Draft version – not to be cited without permission.

There’s more to Contradictions than Explosion

The concept of contradiction plays an important role in philosophical

discourse. Thus, it’s surprising that little effort has been exerted in evaluating

the plausibility of the various prevalent non-equivalent definitions of

‘contradiction’ in the literature. This paper continues the project of evaluating

available definitions by considering the explosion account of contradiction, an

account which constitutes an apparent challenge to the plausibility of

dialetheism by defining contradictions as (sets of) propositions that imply

every proposition, thereby committing the dialetheist to the truth of every

proposition. Two versions of the account are considered: ExS), which defines

contradictions as explosive sets of propositions and ExP), which defines

contradictions as explosive propositions. While ExS) is found to be

inadequate for failing to facilitate important distinctions between sets of

propositions and conceiving of contradictions as sets, ExP) is found to

miscategorise certain logical falsehoods and non-logical falsehoods as

contradictions, fail to respect the substance of the law of non-contradiction,

and preclude the term ‘contradiction’ from entering the vocabulary of

paraconsistent logics. The paper concludes that these criticisms give us good

reason to believe that the explosion account, in both its forms, is an

implausible definition of ‘contradiction’, and consequently isn’t a viable threat

to the dialetheist’s thesis.

Keywords: Contradiction; Explosion; Dialetheism; Law of non-contradiction.

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Numerous non-equivalent definitions of ‘contradiction’ can be found in the

philosophical literature. For example, the definition of contradictions as the conjunction of a

proposition and its negation found in Kalish et al. [1980: 18] doesn’t categorise propositions of

the form ~x(x = x) as contradictions, while the definition of contradictions as logically false

propositions [Detlefsen et al.. 1999: 27-28] does. Consequently, as the definitions categorise

different propositions as contradictions, not all prevalent definitions of ‘contradiction’ are

equally plausible. Given the importance of the concept of contradiction within philosophical

parlance, this non-equivalence between the prevalent definitions should suffice to motivate a

consideration of their plausibility. We should consider the candidates with a view to endorsing

the most plausible one.

In addition, some of these prevalent definitions are also prima facie challenges to the

plausibility of dialetheism, the thesis that some contradictions are true. For example, Slater

[1995: 453] famously defined contradictories as propositions that cannot be true together with a

view to demonstrating that dialetheism was nonsense. In order to rebuke this accusation, the

dialetheist must show either that the proposed definition is faulty or that, contrary to initial

appearances, the definition doesn’t preclude the truth of contradictions; something Priest [2006:

469-471] has attempted with regards to Slater’s definition. Consequently, evaluating the

plausibility of prevalent definitions of ‘contradiction’ isn’t simply a means to establishing the

most reasonable definition; in some cases we are committed to evaluating a definition to

determine whether it precludes the truth of that most intriguing philosophical position,

dialetheism.

In this paper we concentrate on a particularly interesting definition of ‘contradiction’

found in the literature, the explosion account, which defines contradictions as (sets of)

propositions that imply every proposition. If the dialetheist were compelled to accept this

account then she would be forced to admit that her thesis committed her to the truth of every

proposition, otherwise known as trivialism. As this is a commitment the dialetheist famously

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wishes to avoid [Priest 2006: ch. 3], she is obliged to combat the explosion account head on and

demonstrate its implausibility. Consequently, the explosion account is worth evaluating for both

the dialetheist’s sake and as part of the larger project of finding an adequate definition of

‘contradiction’.

1. The Explosion Account of Contradiction

The most recent endorsement of the explosion account of contradiction is found in Field [2005:

24],

I think a better use of the term ‘contradiction’ would be: sentence that implies

every other.

However, earlier versions of the account are found in Quine [Quine & Carnap 1990: 79],

A sentence or set of sentences is contradictory if every sentence is a

consequence of it,

and Carnap [1937: 183],

[A sentence s] is contradictory if and only if every sentence of S is an L-

consequence of s in S.

From these three definitions, two versions of the explosion account are clearly distinguishable.

The first is explicitly only interested in singular explosive sentences or propositions,

ExP) A proposition p is a contradiction if and only if p implies every proposition

[B p ⊨ B],

while the second talks instead in terms of sets,

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ExS) A set of propositions is a contradiction if and only if implies every

proposition [B ⊨ B].

In evaluating the philosophical plausibility of ExP) and ExS), we will rely on the usual criteria for

a successful scientific definition [Belnap 1993]. Namely, that the definitions are neither too lax

nor strict in their categorisation of propositions as contradictions, that the definiens and the

definiendum of the definitions can be used interchangeably in a sentence without altering the

sentence’s meaning, or trivialising the truth or falsity of a substantive principle, and that the

definitions reflect the actual use of the term by the philosophical community.

2. The Explosion Account and Logical Falsehoods

As classical semantics are by far the most accepted semantics in the philosophical literature, a

requirement of any definition of ‘contradiction’ is that it’s at least plausible when assuming a

classical semantics. However, unfortunately for ExP) and ExS), the standard semantic account of

logical consequence used in classical logic entails that certain propositions and sets of

propositions which don’t seem to be contradictions are explosive.

