Is Graham Priest's Dialetheism a Logic of Quantum Mechanics?

Graham Priest tells us that in certain quantum-mechanical experiments “the law of excluded middle tells us that [an atom, particle, etc.] is one or the other”. He also claims that quantum mechanics (or at least its interpretations) suggests otherwise.

It’s tempting to think that the nature of quantum mechanics is the primary reason why the philosopher and logician Graham Priest (1948-) defends, accepts and uses dialetheic logic (or dialetheism generally).¹ Indeed, he mentions quantum mechanics (QM) a few times in various papers and in interviews.

It can now be asked if his views have any impact on the “classical world” or macro-reality (or, at the least, the world as it’s perceived and/or experienced). Perhaps, if dialetheic logic (DL) is truly dependent on the findings of quantum mechanics, it can also be asked if dialetheism is applicable to the “classical world” at all. Of course I may be barking up the wrong tree here. That is, why assume that this macro-world/micro-world opposition has any relevance to dialetheic logic?²

In any case, what has Priest got to say about what he calls “[u]nobservable realms”? (A term which needn’t only refer to the quantum world; but also to historical events, events beyond our galaxy, numbers/abstract objects, etc.) Take this passage from Priest:

“[I]t would sometimes (in the well-known two-slit experiment) appear to be the case that particles behave in contradictory fashion, going through two distinct slits simultaneously.”

Dialetheic logic would be an ideal logic to describe (or capture) such a phenomenon. Nonetheless, Priest’s clause “it would sometimes… appear” does seem to qualify things a little. In other words, since Albert Einstein, hasn’t the supposed “contradictory” (or “paradoxical”) nature of quantum phenomena — or states — been questioned by various physicists (including David Bohm, many-worlds theorists, etc.)?

More broadly, Priest says that “inconsistent theories may have physical importance too”. At first blush, the following can now be asked:

Does inconsistency necessarily have a connection with dialetheism?

And, if so, what’s the connection in particular instances?

What Priest continues to say may mean that not only is the statement above unconnected to dialetheism; it may not be connected to quantum mechanics either. Priest himself continues:

“An inconsistent theory, if the inconsistencies are quarantined, may yet have accurate empirical consequences in some domain. That is, its predictions in some observable realm may be highly accurate.”

Since Priest is talking about an “observable realm”, this mustn’t be about quantum mechanics. (Then again, some of the clues as to the reality of the quantum world are indeed observable.) Not only that: talk of the “accurate empirical consequences” of suspect, questionable and even false theories is something that’s widely accepted throughout the sciences and in the philosophy of science. Indeed Priest finishes off by saying that “one may take the theory, though false, to be a significant approximation to the truth”.³

Priest’s Examples From Quantum Mechanics

Here’s a passage from Priest on an aspect of quantum mechanics that’s relevant — or not! — to dialetheism:

“Unobservable realms, particularly the micro-realm, behave in a very strange way, events at one place instantaneously affecting events at others in remote locations.”

It’s difficult to see how this physical phenomenon has any direct relevance to dialetheism. Nothing contradictory (or paradoxical) is happening here; unlike the two-slit experiment. What’s does happen, however, does indeed go against common-sense views of causation and also against “classical” (or “local”) physics. Nonetheless, that doesn’t automatically entail contradiction or paradox.

Priest also gives the example of radioactive decay. He writes:

“[S]uppose that a radioactive atom instantaneously and spontaneously decays. At the instant of decay, is the atom integral or is it not?”

Now for the traditional logic of this situation. Priest continues:

“In both of these cases, and others like them, the law of excluded middle tells us that it is one or the other.”

Couldn’t the atom be neither integral nor non-integral when it instantaneously and spontaneously decays? (Priest talks of either/or or “one or the other”; not neither/nor.) Or, alternatively, at that point it may not be an atom at all!

