Logical Pluralism
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Abstract: A widespread assumption in contemporary philosophy of logic is
that there is one true logic, that there is one and only one correct answer as to
whether a given argument is deductively valid. In this paper we propose an
alternative view, logical pluralism. According to logical pluralism there is not one
true logic; there are many. There is not always a single answer to the question
“is this argument valid?”
Logic, Logics and Consequence
Anyone acquainted with contemporary Logic knows that there are manyso-called logics.1 But are these logics rightly so-called? Are any of themenagerie of non-classical logics, such as relevant logics, intuitionisticlogic, paraconsistent logics or quantum logics, as deserving of the title‘logic’ as classical logic? On the other hand, is classical logic really as de-serving of the title ‘logic’ as relevant logic (or any of the other non-classicallogics)? If so, why so? If not, why not? Logic has a chief subject matter: Logical Consequence. The chief aim of logic is to account for consequence — to say, accurately and systematically,what consequence amounts to, which is normally done by specifying whicharguments (in a given language) are valid. All of this, at least today, iscommon ground.
Logic has not always been seen in this light. Years ago, Logic was dom- inated by the Frege–Russell picture which treats logical truth as the leadcharacter and consequence as secondary. The contemporary picture re-verses the cast: consequence is the lead character. For example, Etchemendywrites: Throughout much of this century, the predominant conceptionof logic was one inherited from Frege and Russell, a conceptionaccording to which the primary subject of logic, like the pri-mary subject of arithmetic or geometry, was a particular bodyof truths: logical truths in the former case, arithmetical or geo-metric in the latter . . . This conception of logic now strikes us asrather odd, indeed as something of an anomaly in the historyof logic. We no longer view logic as having a body of truths,the logical truths, as its principal concern; we do not, in thisrespect, think of it as parallel to other mathematical disciplines.
If anything, we think of the consequence relation itself as theprimary subject of logic, and view logical truth as simply the de-generate instance of this relation: logical truths are those thatfollow from any set of assumptions whatsoever, or alternatively,from no assumptions at all. [16, page 74]2 1Except where grammar dictates otherwise, ‘Logic’ names the discipline, and ‘logic’ names 2For a more detailed discussion of the centrality of consequence in logic, see Chapter 2 of Stephen Read’s Thinking About Logic [39].
But what is logical consequence? What is it for a conclusion, A, to logicallyfollow from premises Σ? There is a tradition to which almost everyonesubscribes. According to this tradition the nature of logical consequence iscaptured in the following principle: (V) A conclusion A follows from premises Σ if and only if any case in which
each premise in Σ is true is also a case in which A is true. Or equiv-alently, there is no case in which each premise in Σ is true, but inwhich A fails to be true.
Here is one example of the use of this principle to introduce validity. Thequotation is taken from Richard Jeffrey’s text Formal Logic: its scope and itslimits.
Formal logic is the science of deduction. It aims to provide sys-tematic means for telling whether or not given conclusions fol-low from given premises, i.e., whether arguments are valid orinvalid . . .
A valid argument is one whose conclusion is true inevery case in which all its premises are true.
Then the mark of validity is absence of counterexamples, casesin which all premises are true but the conclusion is false.
Difficulties in applying this definition arise from difficulties incanvassing the cases mentioned in it . . . [19, page 1] Notice that (V) does not give us a complete account of logical consequence.
To construct a logic we need an accurate and systematic account of whicharguments are valid. (V) by itself does not give us an account of the casesinvolved. Jeffrey’s last line is significant: “Difficulties in applying this def-inition arise from difficulties in canvassing the cases mentioned it.” Inthis paper we present a view that takes such “difficulties” very seriously.
The view is logical pluralism — ‘pluralism’, for short. Pluralism, we believe,makes the most sense of contemporary work in Logic.
Pluralism in Outline
To be a pluralist about logical consequence, you need only hold that thereis more than “one true logic”. There are hints of pluralism in the literaturein philosophy of logic, but it has not been given a systematic sympathetictreatment.3 In this paper we wish to introduce and defend a particular spe-cific version of logical pluralism. This pluralism comes with three tenets: 1. The pretheoretic (or intuitive) notion of consequence is given in (V).
2. A logic is given by a specification of the cases to appear in (V). Such a specification of cases can be seen as a way of spelling out truth condi-tions of the claims expressible in the language in question.