The standard semantic account of logical consequence (SSC) used in classical logic states

that,

⊨ B if and only if for any v, if A , v(A)=1, then v(B)=1.

Thus, according to SSC, unless the logic in question is trivial,1 a premise set is explosive if and

only if there is no truth-value assignment under which every member of receives the truth-

value true. Given that in classical logic a proposition p’s being false is equivalent to p’s not being

true, a premise set is explosive if the conjunction of all the members of is a logical

1 A logic L is trivial if and only if B ⊨L B, which given weakening ensures AB A ⊨L B.

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falsehood. Similarly, a proposition p is explosive if p is a logical falsehood. Consequently,

according to SSC there are two circumstances in which a set of propositions is explosive:

C-ExS) The conjunction of all the members of is a logical falsehood

S-ExS) A member of is a logical falsehood.

S-ExS) covers those singleton cases that C-ExS) doesn’t. Thus,

A set of propositions is explosive if C-ExS) or S-ExS) are true of .

Consequently, assuming the SSC, ExS) entails,

A set of propositions is a contradiction if C-ExS) or S-ExS) are true of .

Similarly, assuming the SSC, as a proposition p is explosive if p is a logical falsehood, ExP)

entails,

A proposition p is a contradiction if p is a logical falsehood.

Thus, assuming the SSC, ExP) categorises as contradictions at least all logical falsehoods, and

ExS) categorises as contradictions at least those sets of propositions that either contain a

logical falsehood or form a logical falsehood when its members are conjoined. These

consequences of the SSC are problematic for both versions of the explosion account, as there

are clear cases of logically false propositions, and clear cases of sets of propositions that fulfil

either C-ExS) or S-ExS), that aren’t commonly considered contradictions by philosophers. The

remainder of this section demonstrates this troublesome consequence for ExP), and the next

demonstrates the same for ExS).

Contrary to the commitments of ExP), when assuming the SSC, there are propositions

that are logically false but which we wouldn’t normally consider contradictions. Here we will

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concentrate on cases within zero-order classical logic, but cases are also forthcoming in first-

order logic. In a zero-order logic, logically false propositions are those propositions that are of a

logical form F that outputs the truth-value false under all truth-value assignments. This ensures

that, according to ExP), any propositions of the form ~(p ~p) or ~(p p) are a contradiction.

However, while it is clear that propositions of the form ~(p ~p), such as ‘It’s not the case that

either it’s raining or it’s not raining’, are logical falsehoods in classical logic, we have good

reason to doubt that these propositions are contradictions given that our formal derivation

systems fail to recognise them as such.

When constructing a formal reductio we attempt to demonstrate that a contradiction is

entailed by an assumed premise, perhaps in conjunction with other premises. From this, the

truth of the negation of the assumed premise is entailed, given the truth of the other premises.

Only when formulae symbolizing contradictions occur, however, are we allowed to derive the

negation of the assumption. In classical logic these formulae take the form A ~A or ~A A.

Consequently, to complete a reductio in classical logic one must derive a formula of the form

A ~A or ~A A (or, assuming the admissibility of adjunction, A and ~A separately in the

sub-proof). Unfortunately for ExP), the formula ~(p ~p), not being of the form A ~A or

~A A, doesn’t permit the completion of a formal reductio, and as such isn’t recognised as

symbolising a contradiction within our derivation systems. While the formula ~(p ~p) could

be conjoined with the classical theorem p ~p to produce a contradiction, this doesn’t show that

propositions of the form ~(p ~p) are themselves contradictions, for this would require one to

accept the principle that the property of being a contradiction is preserved backwards through

entail; a principle we have good reason to reject.

Allowing for the property of being a contradiction to be backwardly preserved would

significantly restrict the expressive power of the term ‘contradiction’ in at least two significant

respects. Firstly, we would no longer be able to distinguish those theories or sets of beliefs that

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contained a contradiction from those that entailed a contradiction, a distinction we greatly

cherish. It is one thing for a theory to entail a contradiction, but another for a theory to outright

contain a contradiction. In scrutinising an individual’s set of beliefs we will be far less damning

of the individual whose beliefs entail a contradiction than the individual who actually explicitly

believes a contradiction. After all, unless we are unrealistically conservative with our beliefs, our

beliefs are bound to entail a contradiction somewhere.

Secondly, we want to be able to distinguish between the theory that one endorses

containing a contradiction and one’s statement endorsing the theory being itself a contradiction.

However, if the property of being a contradiction is backwardly preserved, then any

endorsement of a theory containing a contradiction will itself be a contradiction as the

proposition ‘Theory T is true’ will entail a contradiction if T contains a contradiction. If we

accept the principle that the property of being a contradiction is backwardly preserved, then we

must admit that any statement of the form ‘X entails a contradiction’ says nothing more than

our consequence relation is reflexive. However, we seem to mean something far more

substantive with statements of this form.

