of mathematical statements.

In reference to all the above, an anti-realist philosopher of mathematics would say that mathematical truth depends on mathematical proof. And mathematical proof is itself a question of decidability. That is, from proof comes truth. And proof, in mathematics, is very much like verification when it comes to the observable realm. That means that proof, in intuitionist mathematics, satisfies the anti-realist position.

Nonetheless, the realist would ask about the situation in which there is no proof of a mathematical statement or equation. He would follow that by saying that such a statement would still be determinately true regardless of whether or not it had been proved. (This is/was the case with Goldbach's Conjecture and Fermat's Theorem. The latter was proved by Andrew Wiles.)

Intuitionism

Objectivity in mathematics isn't a question of objects. Instead, objectivity can be said to be about the objectivity of the procedures and proofs which lead to truth. On the other hand, if this were a question of truth-conditions, then it may have been the case that objects do enter the equation. As it is, according to the intuitionist, this isn't the case for mathematics.

Can't we be realists about numbers, functions,sets, etc.? Aren't they abstract objects? And if they are abstract objects, then don't we have truth-conditions (of some kind) because we have objects (of some kind)?

On the other hand, what if numbers are simply “free creations”, as Richard Dedekind believed? In that case, numbers are created or constructed by the mathematician. The free creation of a number would still be the creation of a determinate something; just as when a person makes (or “constructs”) a toy dog whose nature becomes determinate.

In terms of an intuitionist/anti-realist position on free creations. There must still be decision-procedures which can come up with definite results even if numbers are “constructed”. The toy dog just mentioned was indeed constructed. Still, we have various ways of deciding its nature. That is, like the constructed number or equation, we have ways of determining the nature of the toy dog. The realist, on the other hand, would say that any truths about the toy dog would hold even though no one could ever gain access to it (say, after the creator died, etc.).

Brouwer/Heyting on Maths as a Human Activity

L.E.J. Brouwer, according to Dirk van Dalen and Mark van Atten [2007, 513], thought of mathematics as an “activity rather than a theory” . In that simple sense, truth-conditions or a Platonic reality don't give maths a realist foundation. More importantly, perhaps, “[m]athematical truth doesn't consist in correspondence to an independent reality”. In a certain sense, this means that such a construal of mathematics is beyond the anti-realism/realism debate in that verification or observation isn't even possible in principle.

The basic intuitionist position on mathematical truth was also put by Arend Heyting. Michael Dummett puts Heyting's position this way:

... the only admissible notion of truth is one directly connected with our capacity for recognising a statement as true: the supposition that a statement is true is the supposition that there is a mathematical construction constituting a proof of that statement.” [1982]

In terms of mathematics, that “capacity for recognising a statement as true” would depend on a decision-procedure (or construction) for determining a proof of that statement. As can be seen, everything in the quote above seems to refer to human actions – even if human cognitive actions. A mathematical construction is a cognitive activity. A proof is also a result of a cognitive activity. Indeed recognising a statement to be true is a cognitive activity. What we don't have (in the quote above) is any reference to anything outside these cognitive actions (such as truth-conditions, states of affairs, facts, etc.). Indeed there isn't even a mention of abstract objects as such. However, that doesn't automatically mean that abstract objects aren't illicitly or tacitly referred to.

For example, the word “truth”may refer to the end result of a mathematical construction which works as a proof that a statement is true. But what does the word “true” refer to or mean? To the proof itself? To the construction of the proof itself? In that case, perhaps we have:

truth = proof.

Or alternatively:

a construction (or decision-procedure) = (a) truth

Dummett's Problem With Intuitionism

It may seem odd that the intuitionists' rebellion against “mystical Platonism” should rely, instead, on the happenings which go on in the privacy of a mathematician's head. At least that's how Dummett saw it. Dummett's position is, of course, Wittgensteinian [Dummett, 1978, 215-247]. Yet, in a sense, that's exactly what intuitionists or mathematicians do. That is, even if we supply a retrospective externalist (or “broad content”) account of the meaning/s, etc. of those mental constructions, and also argue that they have externalist/broad features (even during the private acts of mental construction), at the initial stage this mathematical mental activity is still individualistic (or private). The functions, numbers and symbols are of course public or communal. Nonetheless, the mental/cognitive acts of construction are, in an obvious sense, private. So it remains the case that, on the one hand, mental objects and actions aren't really private in that their meanings, senses, extensions or whatever are externally determined. Nonetheless, it's still the case that the mental constructions of numbers or mathematical statements are private. And surely no intuitionist would have denied any of that. (Perhaps we can say said that Dummett was trying too hard to be a Wittgensteinian.)