3The most extensive treatment to date is given by Resnik [40]. But even that systematic es- say, the focus is primarily on non-cognitivism about logical consequence, an issue orthogonalto the concerns of this paper.
3. There are at least two different specifications of cases which may ap- Point (1) is self-explanatory: Using (V) to determine logical consequence isby no means idiosyncratic. We will not attempt an extensive search of theliterature, though evidence for the centrality of an analysis like (V) is nothard to find.4 Logic is a matter of preservation of truth in all cases. This isthe heart of logical consequence.
However, this is not the end of the matter. To use (V) to construct a logic you need to spell out what these cases might be. To give a systematicaccount of logical validity, you need to give an account of the cases inquestion, and you need to tell a story about what it is for a claim to be truein a case. Without an answer to these questions, you have not specified alogic. This truism is given in point (2) of our account of logical pluralism.
To use (V) to develop a logic you must specify the cases over which (V)quantifies, and you must tell some kind of story about which kinds ofclaims are true in what sorts of cases. For example, you might give anaccount in which cases are possible worlds. (Furthermore, you might goon to tell a metaphysical story about what sorts of entities possible worldsare [23, 24, 48, 53].) On the other hand, you might spell out such casesas set-theoretic constructions such as models of some sort. However this isdone, it is not the sole task. In addition, you must give an account of truthin a case. Here is an example of how you might begin to spell this out.
Your account of cases and truth in cases might include this condition: A ∧ B is true in x iff A is true in x and B is true in x.5 where A and B are claims and x is a case. Such an assertion tells us thata conjunction is true in a case if and only if both conjuncts are true inthat case. This gives us an account of truth in cases which not only tellsyou how conjunction works, but it also gives you some data about validity.
Once we have this connection, we have the validity of the argument fromA ∧ B to A. For any case x, if A ∧ B is true in x then A is true in x, by thecondition given above. This is but one example of how you might beginto systematically spell out the conditions under which claims are true incases. To do this is to do logic.
None of this so far is particularly controversial.6 The controversy in our position comes from point (3). According to the third and final claimthere are different ways to specify the “cases” appearing in (V). There is nocanonical account of cases to which (V) appeals. There are different, equallygood ways of spelling out (V); there are different, equally good logics. Thisis the heart of logical pluralism.
4Here is one more case: W. H. Newton-Smith, in his popular introductory text, writes that some arguments “have true conclusions whenever they have true premises. We will say thatthey are valid. That means that they have the following property: In any case in which thepremise (premises) is (are) true, the conclusion must be true.” [33, page 2] 5Here, as elsewhere, ‘iff’ is shorthand for ‘if and only if.’6Well, one part is controversial. We have privileged the model-theoretic or semantic account of logical consequence over and above the proof-theoretic account. A version of pluralism canbe defended which does not privilege “truth in a case.” However, most of the current debateswith which we are interacting lie firmly within this model-theoretic tradition, and we arecomfortable with that tradition, so we develop pluralism in this way.
We will begin our elaboration of (3) by examining different ways (V) has been filled out. We start with a well-known way of filling out (V):Models for classical first-order logic.
Tarskian Models, and Classical Logic
There are many ways in which you might give an account of (V) whichrenders valid all of the theses of classical logic. One way is to treat thecases of (V) as possible worlds. Then your clauses for truth in a case, ortruth in a world, will look like this.
A ∧ B is true in w iff A is true in w and B is true in w.
A ∨ B is true in w iff A is true in w or B is true in w.
∼A is true in w iff A is not true in w.
It is a little harder to give an account of the truth of quantified claims inpossible worlds, but if we allow each object in each world to have a namein our language, then the clauses are trivial.
• ∀xA(x) is true in w iff for each object b in w, A(b) is true in w.
• ∃xA(x) is true in w iff for some object b in w, A(b) is true in w.
Now, with no further analysis of what a world w might be, or how manythere might be, a story of consequence can be told. We have already seenthat this account validates the inference from A ∧ B to A. It also validatesthe inference from A to A∨B, from A∧(B∨C) to (A∧B)∨C, from x(A∨B)to xA ∨ xB, and many more besides.