Thus, we have good reason to reject the principle that the property of being a

contradiction is backwardly preserved through the consequence relation. Consequently,

propositions of the form ~(p ~p) aren’t contradictions in virtue of formulae that allow the

completion of a formal reductio in classical logic being derivable from ~(p ~p). ExP) and our

derivation systems, therefore, are in tension over the categorisation of certain propositions as

contradictions. While ExP) recognises propositions of the form ~(p ~p) as contradictions, our

formal derivation systems don’t. Either our derivation systems are at fault here, or ExP) has

miscategorised certain propositions as contradictions. Given how well-established and

successful our derivation systems are, this disagreement would seem to be a slight upon ExP)

and not our derivation systems.

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The failure of ExP) to adequately differentiate contradictions and logical falsehoods is

also manifested in its inability to distinguish two forms of explosion that we can distinguish in

our formal systems:

C-explosion: {A ~A} ⊨ B.

F-explosion: ⊥ ⊨ B.

Not only are C-explosion and F-explosion non-equivalent, but C-explosion isn’t even a special

case of F-explosion in some logics [Marcos 2005: ch. 1]. For example, with the introduction of

the bottom particle into da Costa’s [1974] C-Systems that allow formulae of form A ~A to

be assigned the truth-value true, the logics would contain F-explosion but not C-explosion as

the C-Systems use the SSC while assigning some contradictions the truth-value true. Thus, we

are able to formally distinguish and model cases of explosion involving contradictions, as they

are normally formally conceived, and cases of explosion involving arbitrary logical falsehoods.

By defining contradictions as explosive propositions, however, ExP) is unable to accommodate

this formally interesting distinction. As far as ExP) is concerned, the premise sets of both

C-explosion and F-explosion are contradictions, and thus there’s no distinction to be made

between them. ExP), therefore, suffers from the weakness of miscategorising certain non-

contradictory logically false propositions as contradictions which, consequently, blinds us to

theoretically interesting formal results.

3. What’s Wrong with Contradictions as Explosive Sets?

In the previous section we showed that ExP) miscategorises certain propositions as

contradictions. In this section we show that ExS) is inadequate for both failing to facilitate

philosophically important distinctions between sets of propositions, and miscategorising

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innumerable sets of propositions as contradictions. We will discuss each concern in turn,

beginning with the former.

By defining a contradiction as a set of propositions that either contains a logical

falsehood or forms a logical falsehood when its members are conjoined, ExS) fails to facilitate

philosophically important distinctions between sets constituted exclusively by a contradiction,

sets constituted by contradictory pairs, sets that entail a contradiction, and those that contain a

contradiction. Once we appreciate the roles that these distinctions play in our discussion of

theories and beliefs, this criticism appears pretty damning of ExS).

As sets of propositions of the form {p, ~p} fulfil C-ExS), sets of this form are as much

contradictions, according to ExS), as sets of the form {p ~p}. Consequently, as it stands, ExS)

cannot adequately distinguish sets constituted of contradictory pairs and sets constituted

exclusively of a contradiction. Indeed, as far as ExS) is concerned, there is no conceptual space

available to accommodate contradictory pairs as distinct from contradictions. However, when

we conceive of sets of propositions as belief sets, the distinction between a set of contradictory

pairs and a set constituted exclusively of a contradiction becomes indispensable. After all, given

that beliefs aren’t closed under entailment, we need to be able to distinguish between cases in

which an individual believes contradictories, B(p) and B(~p), and cases in which an individual

believes a contradiction, B(p ~p).2

2 ExP) also has trouble accommodating contradictories in some cases. Contradictories are the

constituents of contradictions, however some of the propositions that ExP) categorises as

contradictions, such as those of the form ~(p p), have no clear constituents which could be

identified as contradictories. Consequently, either we must admit that some contradictions fail to be

constituted of contradictories, a fundamental element of the concept of contradiction, or admit that

ExP) fails to accommodate the important concept of contradictories into its account of contradictions.

Without independent reason to doubt that all contradictions are constituted of contradictories, this is

additional evidence against ExP)’s plausibility.

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For the same reasons, the distinction between an individual believing a set of

propositions that entail a contradiction and an individual believing a contradiction is equally

important. This is another distinction, however, that ExS) cannot accommodate. Consider a set

of propositions of the form {q, p ~q, p}. While this set doesn’t contain a contradiction

formally conceived, the conjunction of its members forms a logical falsehood, ensuring that the

set fulfils C-ExS). Consequently, the set of propositions is explosive and, according to ExS), is as

much a contradiction as propositions of the form q ~q that can be derived from the set. This

consequence is particularly troublesome for ExS) given that the distinction between believing

sets of propositions that entail, but don’t contain, contradictions, and believing a contradiction,

is necessary for making sense of the well-circulated claim that we can rationally hold a set of

beliefs that entail a contradiction but not a contradiction itself [Foley 1979].

Finally, when talking about contradictory theories, we often wish to distinguish between

the whole contradictory theory and that element of the theory which is itself a contradiction. By

defining contradictions simply as sets of propositions that explode, however, ExS) fails to

respect the important distinction between sets of propositions containing a contradiction and

sets constituted exclusively by a contradiction. This can be demonstrated through weakening:

for any proposition p, if p ⊨ q, then for any set of propositions , {, p} ⊨ q. Given weakening,

if a proposition p is explosive, any sets containing p are also explosive. Thus, any set

containing an explosive proposition is, according to ExS), a contradiction. There is no

conceptual room for the distinction between a contradiction and a larger set of propositions

containing a contradiction.