It is still the case that private mental constructions can't be the subject of a decision-procedure by other mathematicians (even if the mathematical symbols, functions, etc. are bone fide externalist items). This of course ceases to be the case once the mathematical statements are written down or notated in some other way.

References

Dalen, Dirk van and Mark van Atten (2007) 'Intuitionism'.
Dummett, Michael. (1982) 'Realism'.




Thursday, 10 November 2016

Paraconsistent Logic: Inconsistency, Explosion and Relevance




In the following I argue against the paraconsistent acceptance of inconsistencies. I also question the acceptance of logical explosion and logical triviality by classical logicians. 

As C.I. Lewis once claimed (quoted by Bryson Brown), no one “'really accepts contradictions'”. From that it can be said that the prime motivation for paraconsistency - as can be gleaned from some paraconsistent logicians themselves (at least implicitly) - is entirely epistemological and, sometimes, also based on positions and findings within science (specifically physics).

The following positions are also seemingly at one with the position of David Lewis who argued that it's impossible for a statement and its negation to be true at one and the same time. Having said that, all this depends on what's said about the embracing of both A and ¬A; as well as how it's 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”. Bryson Brown and Graham Priest also call negation (or it) “denial”. Indeed Priest explicitly states that paraconsistent negation isn't (Boolean) negation.

Thus if the acceptance of inconsistencies is an epistemological move (as I argue), then that move isn't “really” an acceptance of both A and its negation at one and the same time.

(All the above may or may not be ironic when one bears in mind that relevance logic is a sub-branch/type of paraconsistent logic.)

The Acceptance of A¬A


As already stated, Bryson Brown says that “a defender of [C.I.] Lewis's position might argue that we never really accept inconsistent premises” (630). I'm prone, prima facie, to agree with that. Indeed Bryson immediately follows that statement with a defence of inconsistency which doesn't seem to work. Brown says that “[a]fter all, we are finite thinkers who do not always see the consequences of everything we accept”. Perhaps C.I. Lewis's reply to that might have been that we don't “accept inconsistent premises” that we know - or that think we know - to be inconsistent. Sure, having finite minds is a limit on what we can know. Nonetheless, we still don't accept the conjunction A ¬A; at least not in its (empty) symbolic form. Though it needn't always be entirely a problem of symbolic autonyms. Paraconsistent logicians don't accept 1 = 0 either; and we don't accept the disjunctive statement “John is all black and John is all white”.

As for not seeing consequences of our premises: no, we don't – not of every premise (or proposition) we accept or even know. Though we do know the consequences of some of the premises we know! The finiteness of mind doesn't stop us accepting certain premises - or even whole arguments - either. Still, Brown may only be talking about premises which we're not sure about. In such cases then, sure, the limitations of our minds is salient – we can't know all the consequences of all the premises we accept and we can't know if they're all truly consistent.

Similarly, do we accept inconsistent premises for pragmatic reasons, as Bryson suggests? 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”. Talk of pragmatics 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/relevance, such salience/relevance will also (partly) determine our choice between two contradictory premises. What's more, talk of “how best to respond to our observations and to the consequences of what we have already accepted” (Brown's words) will, again, make it less likely that we would accept inconsistent premises, not more likely. That is, p may have observational consequences radically at odds with the observational consequences of ¬p. Thus why would we accept both – even provisionally? (Unless accepting both p and ¬p is a way of hedging one's bets.)

Logical Explosion

Dale Jacquette puts the paraconsistent position when he says that “logical inconsistencies need not explosively entail any and every proposition” (7). What's more, “contradictions can be tolerated without trivialising all inferences”. There we have the twin problems (for paraconsistent logic) of logical explosion ( or ex contradictione sequitur quodlibet) and logical triviality.

To be honest, I never really understood the logical rule 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?)” (Brown, 628). I was never sure how this worked. That is, I was never sure of the logic - or is it the philosophy? - behind it. In other words, how does anything follow from an inconsistent pair of premises (or propositions), let alone everything? An inconsistent pair surely can't have any consequences – at least not any obvious ones. (You can derive, I suppose, trivialities such as ¬¬A ¬A and similar conclusions or premises.) In terms of truth conditions (if we take our symbols - or logical arguments – to have truth conditions or extensional interpretations), how could we derive anything from the premises “John is a murderer” and “John is not a murderer”? Again, there are (indirect) trivial consequences; though none of interest. (E.g. that “John is a person”?)