If the cases in our account encompass all possible worlds, then an ar- gument is valid if and only if in any world in which the premises are true,so is the conclusion, or equivalently, if it is impossible for each premise tobe true but for the conclusion to not be true. Call this the necessary truthpreservation account of validity. This is one way to elaborate (V), but it isnot the only one. In fact, it is not at all the traditional picture of logicalconsequence. The possible worlds account is not formal because it makesno essential use of the forms of the claims analysed. To be sure, our elucida-tion has picked out conjunctions, disjunctions, negations and quantifiers,but there was no need at all to do this. We could just as well have givenclauses for colour terms.
a is red is true in w iff a is red in w.
a is coloured is true in w if and only if a is coloured in w.
This explains why the necessary truth preservation account of validity ren-ders the argument from a is red to a is coloured valid. It is valid because inany case (that is, in any possible world) in which something is red, it isalso coloured. It is impossible that something be red and for it to fail to becoloured.
This is not the only way to account for logical consequence, and, as we have mentioned, it is not the mainstream tradition. According to logicalorthodoxy, the argument from a is red to a is coloured is invalid, because it is not formal. It does not exploit any logical form: it has the form Fa and this form is invalid. We can give an account of this form of validityby varying the cases over which (V) quantifies. Now validity is a matterof form, and cases interpret formal languages. In our case, the languagesof first-order logic, in which we have simple predicates, names, variables,quantifiers and connectives. Then sentences in such a formal language areinterpreted in a model. These are Tarskian models of first-order logic. ATarskian model, M, is a structure that comprises the following: 1. A nonempty set D, the domain; and 2. A function I, the interpretation, satisfying the following conditions: (a) I(E) is an element of D, if E is a name (in the given language); (b) I(E) is a set of ordered n-tuples of D-elements, if E is an n-place Then we use a model to interpret the language.7 If α is an assignment of D-elements to variables, then Iα(x) = α(x). If Ft1 · · · tn is true in M, α iff Iα(t1), . . . Iα(tn) I(F).
A ∧ B is true in M, α iff A is true in M, α and B is true in M, α.
A ∨ B is true in M, α iff A is true in M, α or B is true in M, α.
∼A is true in M, α iff A is not true in M, α.
• ∀xA is true in M, α iff A is true in M, α for each x-variant α of α.
• ∃xA is true in M, α iff A is true in M, α for some x-variant α of α.
We take models to be cases, and we have defined truth in a model, forsentences of a formal language, by the standard recursive clauses. This ac-count then tells us about validity for arguments in the formal language,by way of (V). An argument is valid if and only if in every model in whichthe premises are true, so is the conclusion. For arguments of our natu-ral language, validity is inherited by way of formalisation. We can definetruth-in-a-model for claims of English by the standard processes of regi-mentation of those claims, and therefore we can define validity for naturallanguage arguments. Call this account the Tarskian account of validity ofarguments in natural language.
We now have our first dimension of plurality. Consider the question: Is the argument from a is red to a is coloured valid? We have seen that theanswer is yes for validity as necessary truth preservation. The answer is nofor the Tarskian account of validity. This argument has the form Fa and there are many models in which the premise is true and the conclusion 7We use assignments of values to variables, in order to interpret sentences with free vari- false. So, we have at least two different accounts of validity. One mightnow wonder: Is there any basis upon which to choose between these twoaccounts? Is there any reason you might prefer one to the other? Theanswer here is a resounding yes. Tarskian validity is formal; necessary truthpreservation is not. Tarskian validity can (perhaps) be known a priori, butnecessary truth preservation (probably) cannot. If Kripke is correct [21],the argument from a is water to a is H2O is necessarily truth preserving,but this cannot be known a priori.
On the other hand, validity as necessary truth preservation does not rely on a choice of the family of logical constants. Colour connections,temporal, spatial and other modalities, part–whole relations, and manyother forms of necessary connections are equally encompassed by this ac-count. The Tarskian account, on the other hand, makes a choice of logicalconstants, the privileged parts of language which can contribute to logi-cal form, and hence, logical validity. Not all Logic is simply a matter ofform. (This is one part of Etchemendy’s criticism of the Tarskian accountof logical validity [17].) A pluralist on the question of formality will call both accounts logic.
Thoroughgoing pluralists will be happy to call the result of both Tarski’saccount, and the necessary truth preservation account, logic, for both areways of spelling out the pretheoretic account (V) of logical consequence.