Consequently, ExS) fails to adequately distinguish between contradictions, contradictory

pairs, sets of propositions that entail a contradiction, and sets of propositions that contain

contradictions; all distinctions that we cherish in our discussion of beliefs and theories.

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To show that an account of contradiction can respect these important distinctions, and

thus sustaining this expressive power is something to be expected of a definition of

‘contradiction’, consider the definition of a contradiction as the conjunction of a proposition

and its negation. By defining contradictions so, sets of contradictories can be identified as

distinct from contradictions, as the two conjuncts of a contradiction which remain as solely

contradictories until they are conjoined. Similarly, as proposition sets of the form

{q, r q, p ~p} contain other propositions in addition to a conjunction of a proposition and

its negation, the account allows us to distinguish sets of this form from sets constituted solely of

a contradiction, and as sets of propositions of the form {q, p ~q, p} don’t contain a

conjunction of a proposition and its negation, the account equally allows us to distinguish

between these sets which entail, but don’t contain, a contradiction, and contradictions

themselves. Not only, therefore, does ExS) fail to respect the important distinctions between

contradictions, contradictory pairs, sets of propositions containing contradictions, and sets of

propositions that entail contradictions, but at least one of the account’s competitors successfully

sustains these important distinctions.

The criticism in this section of ExS) has so far concentrated on its expressive failure, a

failure which some competing definitions of ‘contradiction’ do not suffer from. Our second

criticism of ExS), however, is even more devastating. Sets cannot be contradictions. Any

plausible definition must ensure that the meaningfulness of a sentence containing the definiendum

must be preserved when it is replaced by the prospective definiens. Sets, however, are not the

right kind of entity to be true or false. Yet, it certainly makes sense to say ‘Contradictions are

false’. Consequently, ExS) fails to fulfil one of the fundamental criteria for any definition of

‘contradiction’, namely preserving the meaningfulness of sentences containing the definiendum.

No definition of ‘contradiction’ in terms of sets can be adequate. Therefore, if any version of

the explosion account is to be ultimately adequate, despite its weaknesses outlined in section 2,

then it will have to take the form of ExP) and not ExS).

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4. The Explosive Statement

In section 2 we found that ExP) miscategorised certain non-contradictory logical falsehoods as

contradictions. In addition to miscategorising these logical falsehoods, there is at least one

explosive proposition that ExP) categorises as a contradiction which is neither commonly

considered to be a contradiction, nor a logical falsehood in propositional or first-order logic:

) Every proposition is true.

This criticism is putatively distinct from that in section 2, that ExP) doesn’t adequately

distinguish contradictions and logical falsehoods, as ) putatively isn’t a logical falsehood.

Indeed, for ) to count as a logical falsehood, we would need to admit a logic that allowed us to

quantify over propositions and included truth and falsity predicates.3 Therefore, if one didn’t

admit any logic beyond first-order plus identity, ) would constitute a new challenge to ExP).

Two responses seem available to the advocate of ExP) to combat ), though neither are

ultimately successful.

Firstly, the advocate of ExP) could appeal to the principle of reflective equilibrium and

bite the bullet with this example, thereby considering ) a contradiction. It is, after all, only one

3 Let T be the truth predicate and F the falsity predicate:

1. pTp Assumption

2. Tp {1} Universal Elimination

3. p {2} T-predicate Release [Tp p]

4. T(Fp) {1} Universal Elimination

5. Fp {4} T-predicate Release

6. ~p {5} F-predicate Release [Fp ~p]

7. ~pTp {1-6} Reductio

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example. Yet, for a theory to be justified in absorbing anomalies we need to have independent

support for the theory. There is no justification at all for an unsuccessful theory to absorb

anomalies. Given that we aren’t at present aware of any independent support for ExP), the

account would be unjustified in absorbing the anomaly to deflect the counterevidence against it.

Additionally, arguing for ) to be considered a contradiction isn’t an option for the advocate of

ExP). If contradictions are just explosive propositions, then there shouldn’t be any other

properties for the advocate of ExP) to appeal to in arguing that ) is indeed a contradiction,

contrary to appearances. If the proposition is explosive, then it’s a contradiction as far as ExP) is

concerned; there are no further identifying properties of contradictions.

Secondly, ExP) could be adapted so that contradictions aren’t simply propositions which

explode, but instead propositions which explode due to their logical falsity. Thus, a proposition

would only be categorised as a contradiction if it were explosive because of its logical form and

the SSC. This response would successfully block ) from being categorised as a contradiction by

ExP) for, as stipulated by the counterexample, it isn’t a logical falsehood. In giving this reply,

however, the advocate of ExP) is entering dangerous ground. By adding the condition that

contradictions must explode because of their logical form, she ensures that contradictions are of

a logical form that no other propositions are. Now, if contradictions are of this unique logical

form, it’s bemusing why we should define contradictions as those propositions that explode due

to their logical form, rather than directly in terms of their unique logical form. After all, if we

defined contradictions in terms of their logical form, we would be able to explain why

contradictions are often considered to be explosive, rather than simply stipulating that they are.