This appears to be a temporal problem which must surely incorporate definitions — or philosophical accounts — of the concepts [instantaneously] and [spontaneously].⁴ Nonetheless, if they define time instants that don’t exist (the period from t¹ to t² doesn’t exist), then Priest and others may have a point. However, can an atom — or anything else — “decay” (or do anything) in a “timespan” which doesn’t actually exist? How can decay — or anything else for that matter — occur if there’s no time in which it can occur?…

So what about Priest’s own conclusion when it comes to atomic decay?

Priest claims that the aforementioned atom “at the point of decay is both integral and non-integral”. This isn’t allowed — Priest says — if the law of excluded middle is globally accurate or true. The law of excluded middle tells us that the said atom must either be integral or non-integral; not “both integral and non-integral”.

Again, none of this (basically) philosophy of science — on Priest’s part — is solely applicable to quantum mechanics or dialetheism. The word “solely” was just used because Priest does indeed give another example from QM. He states that

“those who worked on early quantum mechanical models of the atom regarded the Bohr theory [as] certainly inconsistent”.

And “yet its empirical predictions were spectacularly successful”.

Needless to say, it must be stressed here that the word “inconsistent” is very different to the word “contradictory” (or “paradoxical”). Something can indeed be inconsistent because contradictory. However, can’t something also be inconsistent without being (logically) contradictory?

Abstract objects were mentioned in brackets earlier and Priest himself “move[s] away from the empirical realms” to “the realm of sets”.

Dialethic Logic and the Paradoxes of Set Theory

Priest claims that this realm “appears to be inconsistent” too.

Here again Priest uses the word “appears”. There are indeed paradoxes in set theory. However, haven’t logicians and mathematicians — like quantum physicists — attempted to rectify those inconsistencies or paradoxes? True, unsolved, these inconsistencies or paradoxes (again like QM) are perfect specimens to be dealt with — and captured by — dialethic logic. Nonetheless, it can be said that what’s captured isn’t a world (or reality) of any kind: it’s simply an unsolved paradox or inconsistency. In addition, just as it was asked earlier about how dialethic logic could be applied to the empirical world, so it can now be asked what the connection is between the paradoxes of set theory and the empirical world. That is, it can be argued that set theory needn’t have a necessary connection to such a world. (This will depend on a whole host of factors; such as what one’s take is on the reality — or existence — of numbers, sets, the members of sets, etc.)

If one believes in abstract objects such as sets, then they must exist in an abstract world. Thus dialethic logic, in this case at least, may be applicable to a world — an abstract world. (These vaguely Platonist announcements about sets and abstract objects will, of course, be rejected by certain philosophers, logicians and mathematicians.) ⁵

Conclusion

If dialetheic logic is all about quantum mechanics, then why not call it a logic of quantum mechanics? This means that if dialetheic logic is (as it were) justified by the nature of quantum reality, then it must also depend on that reality.

That said, there’s a tradition in the philosophy of logic which states that logic doesn’t depend on anything — least of all on the nature of the (or a) world. (Ludwig Wittgenstein, at one point, stated this position; and the early Bertrand Russell took the contrary view.) If that were the case, then it may also be the case that dialetheic logic isn’t dependent on the nature of quantum mechanics. It just so happens that QM sometimes — or many times — behaves in a way which can be captured by dialetheic logic. Yet that doesn’t also mean that dialethic logic is dependent on QM. And it doesn’t mean that dialethic logic is derived — in any way — from QM either.

In any case, Priest does clarify his position by arguing that

“the micro-realm is so different from the macro-realm that there is no reason to suppose that what holds of the second will hold of the first”.

True. However and again, is dialetheism independent of quantum mechanics? What’s more, is dialethism applicable to the “classical world”?

As already hinted at, perhaps Priest would reply:

What does it mean to ask if dialethic logic is applicable to the classical world?

Indeed need logic be applicable to the (or even a) world at all?

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Notes

(1) This case parallels — at least to some extent — ontic structural realism, which (it can be argued) is similarly motivated by the reality of quantum mechanics. Or to put that another way: what relevance does much of the ontic-structural-realist position have to the macro/classical world? (See here for my discussion of this subject.)