In terms of the logic of explosion, let's take it step by step so it can be shown where the problems are. One symbolisation can begin in this way:

i) If P and its negation ¬P are both assumed to be true,
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 (which paraconsistent logicians accept, though others don't), then of course P (on its own) must also be true. Here, the inference itself is (kind of) classical; even though the conjunction of P and its negation isn't.

Following on from that, we have the following:

ii) From i) above, it follows that at least one of the claims, P, and some other (arbitrary) claim A, are 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 that consequence (A) be arbitrary? An arbitrary A doesn't follow from P. Or, more correctly, some arbitrary A may well follow; though not any arbitrary A (this is regardless of whether or not A, like P, is true).

Thus perhaps this isn't about consequence.

A, instead, may just sit (i.e. be consistent or coherent) with P without being a consequence of P; or without it following from P. Thus if A isn't a consequence of P (or it doesn't follow from A), then the only factor of similarity it must have with P is that it's true. Though, if that's the case, why put A 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 follow as a consequence of P, then why state (or mention) A at all? Moreover, why does A follow from P?

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

iii) If we know that either P or A is true, and also that P is not true (that ¬P is true), we can conclude that A - which can be anything - is true.

This is where the inconsistent conjunction is found again. Here there's a (part) repeat of ii) above. That is, P is both truth 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 A following ¬P (i.e. separately). Though, again, why an arbitrary A? Instead of any A following from an inconsistent conjunction, why not say that no A can follow from an inconsistent conjunction? However, the gist seems to be that because we have both P and ¬P, it's necessarily (or automatically) the case that any arbitrary A must follow from any inconsistent conjunction.1

Logical Triviality

Rather than explosion, we now encounter logical triviality; to which it is very similar. Instead of dealing with any (arbitrary) proposition/theorem following from an inconsistent conjunction, we now have every proposition/theorem (in a theory) doing so. It goes as follows:

Thus if a theory contains a single inconsistency, it is trivial — that is, it has every sentence as a theorem.

There are two problems here, both related to the points I've already made about logical explosion. Why does an “inconsistency” have “every sentence as a theorem”? Sure, if this is true, then one can see the triviality of the situation. Nonetheless, I simply can't see how the conjunction P ¬P generates every sentence as a theorem; or, indeed, even a single sentence. Strictly speaking, the conjunction P ¬P generates nothing!

This isn't to say that inconsistencies aren't a problem for theories. Of course they are. Though saying 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 I confessed to rarely having seen a logical 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 says that “this defence [of Triv] is just a rhetorical dodge”. And that's how I saw it. It seemed that the logical rule that “from any inconsistent premise set, every sentence of the language follows” was rhetorical in nature.

In terms of the logical notion of the unsatisfiable nature of premise sets, things seem to be much more acceptable. Firstly, this is Brown's formulation:

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 book.)

Logical Relevance

If relevance logic is a type of paraconsistent logic, then that may well be relevant to some - or many - of the points raised above about explosion and triviality. My point is that if relevance is a logical stance, then nothing explodes from accepting both P and ¬P 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, it trivially brings about every sentence as a theorem.

On the other hand, if we accept 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. 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 either P or ¬P; though not when taken together as jointly true.

In symbols, the above expressed in the following way:

If AB is a theorem,
then A and B share a non-logical constant (sometimes called a propositional parameter).

On the other hand, that (indirectly) means that

(A ∧ ¬A) → B
Therefore B.

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 everyday statements rather than - 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 logical point that two true premises necessarily engender a true conclusion regardless of the propositional parameters of the premises and conclusion . In the classical case, the premises can be genuinely true, along with the conclusion. In addition, there's no contradiction even if the premises and conclusion aren't semantically related.

Now take logical explosion. In this case, the premises above engender all statements or theorems because they're contradictory. The propositional parameters are irrelevant, only the truth-values of the premises and conclusion matter. Not only that: we have “proved” that Geezer Butler is a Brummie from the premises “Jesus H. Corbett is dead” and “Jesus H. Corbett is not dead”.

References

Brown, Bryson. (2007) 'On Paraconsistency'.
Jacquette, Dale. (2002) A Companion to Philosophical Logic, Introduction.
Lewis, David (1998) [1982]. 'Logic for Equivocators', Papers in Philosophical Logic. Cambridge: Cambridge University Press. pp. 97–110.

Slater, B. H. (1995). 'Paraconsistent Logics?', Journal of Philosophical Logic. 24 (4): 451–454.