When it is a choice between stipulation and explanation, the account that offers the explanation

seems the more theoretically fruitful option. Consequently, by adding this condition to the

explosion account its advocate is almost admitting that there are other more explanatorily

fruitful accounts of contradiction available, such as the definition of contradictions as the

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conjunction of a proposition and its negation, or as propositions of the form A ~A [Haack

1978: 244]. To this extent, it hardly seems a viable solution to ) for the explosion account.

Thus, unless the advocate of ExP) can admit ) as a logical falsehood, it seems that ExP)

miscategorises at least one non-logically false proposition as a contradiction, in addition to

certain non-contradictory logical falsehoods.

5. The Law of Non-Contradiction

So far we have argued that ExP) miscategorises certain logical falsehoods, and at least one non-

logical falsehood, as contradictions. Unfortunately for advocates of ExP) there is a further,

philosophically more fundamental, concern with ExP), which is that the account seems to entail

the law of non-contradiction (LNC) is insubstantial.

Any adequate definition of ‘contradiction’ must take into account the perceived

philosophical importance and substance of the LNC. The significance of this feature of an

account of contradiction to its plausibility is ensured by a) the perceived philosophical

importance of the LNC, and b) the fact that the LNC contains the concept of contradiction.

Evidence for the former point comes from the historically elevated position that the

LNC has held as one of the most important logical laws, with Aristotle [1984: 1005b 22]

considering it to be the most certain of all principles. The continued vaulted status of the law in

contemporary philosophy is demonstrated both by the general distaste for contradictions within

philosophical theories, and by the fact that philosophers still bother to propose formulations of

the law while mentioning it as a fundamental logical principle in introductory logic texts.

The latter point, that the LNC contains the concept of contradiction within it, if not

obvious from the occurrence of the term ‘contradiction’ in its name, can be shown by

formulations of the law in the literature:

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Aristotle1 [1984: 1011b 13-14]: Contradictory statements are not at the same time true.

Aristotle2 [1984: 1005b 19-20]: The same attribute cannot at the same time belong and not

belong to the same subject in the same respect.

Brown1 [2004: 126]: No contradictory sentence is ever correctly assertable.

Dummett1 [1978: xix]: Not both A and not A.

Englebretsen1 [1981: 5]: A sentence and its negation cannot both be true.

Prior1 [1967: 461]: A statement and its direct denial cannot be true together.

Russell1 [2001: 46-47]: Nothing can at the same time have and not have a certain property.

Smith1 [2003: 101]: A proposition and its negation cannot both be true.

While only two of the definitions above explicitly use the terms ‘contradiction’ or

‘contradictory’, those that don’t can be shown to include the concept of contradiction under the

guise of a particular account of contradiction. For example, Englebretsen1 and Smith1 speak of

truthbearers and their negations rather than contradictions, however in Kalish et al. [1980: 18]

we find very similar language being used to define ‘contradiction’. Similarly, the “statement and

its direct denial” of Prior1 can either be linked to the Kalish et al. definition, interpreting Prior’s

“denial” as equivalent to negation, or it can related to the pragmatic account of contradiction in

terms of assertion and denial found in Strawson [1993: 21]. Equally, the “A and not A” of

Dummett1 could be a version of Kalish et al.’s account of contradiction, or a partial natural-

language translation of the formula p ~p used by Haack [1978: 244] as a definition of

‘contradiction’. Finally, while Aristotle2 and Russell1 speak of a subject both possessing and not

possessing a property, rather than directly of contradictions, we find a definition of

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‘contradiction’ in similar terms endorsed by Routley and Routley [1985: 204]. Consequently, it

seems that the concept of contradiction is contained within formulations of the LNC. While we

have only considered eight examples here, out of the thousands available in the literature, we

should be confident that any plausible version of the LNC can be shown to include the concept

of contradiction within it.

The LNC, therefore, appears to be a logical law of great philosophical import that

contains the concept of contradiction. If an account of contradiction is to be philosophically

plausible, it must respect both of these facts, either by ensuring that the definition being

advanced can be embedded into the LNC while maintaining the law’s substance, or by

explaining away its perceived substance as illusory.

There are three broad interpretations of the LNC. The first is as a semantic law, as in

Englebretsen1, stating a semantic fact about contradictions, the second is as a pragmatic law,

stating what one shouldn’t assert or believe, as in Brown1, and the third is as a metaphysical law,

asserting what cannot obtain, as in Russell1. Given that ExP) conceives of contradictions as

propositions, and not states of affairs, a metaphysical interpretation of the LNC is clearly

inappropriate to embed ExP). This leaves the account with either the semantic or pragmatic

interpretation of the law to demonstrate that it can be meaningfully embedded into the LNC.

We will explore both possibilities in turn, beginning with the pragmatic LNC.

The pragmatic interpretation of the LNC states that one shouldn’t assert or believe

contradictions. In what follows, for expediency, we shall concentrate here on the assertion

version of the pragmatic LNC,

LNCA) One shouldn’t assert contradictions.