(2) Graham Priest is also inspired by Buddhist logic (or simply by Buddhist thought — see here). So, conceivably, this piece may just as justifiably have been entitled ‘Is Graham Priest’s Dialetheism a Logic of Buddhism?’.

(3) This chimes in with Karl Popper’s verisimilitude in which scientific theories have quantifiable truth-to-falsity contents.

(4) Perhaps, as hinted at earlier, what’s needed is some good old-fashioned conceptual analysis of the words “integral”, “non-integral”, “decay”, “spontaneously” and “instantaneously”. Not, of course, the kind of conceptual analysis which historically disappeared up its own backside (i.e., by ignoring science completely). Nonetheless, if quantum reality is the way many physicists say it is, then much conceptual analysis on this matter will either be wrong or simply inapplicable to the quantum domain.

(5) What is — or what constitutes — a world anyway?

Reference

Priest, Graham. (2002) ‘Logicians setting together contradictories: A perspective on relevance, paraconsistency, and dialetheism’.

Material Logic vs. Formal Logic?

In a purely logical argument, even if the premises aren’t in any way (semantically) connected to the conclusion, the argument may still be both valid and sound.

Professor Edwin D. Mares displays what he sees as a problem with purely formal logic when he offers us the following example of a valid argument:

The sky is blue.

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∴ there is no integer n greater than or equal to 3 such that for any non-zero integers x, y, z, xn = yn + zn.

Edwin Mares says that the above “is valid, in fact sound, on the classical logician’s definition”. It’s the argument that is valid; whereas the premise and conclusion are sound (i.e., true). In more detail, the

“premise cannot be true in any possible circumstance in which the conclusion is false”.

Clearly the content of the premise isn’t semantically — or otherwise — connected to the content of the conclusion. However, the argument is still valid and sound.

That said, it’s not clear from Edwin Mares’ symbolic expression above if he meant this: “If P, therefore Q. P. Therefore Q.” That is, perhaps the premise “The sky is blue” with a line under it, followed by the mathematical statement, is used as symbolic shorthand for an example of modus ponens which doesn’t have a sematic connection between P and Q. In other words, Mares’ “P, therefore Q” isn’t (really) an argument at all. However, if both P and Q are true, then, logically, they can exist together without any semantic connection and without needing to be read as shorthand for an example of modus ponens.

Whatever the case is, what’s the point of the “The Sky is blue” example above?

Perhaps no logician would state it for real. He would only do so, as Mares himself does, to prove a point about logical validity. However, can’t we now ask why it’s valid even though the premise and conclusion are true?

Perhaps showing the bare bones of the “The sky is blue” example will help. Thus:

P
∴ Q

Does that look any better? Even though we aren’t given any semantic content, both P and Q must have a truth-value. (In this case, both P and Q are true.) It is saying: P is true. Therefore Q is true. The above isn’t saying: Q is a consequence of P. (Or: P entails Q.) Basically, we’re being told that two true and unrelated statements can (as it were) exist together — as long as they don’t contradict each another. (Or on the aforementioned alternative reading: “If P is true; then Q is true. P is true. Therefore Q is true.”)

So there are cases in which the premises of an argument are all true, and the conclusion is also true; and yet as Professor Stephen Read puts it:

“[T]here is an obvious sense in which the truth of the premises does not guarantee that of the conclusion.”

Ordinarily the truth of the premises is meant to “guarantee” the truth of the conclusion. So let’s look at Read’s own example:

i) All cats are animals
ii) Some animals have tails
iii) Therefore some cats have tails.

Clearly, premises i) and ii) are true. Indeed iii) is also true. (Not all cats have tails. And, indeed, according to some logicians, “some” also implies “all”.)

So why is the argument above invalid?

It’s invalid not because of the assigned truth-values of the premises and the conclusion; but for another reason. The reason is that the sets used in the argument are (as it were) mixed up. Thus we have the distinct sets [animals], [cats] and [animals which have tails].