The proper answer to the question ‘is the argument from a is red to a iscoloured really valid?’ is to say ‘yes, it is necessarily truth preserving, and no,it is not valid by first-order logical form.’ A pluralist account of disagreement about logical form goes as follows: It is not fruitful to debate which of these things is logic. Both are fleshout (V), so both are logic. Given an argument which is necessarily truthpreserving but not Tarski-style valid, it is surely more informative to say:yes, there is no possibility in which the premise is true and the conclusionfalse, but there is a Tarski-style model in which the premise is true andthe conclusion false, and this shows the necessary truth preservation isnot in virtue of the first-order logical form of the claims involved. That isinformative analysis. A debate about which of these is logic adds nothing.
However, this is not the only kind of problem people might have with the Tarskian analysis of logical consequence and first-order logic. Considerthe zero-premise arguments to conclusions such as x(x = x) or to x(Fx ∨ ∼Fx). If cases comprise Tarskian-style models these arguments are valid.
Famous debates have raged over this result. A long and rather formidabletradition claims that neither x(x = x) nor x(Fx ∨ ∼Fx) is Really Valid; logic, in this tradition, allows for the empty case, but Tarskian-style casesare never empty.8 8Another famous objection is voiced by Kreisel, Boolos, and McGee [9, 20, 26], to the effect that the models given in the traditional Tarskian account of validity are too limited. Why notallow for domains too “big” to be sets? Logic, alone, seems not to impose this restriction, buttraditional Tarskian cases do.
There is also a philosophically illuminating independent justification for pluralism on the matter of the domain of quantification. Phillip Bricker has developed an account of modalrealism which deals with the ‘isolated universes’ problem by allowing not only concrete pos-sible worlds as units of evaluation, but also classes of possible worlds. [10] A class of worldsdoes duty for a ‘world’ with spatio-temporally disconnected parts. Now, of course classes maybe empty. If we allow the empty class, and we define validity as truth preservation in all The foregoing concerns fit a pattern in this way: Let C be an account of consequence, or some precisification of ‘validity.’ Then C is said to under-generate, with respect to some argument, if that argument is Really Validbut not C-valid. The problem, in this case, is that C gets things wrong byfailing to call the argument ‘valid’ when “in fact” it is valid — Really Valid.
C is said to overgenerate, with respect to some argument, if the argument isnot Really Valid but is C-valid. In this case, C gets things wrong by callingthe argument ‘valid’ when “in fact” it is not.
The undergeneration–overgeneration pattern is ubiquitous in philoso- phy of logic; indeed, it may well be the central pattern of dispute in thefield.9 The important point here is that our pluralism can make sense ofthe debate, though in general it refrains from blessing only one side of thedebate with the title ‘logic’. In particular, a pluralist response to these is-sues goes as follows: Many appeals to “Real Validity” are appeals to realvalidity; they are not, however, appeals to the only real validity. Real valid-ity comes from a specification of cases which appear in (V). According topluralism, there are at least two such specifications of cases. So far, we haveseen two different approaches within classical logic — the worlds approach,and the Tarskian models approach. But these are just the beginning.
Situations, and Relevant Consequence
Each of the accounts of interpretations or truth conditions seen so far havebeen classical with respect to negation. For any cases x seen so far, be theyworlds, Tarskian models, class-size models, or even models with emptydomains, ∼A is true in x iff A is not true in x.
Call this the classical negation clause. There are many good reasons for usinga classical negation clause in constructing an account of truth in cases. Themost obvious reason is the way that we use negation, and the conditionsunder which negations are, in fact, true: ∼A is true just when A is not true.
This is simply what ‘not’ means.10 But to infer from this truism that theclassical negation clause is the only one worth using in elaborating (V)’scases would be far too swift. To do so would be to assume that the onlyacceptable use of cases is to model consistent, complete worlds. But manyhave questioned this assumption. There are other ways to give an accountof cases, or conditions under which claims might be true or false. Onesuch account is the situation theory of Barwise and Perry [1, 2, 3].
The world is made up of situations. They are simply parts of the world.