However, given the similarity between versions of the pragmatic interpretation of the LNC, the

concerns raised over embedding ExP) into LNCA) will also be relevant to other versions of the

pragmatic interpretation such as,

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LNCB) One shouldn’t believe contradictions,

and

LNCD) One should deny contradictions.

There are two reasons to believe that ExP) cannot be embedded into LNCA) while maintaining

the law’s substance. Firstly, by substituting the definiens of ExP) for ‘contradiction’ in LNCA) we

obtain,

LNCA’) One shouldn’t assert propositions that imply every proposition.

So far we merely replaced the definiendum with the definiens of ExP). A further rather simple

inference, however, shows how inconsequential this interpretation of the LNC is. If one

shouldn’t assert propositions that imply every proposition, then this cannot be because it

implies true propositions, after all surely it is a good thing to be rationally committed to true

propositions, rather it must be because it implies falsehoods (in fact, every falsehood). Thus, the

imperative behind LNCA’), if it’s to be a reasonable philosophical principle, must be,

LNCA’’) One shouldn’t assert propositions that imply falsehoods.

Yet, LNCA’’) is hardly the impressive philosophical law. Propositions that imply falsehoods are

themselves falsehoods. So, LNCA’’) is a special instance of the more general epistemic maxim

‘One shouldn’t assert falsehoods’. By embedding ExP) into LNCA), the pragmatic interpretation

of the law becomes a rather convoluted way of stating that one shouldn’t assert propositions

which imply falsehoods; a special instance of the epistemic maxim that one shouldn’t assert or

believe falsehoods. Consequently, embedding ExP) into LNCA) demotes the LNC from the

status of a fundamental logical law to a special instance of an epistemic principle.

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Our second reason to doubt that ExP) can be embedded into LNCA) while maintaining

the law’s substance is that the LNC’s well-established formalisation, ~(A ~A), is inconsistent

with LNCA’). Making a claim about whether one should assert contradictions, LNCA) requires

the use of a force operator to be adequately formalised. As it is, the standard formalisation of

the law contains two tildes, with the outer tilde functioning on contradictions formalised as

formulae of the form A ~A. Consequently, to incorporate the standard formalisation of the

law into the pragmatic interpretation of the LNC requires interpreting the outer tilde as a force

operator. However, given that we are formally required in a calculus to interpret identical

symbols consistently, interpreting tilde as a force operator would produce absurd consequences

given that force operators cannot be meaningfully embedded. Thus, acceptance of LNCA) as an

adequate interpretation of the law would require a reevaluation of our standard formalisation of

the law, and consequently a reevaluation of the law’s status as a logical truth, given that

formulae symbolising the law wouldn’t be theorems of zero- or first-order classical logic as

neither logic contain force operators.

Even if we were justified in changing our formalisation of the LNC, so that it were

consistent with the pragmatic interpretation of the law, there are good reasons to believe that

the formalisation would be inadequate to capture the whole content of LNCA’), or any of the

other versions of the pragmatic interpretation containing ExP). Take as an example Brady’s

[2004] suggestion that we change our formalisation of the LNC from ~(A ~A), on the

grounds that it fails to serve the function in formal systems expected of the LNC by not

precluding the truth of contradictions, to ┤A ~A in a formal rejection system containing a

dual symbol ‘┤’ for rejection in addition to the turnstile for theoremhood. With this new

formalisation of the law it may be possible to adequately model LNCD) within a formal system,

19

using ‘┤’ as a denial operator.4 However, given that there are propositions of different logical

forms that are explosive according to SSC, and at least one explosive proposition that isn’t a

logical falsehood in zero- or first-order logic, formalising the LNC as ┤A ~A wouldn’t

adequately capture all of LNCD) when it contained ExP),

LNCD’) One should deny propositions that imply every proposition.

As ┤A ~A only informs us to deny or reject propositions of the form A ~A, and not all

explosive propositions, it is incomplete in its expression of LNCD’). To effectively formally

express LNCD’) would require the introduction of an explosion operator which could be

prefixed to any formula A symbolising an explosive proposition, so that ‘┤A’ becomes ‘Deny

explosive propositions’. This accurate formalisation of LNCD’), however, seems far removed

from our common conception of the law, and requires us to admit that the formal

representation of the LNC isn’t a theorem of, and cannot be expressed within, classical zero- or

first-order logic. This is a very high price to pay for an account of contradiction which so far has

very little to recommend it.

Consequently, it seems that the pragmatic interpretation of the LNC is unsuitable to

contain ExP). Combining ExP) with the pragmatic interpretation both demotes the LNC from

the status of a fundamental logic law to a special instance of an epistemic principle, and ensures

that a complete reevaluation of our formalisation of the LNC is necessary; a reevaluation which

seems unmotivated. Thus, if ExP) is to be meaningfully embeddable into the LNC while

4 There are independent reasons to doubt that LNCD) can be adequately modelled by formulae of this

form, as there is good reason to believe that the turnstile, and the reverse turnstile ‘┤’, cannot be

meaningfully interpreted as force operators (see X). However, as a discussion of these reasons is far

beyond this paper’s scope, we will admit the possibility here of adequately modelling LNCD) with

┤A ~A.