It doesn’t logically follow from “some animals have tails” that “some cats have tails”. If some animals have tails it might have been the case that cats are animals which don’t have tails. Thus iii) doesn’t necessarily follow from ii). (iii) doesn’t follow from i) either.) ii) can be taken as an existential quantification over animals. iii), on the other hand, is an existential quantification over cats. Thus:

ii) ((Ǝx) (Ax)
iii) (Ǝx) (Cx))

Clearly, Ax and Cx are quantifications over different sets. It doesn’t follow, then, that what’s true of animals is also generally true of cats; even though cats are members of the set [animals]. Thus iii) doesn’t follow from ii).

To repeat: even though the premises and the conclusion are all true, the above still isn’t a valid argument. Read himself helps to show this by displaying an argument-form with mutually-exclusive sets — namely, [cats] and [dogs]. Thus:

i) All cats are animals
ii) Some animals are dogs
iii) Therefore some cats are dogs.

This time, however, the conclusion is false; whereas i) and ii) are true. It’s the case that the subset [dogs] belongs to the set [animals]. Some animals are indeed dogs. However, because some animals are dogs, it doesn’t follow that “some cats are dogs”. In other words, because dogs are members of the set [animals], that doesn’t mean that they’re also members of the subclass [cats] simply because cats themselves are also members of the set [animals]. Cats and dogs share animalhood; though they’re different subsets of the set [animal]. In other words, what’s true of dogs isn’t automatically true of cats.

The importance of sets, and their relation to subsets, may be expressed in terms of brackets. Thus:

[animals [[cats [[[cats with tails]]]]
not-[animals [[cats [[[dogs]]]]

Material Validity and Formal Validity

Stephen Read makes a distinction between formal validity and material validity. He does so by using this example:

i) Iain is a bachelor
ii) So Iain in unmarried.

(One doesn’t usually find an argument with only a single premise.)

The above is materially valid because there’s enough semantic material in i) to make the conclusion acceptable. After all, if x is a bachelor, he must also be unmarried. Despite that, it’s still formally invalid because there isn’t enough content in the premise to bring about the conclusion. That is, one can only move from i) to ii) if one already knows that all bachelors are unmarried. We either recognise the shared semantic content or we know that the term “unmarried man” is a synonym of “bachelor”. Thus we have to add semantic content to i) in order to get ii). And it’s because of this that the overall argument is said to be formally invalid. Nonetheless, because of what’s already been said, it is indeed still materially valid.

The material validity of the above can also be shown by its inversion:

i) Iain is unmarried
ii) So Iain is a bachelor.

Read makes a distinction by saying that its

“validity depends not on any form it exhibits, but on the content of certain expressions in it”.

Thus, in terms of logical form, it’s invalid. In terms of content (or the expressions used), it’s valid. This means that the following wouldn’t work as either a materially or a formally valid argument:

i) Iain is a bachelor.
ii) So Iain is a footballer.

There’s no semantic content in the word “bachelor” that can be directly tied to the content of the word “footballer”. Iain may well be a footballer; though the necessary consequence of him being a footballer doesn’t follow from his being a bachelor. As it is, the conclusion is false even though the premise is true.

Another way of explaining the material (i.e., not formal) validity of the argument above is in terms of what logicians call a suppressed premise (or a hidden premise). This is more explicit than talk of synonyms or shared content. In this case, what the suppressed premise does is show the semantic connection between i) and ii). The actual suppressed premise for the above is the following:

All bachelors are unmarried.

Thus we should actually have the following argument:

i) Iain is a bachelor.
ii) All bachelors are unmarried.
iii) Therefore Iain is unmarried.

It may now be seen more clearly that

i) Iain is unmarried.
ii) So Iain is a bachelor.

doesn’t work formally; though it does work materially.

What about this? -

i) All bachelors are unmarried.
ii) So Iain is unmarried.

To state the obvious, this is clearly a bad argument. (It’s called an enthymeme.) Indeed it can’t really be said to be an argument at all. Nonetheless, this too can be seen to have a suppressed (or hidden) premise. Thus:

i) All bachelors are unmarried.
ii) [Suppressed premise: Iain is a bachelor.]
iii) So Iain is unmarried.