Claims are true of not only the world as a whole, but some claims at leastare true of situations. We will not spend time on the theory of situationsand their individuation here: we will simply illustrate it. In the situationinvolving Greg’s household as he writes this, it is true that Christine isreading a paper. It is also true that the stereo is playing. It is false that thetelevision is on. It follows from this, and the fact that the television is in classes of worlds, we have a free logic. If we do not, our logic has existential import. Whichshould you choose? The metaphysical view need not constrain you, according to pluralism.
9The terms ‘undergeneration’ and ‘overgeneration’ are found elsewhere [17, 38, 39, 47].
10You might well say, instead, that this is what true means.
fact an inhabitant of the situation, that it is true, in this situation, that thetelevision is off.
Situations “make” claims true and they “make” others false. However, some situations, by virtue of being restricted parts of the world, may leavesome claims undetermined.11 It is not true in this situation that JC isreading. It is also not false in this situation that JC is reading — that is, itis not true in this situation that JC is not reading. JC does not feature inthis situation at all.
It follows that the classical account of negation fails for situations. This treatment of negation is out of place in this context. It seems plausible,however to hold fast to the classical analyses of conjunction and disjunc-tion.
A ∧ B is true in s iff A is true in s and B is true in s.
A ∨ B is true in s iff A is true in s or B is true in s.12 We must emphasise at this point that the non-traditional treatment of negation does not mean that we are modelling a non-classical negation.
Quite to the contrary. Our treatment of negation is not the traditional onesimply because we are entering a new field — the logic of situations. It hasnot been traditional to formally model claims of the form ‘A is true in sit-uation x’. Once you do so, and once you acknowledge that situations arerestricted parts of the world, it becomes clear that you ought reject the clas-sical treatment of negation when applied to situations. This is completelyconsistent with the classical treatment of the truth or falsity of negationsimpliciter. We may maintain that ∼A is true if and only if A is not true.
That is not in question. The situation theoretic analysis of this equivalencewill proceed further: ∼A is true if and only if ∼A is true in some (actual)situation or other. A is not true if and only if A is not true in any (actual)situation whatsoever. The traditional, classical equivalence is maintainedif we agree, then, that if ∼A is true in some (actual) situation, then A is nottrue in any (actual) situation. And this is simple to maintain, given three,plausible, theses.
There is a situation w, of which every actual situation is a part.
If A is true in s and s is a part of s then A is true in s .
If s is an actual situation, and if ∼A is true in s then A is not true in s.
These theses connecting negation and situations ensure the truth of theclassical account of negation. Negation, here, is classical.
The work, however, is not yet done; what is needed is a systematic treat- ment of the truth or falsity of negations in situations. This can be done 11We use shudder quotes around ‘make’ here not that we wish to avoid the use of “truth- making” terminology. To the contrary, we value the recent revival of this terminology andthe analysis of the connections between claims and parts of the world which make themtrue [18, 31, 41]. However, this terminology is not used by situation theorists, and that itwould be a mistake to impute it to them.
12The conjunction clause is never disputed. The disjunction clause sometimes is. However, it seems sound for the intended interpretation. If in this situation the milk is on the table orin the fridge, then either in this situation the milk is on the table or in this situation it is inthe fridge.
in any of a number of ways. You can, for example, take satisfaction anddissatisfaction of relations in situations as primitive, and then inductivelybuild up truth and falsity conditions of complex claims.13 This approachis traditional in situation theory, and it is also used in some varieties ofsemantics for non-classical logics [2, 5, 13, 32].
Here, however, we will favour a different approach — a compatibility semantics, which stems from Dunn’s [14, 15, 43] analysis of of negation.
On this proposal negation is taken to act in situations rather like neces-sity or possibility does in possible worlds. We admit into our semanticsnon-actual situations (or models of non-actual situations) which are con-nected by a binary relation of compatibility, which we write ‘C’. Given thisapparatus negation is definable.
∼A is true in s iff A is not true in s for any s where sCs .
Accordingly, the negation ∼A is true in s just when any situations in whichA is true are incompatible with s. This clause follows fairly immediatelyfrom the meanings of negation and compatibility. If ∼A is true in s and Ais true in s , then s is not compatible with s . Conversely, if A is not truein any s compatible with s, then it appears that s has ruled A out. Thatis, ∼A is true in s. This reading does not rely on a “funny” negation; it iscompletely compatible with a classical view of negation.14 Given such asemantics of situations, a natural reading of (V) emerges: a situated reading.