20

maintaining the law’s substance, it will have to be embeddable into the semantic interpretation

of the LNC.

If we interpret the LNC semantically, the law states a new semantic fact about

contradictions; namely,

LNCF) All contradictions are false.

Replacing ‘contradiction’ with the definiens of ExP), LNCF) becomes,

LNCF’) All propositions that imply every proposition are false.

Yet, LNCF’) is totally uninformative. We already knew that propositions that imply every

proposition are false. Firstly, some propositions are false. Thus, explosive propositions imply

false propositions. However, if a proposition implies a false proposition, it itself is false.

Consequently, of course explosive propositions are false. We don’t need a fundamental logical

law to inform us of this. Secondly, apart from the example of ) given in section 4, assuming

the SSC all explosive propositions are explosive because they are logically false. Thus, by

understanding the SSC, we already know LNCF’) apart from the case of ), which is obviously

false for the first reason given. Consequently, LNCF’) simply reiterates a result of the SSC plus

the uninteresting fact that ) is false – hardly the fundamental logical law.5

While the LNC conceived as LNCF’) could be pedagogically useful, expressing

succinctly the conclusion of a chain of reasoning, this isn’t how the law has been historically

5 As the SSC only informs us that a proposition A is explosive if and only if A is either logically false

or every proposition is logically true, we might think that LNCF’) informs us of something which the

SSC doesn’t. Namely, that not every proposition is a logical truth. However, this clearly isn’t the

perceived content of the LNC by the philosophical community, nor do we need a philosophical

principle to inform us that there are propositions of the form p that aren’t logical truths, for it is a

simple consequence of the SSC.

21

viewed. To move the law from the status of substantial logical principle to mere shorthand for a

chain of reasoning that can be easily explained seems ill-motivated.6 Demonstrating the truth of

the LNC isn’t synonymous with explaining the SSC, the fact that not every proposition is

logically true, and the falsity of ). By embedding ExP) into LNCF) the LNC seems to lose all of

its philosophical significance.

In addition to these concerns over LNCF’) reducing the LNC to a pedagogical tool, this

formulation causes problems when one attempts to find a formalisation of the law. As when

ExP) was embedded within the pragmatic interpretation of the LNC, the standard formalisation

of the law ~(A ~A) fails to express all of LNCF’)’s content. The standard formalisation of the

law tells us that propositions of the form ~(A ~A) are logically true, which given the normal

semantics for negation entails that propositions of the form A ~A are logically false. It

doesn’t, however, tell us that all explosive propositions are false, for not every explosive

proposition is of the form A ~A. Nor is the meta-logical statement ‘⊥ ⊨ B’ an adequate

formal expression of LNCF’), as it only informs us that every logically false proposition is

explosive, not that every explosive proposition is false.

Consequently, as when ExP) was embedded within the pragmatic version of the LNC,

LNCF’) requires us to revise our standard formalisation of the LNC with the introduction of an

explosion operator, a move that we seem to lack motivation for given ExP)’s other faults;

especially given that it would require us to admit that the formula symbolising the LNC isn’t a

theorem of classical zero- or first-order logic, for the reasons already mentioned above.

Neither the semantic nor pragmatic interpretations of the LNC are suitable when ExP) is

embedded within them. Yet, these seem the only viable interpretations of the law for a

6 While Barnes [1969] has argued that Aristotle considered syllogistic demonstrations to have a solely

pedagogical use, there’s no evidence that Aristotle equally considered his presentation of the logical

laws to be solely pedagogical. For example, the fact that Aristotle offers justification for the LNC

suggests he wasn’t merely presenting an already well-established logical or metaphysical fact.

22

definition of ‘contradiction’ in terms of truthbearers. Thus, ExP) has failed to show that it can

be meaningfully embedded into the LNC while maintaining the law’s substance. Consequently,

the advocate of the explosion account has one option with which to rescue her definition from

the accusation that it makes a substantial logical law insubstantial. She must provide us with

reason to believe that, consistent with her definition, the LNC isn’t of philosophical import.

Whether she can achieve this is beyond this paper’s scope. However, given that other

definitions of ‘contradiction’, such as Kalish et al.’s definition in terms of the conjunction of a

proposition and its negation, can maintain the LNC’s status as a substantial logical law, there’s

good reason to doubt we can be persuaded that the LNC isn’t of philosophical import.

6. A Threat to Paraconsistency

We have shown that ExP) miscategorises certain propositions as contradictions, and that the

account cannot be meaningfully embedded within the LNC while maintaining the law’s

substance. We have one final criticism of ExP) – that it precludes the possibility of some

paraconsistent logics through stipulation alone. Paraconsistent logics are logics which invalidate

explosion. Particularly, they invalidate an unconjoined form of C-explosion {A, ~A} ⊨ B.

While some of these logics, such as Jaśkowski’s [1969] discursive logic, only invalidate this

unconjoined form of C-explosion, other paraconsistent logics, such as Priest’s [1979] LP and da

Costa’s [1974] C-systems, also invalidate a conjoined C-explosion {A ~A} ⊨ B. Now, these

latter paraconsistent logics, in invalidating the conjoined C-explosion, are ordinarily understood

to be showing that contradictions are not explosive. Indeed, the whole point of Priest’s LP is to

support the dialetheic hypothesis that some contradictions are true without triviality ensuing.