Now let’s take the classic case of modus ponens:

A, if A then B / Therefore B

That means:

A, if A is the case, then B is the case. A is the case. Therefore B must also be the case.

The obvious question here is: What connects A to B (or B to A)? In terms of this debate, is the connection material or formal? Clearly, if the content of both A and B isn’t given, then it’s impossible to answer this question.

We can treat the example of modus ponens above as having the aforesaid suppressed premise. Thus:

i) [Suppressed premise: Britain’s leading politician is the Prime Minister.]
ii) Boris Johnson is Britain’s leading politician.
iii) Therefore Boris Johnson is Britain’s Prime Minister.

In this instance, premises and conclusion are true. Yet i) is only contingently (i.e., not necessarily) connected to ii) and iii).

Finally, Stephen Read puts the formalist position on logic very clearly when he states the following:

“Logic is now seen — now redefined — as the study of formal consequence, those validities resulting not from the matter and content of the constituent expressions, but from the formal structure.”

We can now ask:

What is the point of a logic without material or semantic content?

If logic were purely formal, then wouldn’t all the premise and predicate symbols — not the logical symbols — simply be autonyms? (That is, all the p’s, q’s, x’s, F’s, G’s etc. would be purely self-referential.) So what would be left of logic if that were the case? Clearly we could no longer say that logic is about argumentation — or could we? Not really. The fact is that we can still learn about argumentation from schemas (or argument-forms) which are purely formal in nature. And that basically means that the dots don’t always — or necessarily — need to be filled in.

'and' and 'tonk'

'and' and Analytic Validity

In order to understand A.N Prior's use of the neologism 'tonk', we firstly need to understand the way in which he takes the connective of conjunction – namely, 'and'.

Prior makes the counterintuitive claim that “any statement whatever may be inferred, in an analytically valid way, from any other” (130). Prima facie, that raises a question: 

Does that mean that any statement with any content can be inferred from any other with any content?

The word 'and' (in this logical sense at least) is understood by seeing its use in statements or propositions.

We start off with two propositions: P and Q; which begin as separate entities in this context. Prior argues that we can “infer” P-and-Q from statements P and Q. The former symbolism, “P-and-Q” (i.e., with hyphens) signifies the conjunction; whereas “P and Q” (i.e., without hyphens) signifies two statements taken separately. However, we can infer P-and-Q from any P and Q. That is, from P on its own, and Q on its own, we can infer P-and-Q. In other words, statements P and Q can be joined together to form a compound statement.

We can now raise two questions. One, do the truth-values of both P and Q matter at this juncture? Two, do the contents of both P and Q matter at this point?

In basic logic, the answer to both questions is 'no'. It's primarily because of this that some of the counterintuitive elements of this account become apparent.

For example, Prior says that “for any pair of statements P and Q, there is always a statement R such that given P and given Q we can infer R” (129). The important word to note here is “any” (as in “for any pair of statements P and Q”). This leads to the conclusion (just mentioned) that the truth-values and/or contents of both P and Q don't matter within the logical context of defining the connective 'and'. It's partly because of this that Prior tells us that “we can infer R” from P and Q. Thus:

(P) The sun is in the solar system.

(P & Q) Therefore the sun is in the solar system and peas are green.

(R/Q) Therefore peas are green.

All those statements are true; yet they have unconnected contents and a conclusion which doesn't follow from (the content!) of the premises. Similarly with two false premises. Thus:

(P) The sun is in the bathroom.

(P & Q) Therefore the sun is in the bathroom and peas are blue.

R/Q) Therefore peas are blue.

It's because of this irrelevance of contents and truth-values that R will follow from any P and any Q.

Thus it's no surprise that Prior also says that “given R we can infer P and can also infer Q”. As in this example:

(R/P) Peas are green

(P and Q) Therefore peas are green and the sun is in the solar system.