The argument from Σ to A is relevantly valid if in any model, inany situation in which all premises in Σ are true, so is A.
To speak loosely but suggestively: To make the premises true you makethe conclusion true too. The relevance of this reading of consequence isimmediate. The inference from A to B ∨∼B fails, since a situation in whichA is true need not be one in which B ∨ ∼B is true.
If we take the relevant tautologies to be those claims true in every situ- ation, then B ∨ ∼B is not among them. This does not mean that we haveadopted a strange non-classical account of negation. We agree with theclassical theorists that B ∨ ∼B is true, that it is true in every world. Ournegation is classical. The invalidity of the argument from A to B ∨ ∼B is arelevant invalidity. The argument, of course, is still classically valid, in thesense that in any world in which A is true, B ∨ ∼B is true.
The move to situations as incomplete parts of the world is natural. It has a more daring generalisation, to consider not only incomplete situa-tions, but also inconsistent situations, or ways things could not be [28, 42,43, 46, 54]. These are situations which fail to be self-compatible. If, forexample, s is not compatible with itself, then it is possible that both Aand ∼A be true at s. This, again, is not terribly non-classical. According to 13For example, you will say that not only is a conjunction true in when both conjuncts are true, but dually, a conjunction is false when one conjunct is false.
14The three minimal conditions cited earlier for a classical treatment of negation have their “compatibility” readings. (1) Any actual maximal, self-compatible situations); (3) if too (compatibility of wholes leads to compatibility of parts).
our account of worlds as consistent, complete situations, such impossibiliacannot be a part of any world. Worlds are consistent, and hence, have noinconsistent parts. This does not mean, of course, that there are no waysthat things could not be; it means, simply, that the worlds are not (and couldnot be) among them.
Given the admission of inconsistent situations, an argument from A ∧ ∼A to B fails the relevant test; for a situation in which A ∧ ∼A is true neednot be one in which B is true. A situation might well be inconsistent aboutA without involving everything. This same situation gives us a counterex-ample to the relevant validity of disjunctive syllogism: the argument fromA ∨ B and ∼A to B. A situation inconsistent about A but not judging B astrue suffices. A ∨ B is true in this situation, as is ∼A, but B fails.
This last case has been the cause of much debate in the literature on relevant logics and relevant inference. Much ink has been spilled on thefailure of disjunctive syllogism and whether it is a virtue or a vice [27,37, 45]. We do not plan to add to the spilling of ink in any depth here.
We will simply note that traditional criticisms of the relevant rejectionof disjunctive syllogism are beside the point, when seen in the light ofpluralism. We will end this section on relevant consequence by explainingwhy this is so.
One cause of concern with the rejection of disjunctive syllogism is that disjunctive syllogism is obviously valid, and we reason with it all the time —we could not do without it in everyday reasoning [6]. Our pluralism willagree: Of course there is a sense in which disjunctive syllogism is valid —and even obviously so. In any possible world in which the premises aretrue, so is the conclusion. In that sense — the sense afforded by cases asworldlike — disjunctive syllogism is valid. The virtue of a pluralist accountis that we can enjoy the fruits of relevant consequence as a guide to infer-ence without feeling guilty whenever we make an inference which is notrelevantly valid. With classical consequence you know you will not makea step from truth to falsehood. With relevant consequence, the stricturesare tighter; you know you will not make a step from one that is true in asituation to something not true in it (but which might be true outside it).
This is a tighter canon to guide reasoning.15 So, the case of incomplete and inconsistent situations motivates a gen- uinely different elucidation of logical consequence — one which differswith the classical account on the validity of inferences down to the propo-sitional level. This account of consequence is still recognisably logic; it isanother way to flesh out our condition (V). It is not a rival in any sense tothe classical, traditional explications of that condition. Instead, it coexistsalongside classical validity as another important variety of logical conse-quence.16 15For more elaboration and defence of this point, see “Defending Logical Pluralism” [4].
16We have restricted our attention here to the conjunction, disjunction and negation frag- ment of relevant logics. More can be done to bring the notion of relevant entailment intothe language. For another approach to relevant logics which motivates two varieties of con-sequence, but from a very different perspective, we refer the reader to Mark Lance’s “TwoConcepts of Entailment” [22].
So now we have more pluralism, a pluralism in which (V)’s cases may be worlds, incomplete situations, and even incomplete and inconsistentsituations.