Yet, by defining contradictions as explosive propositions, ExP) precludes the possibility of this

very serious research area on purely definitional grounds. There can, according to Exp), be no

logics that model non-explosive contradictions, as this would contravene the meaning of

23

‘contradiction’. This consequence of the account seems totally inappropriate. Paraconsistent

logics that invalidate conjoined C-explosion have been successfully constructed, and they

cannot be simply disregarded on definitional grounds.

For a potential solution to this concern, we can look to an actual advocate of the

explosion account. Field [2005] has suggested that paraconsistent logicians aren’t in fact talking

about contradictions when they state that ‘Contradictions aren’t explosive’. After all, they can’t

be, because contradictions explode:

Priest revels in saying that we should accept contradictions. Here

‘contradiction’ is used either in the sense indicated above (a pair of a sentence

and its negation) or in the sense of a sentence of the form B ~B… Talk of

accepting contradictions shows a flair for the dramatic, but I think it tends to

put people off for bad reasons… I think a better use of the term

‘contradiction’ would be: sentence that implies every other. On this alternative

usage, the way to put Priest’s view is that sentences of form B ~B (or pairs

{B, ~B}) aren’t in general contradictory: they don’t imply everything.

[Field 2005: 24]

Field, therefore, allows for paraconsistent logicians to articulate their point while requiring they

don’t use the term ‘contradiction’. The content of the paraconsistent logician’s theories remain,

while their terminology is revised. The problem, one might think, has been avoided. The

possibility of paraconsistent logics that invalidate conjoined C-explosion is no longer precluded.

Unfortunately, the proposed solution has unsavoury consequences. Dialetheists use

paraconsistent logics that invalidate conjoined C-explosion because they wish to admit the truth

of contradictions without triviality. Yet, according to ExP), if a proposition isn’t explosive then it

isn’t a contradiction. It turns out that those propositions dialetheists propose as contradictions,

and propose the truth of, aren’t actually contradictions (as Field admits above). What then are

dialetheists proposing if not the truth of contradictions? According to Field, it would be the

truth of the conjunction of a proposition and its negation. Yet, there is an available account of

24

contradiction that defines contradictions just as the conjunction of a proposition and its

negation (Kalish et al. 1980: 18). What reasons then does Field give us to prefer the explosion

account to this rival definition, when it seems ExP) requires us to unnecessarily distort the

debate between the classical logician and the dialetheist? He proposes that by defining

contradictions as the conjunction of a proposition and its negation, we “put people off [the

dialetheist’s thesis] for bad reasons.” But, one of the reasons why philosophers are concerned

by dialetheism is that it challenges the historic LNC in some form, and what is the LNC about if

not contradictions? Dialetheism could hardly be offering violations of the LNC if it were not

talking about contradictions. Given that we possess available accounts of contradiction that can

explain how dialetheists are proposing the truth of contradictions, and thus proposing violations

of the LNC, we need to be given good reason to consider the term ‘contradiction’ inappropriate

for use by a dialetheist. Field’s reason isn’t one.

It is theoretically damaging for an advocate of explosive logics to preclude the possibility

of ‘contradiction’ entering the vocabulary of non-explosive logics. While it’s an option for

advocates of explosive logics to insist that non-explosive logics just don’t talk about

contradictions, that ‘contradiction’ doesn’t enter these logics’ semantics, it seems theoretically

unfruitful for explosive logics to have a monopoly on the term. It promotes the appearance of

incommensurability between the logics when there is none. We understand the paraconsistent

logician perfectly when she says that contradictions don’t entail every proposition. Any account

of contradiction that demonstrates there is no incommensurability between explosive and non-

explosive logics has a theoretical advantage over the explosion account.

7. Conclusion

This paper has considered the plausibility of the explosion account of contradiction, in both the

form of ExS) and ExP). We have argued that ExS) is implausible for conceiving of contradictions

25

as sets and failing to respect important distinctions between different types of sets of

propositions, and that ExP) miscategorises certain logical falsehoods, and at least one putative

non-logical falsehood, as contradictions. Additionally, as ExP) cannot be meaningfully

embedded within either a pragmatic or semantic interpretation of the LNC, we argued that

advocates of ExP) are committed to demonstrating that the perceived philosophical significance

of the LNC is illusory, a particularly heavy burden for an account of contradiction with very

little independent support. Finally, we argued that ExP) precludes the term ‘contradiction’ from

entering into the vocabulary of paraconsistent logics, a move that unnecessarily distorts the

disagreements between paraconsistent logicians, dialetheists, and classical logicians.

Combined, these criticisms of the explosion account give us good reason to believe that

in both its forms, ExS) and ExP), the account is implausible. A particularly transparent

conclusion given that some competing definitions of ‘contradiction’, such as Kalish et al.’s,

apparently fail to suffer from the same faults. Consequently, we shall have to look elsewhere for

a plausible definition of ‘contradiction’, a welcome conclusion for the dialetheist for whom the

explosion account is no longer a viable threat.

26

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