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.]

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.