The difference here is that a compound statement (i.e., P-and-Q) is derived from an atomic statement (i.e., R/P). (Except, in this instance, R should be P and P and Q should be R.) Nonetheless, contents and truth-values still don't matter. Another way of putting this is (as in the argument-form above) that premises and conclusions can change places without making a difference.

'tonk'

If we still have problems with Prior's 'tonk', that situation arises because we fail to see that the “meaning” of any connective “is completely given by the rules” (130).

Prior gives the following example of this logical phenomenon:

(P) 2 and 2 are 4.

(Q)Therefore, 2 and 2 are 4 tonk 2 and 2 are 5.

(R/Q)Therefore 2 and 2 are 5.

Clearly the connective 'tonk' is doing something to the proceeding 2 and 2 are 4 - but what? Could it be that 'tonk' seems to mean add 1 - at least in this instance? That would mean, however, that 'tonk' is the operation of adding 1, which isn't (really?) a connective of any kind.

The new connective 'tonk' works like the connective 'and'. Or as Prior puts it:

“Its meaning is completely given by the rules that (i) from any statement P we can infer any statements formed by joining P to any statement Q by 'tonk'... and that (ii) from any 'tonktive' statement P-tonk-Q we can infer the contained statement Q.” (130)

Thus, at a symbolic level, 'tonk' works like 'and'. And just as Prior symbolised P and Q taken together as P-and-Q; so he takes P and Q taken together with tonk as P-tonk-Q.

In this case, '2 and 2 are 4' (P) is being conjoined with '2 and 2 are 5' (Q). Thus the conclusion, 'therefore, 2 and 2 are 5' (R) follows from '2 and 2 are 5' (Q), though not from '2 and 2 are 4'. In other words, R only needs to follow from either P or Q, not from both. Thus when P and Q are (as it were) tonked, we get: '2 and 2 are 4 tonk 2 and 2 are 5'. And the conclusion is: 'Therefore 2 and 2 are 5.'

To express all this in argument-form, take this example:

(P) Cats have four legs.

(P & Q) Therefore cats have four legs tonk Cats have three legs.

(R/Q) Therefore cats have three legs.

What is 'tonk' doing in the above? It seems to be cancelling out the statement before (i.e., 'Cats have four legs'). Thus if 'tonk' comes after any P in any compound statement, then Q will cancel out P. If that appears odd (especially with the seeming contradiction), that's simply because, as Prior puts it, “there is simply nothing more to knowing the meaning of ['tonk'] than being able to perform these inferences” (129). In this case, we firstly state P, and then P-tonk-Q (in which Q cancels out P), from which we conclude R.

Nuel D. Belnap helps us understand what's happening here by offering us different symbols and a different scheme. Instead of the argument-form above (which includes P, Q and R), we have the following:

i) A ⊢ A-tonk-B

ii) A-tonk-B ⊢ B

iii) A ⊢ B

Quite simply, one can deduce A-tonk-B from A. Then one can deduce B from A-tonk-B. Finally, this means that one can derive B from A.

In our example, by a simple rule of inference, one can derive 'Cats have four legs tonk cats have three legs' (A-tonk-B) from 'Cats have four legs' (A). And then one can derive 'Cat have three legs' (B) from 'Cats have four legs tonk cats have three legs' (A-tonk-B). Finally, one can derive 'Cats have three legs' (B) from 'Cats have four legs' (A).

Belnap claims that an arbitrary creation of a connective (through implicit definition) could or can result in a contradiction. Thus, the symbol '?' in the following

(a _? c) _{= df}  a + c

(b    d)        b + d.

could result in: 2 ₌ 3

                           3    5

However, doesn't Prior's 'Therefore 2 and 2 are 4 tonk 2 and 2 are 5' also contain a contradiction? Prior seems to be stressing the point that in the definitions of connectives, even counterintuitive ones, such contradictions are to be expected. Isn't that the point? 

*) See also Saul Kripke's 'quus' and Wittgenstein's 'rule-following'.

References

Belnap, Nuel. (1962) 'Tonk, Plonk and Plink'.

Prior, A.N. (1960) 'The Runabout Inference Ticket'.