Constructions, and Intuitionistic Logic
Mathematicians do not, generally speaking, concern themselves with a sit-uated account of logical consequence while reasoning about mathematicalobjects or structures. However, they too can make some distinctions whichare blurred by classical accounts of validity. We have in mind the mathe-matics pursued by mathematical constructivists.
The constructivism of the mathematicians Errett Bishop [7, 8], Dou- glas Bridges [11], Fred Richman [30, 44] and others can best be describedas mathematics pursued in the context of intuitionistic logic.17 In construc-tive mathematics the goal is to gain understanding of mathematical struc-tures, and to prove theorems about them (just as in classical mathematics).
However, the goal is to prove mathematical theorems with constructive,or computational content. If a statement asserting the existence of somemathematical object is proved in a constructive manner (using the rulesof intuitionistic logic) then this proof will contain the means of specifyingthe object or structure in question. Wittgenstein illustrates the advantagesof constructive proof over its classical cousin by drawing out its implica-tions for our understanding.
A proof convinces you that there is a root of an equation (with-out giving you any idea where) — how do you know that youunderstand the proposition that there is a root? [52, page 146] This feature of constructive mathematics is guaranteed by the structureof constructive proofs. We emphasise the fact that this is a new notion ofproof by using the word ‘construction’ for this notion. Constructions obeythe following laws: A construction of A ∧ B is a construction of A together with a con- A construction of A ∨ B is a construction of A or a construction of B.
A construction of A B is a technique for converting constructions There is no construction of .18 A construction of xA is a rule giving, for any object n, a construction A construction of xA is an object n together with a construction of 17Tait provides more explicitly philosophical account which draws very similar distinctions to the work of constructive mathematicians [49, 50].
into absurdity. It shows that there are no constructions of This elucidation is not formal. It is informal because it leaves the centralnotion of a construction undefined.19 For all its informality, however, itgives us an understanding of the behaviour of constructive proof. For ex-ample, the inference from x(A ∨ B) to xA ∨ xB is classically valid, butnot constructively valid. For example, it is easy to demonstrate that everystring of ten digits in the decimal expansion of π is either a string of tenzeros, or it is not. This does not give us a construction of the claim thateither there is a string of ten zeros in π or every string of ten digits in π isnot a string of zeros. Any construction of this claim must either prove thatthere is no string of ten zeros in π or to show where one such string is. Theconstructive content of xA ∨ xB is greater than that of x(A ∨ B).
Theorems of constructive mathematics are simply theorems of mathe- matics proved constructively.20 According to this approach, the theoremsof constructive mathematics are also theorems of classical mathematics.
The difference between constructive and classical mathematics is not oneof subject matter, but one of the required standards of proof. Classicalmathematicians may appeal to the law of the excluded middle, and proofby contradiction. Constructive mathematicians do not, for these movesdestroy constructivity.
A truth conditional semantics may be given for the intuitionistic logic of constructive mathematics, which both does justice to the practice ofconstructive mathematics and opens the way for a pluralist reading of thatpractice. The truth conditional semantics is simply Kripke’s semantics forintuitionistic logic. Truth is relativised to points (which model construc-tions) which are partially ordered by strength (written ‘ ’).
A ∧ B is true in c iff A and B are true in c.
A ∨ B is true in c iff A is true in c and B is true in c.
A B is true in c iff for any d ∼A is true in c iff A is not true in d for any d The points in a Kripke structure for intuitionistic logic do a good jobof modelling constructions ordered by a notion of relative strength. Theclauses for conjunction and disjunction are straightforward transcriptionsof our pre-formalised notion of constructions. The rules for implicationand negation differ somewhat, but can be motivated to follow from thepre-theoretic notion. A construction proves A B if and only if whencombined with any construction for A you have a construction for B. Theassumption guiding Kripke models is that a construction for A B com-bined with one for A will be a stronger construction. A B is true at c ifand only if any stronger construction d for A is also a construction for B.
19See the similarity to the account of worlds at the start of Section 3. We gave an account of what it is for a conjunction to be true in a world. We gave no account of what it is for anarbitrary claim to be true in a world. Similarly here, we give no account of what it is for anarbitrary statement to be given by some construction.
20This position is inconsistent with any position which takes there to be results which conflict with classical mathematics. A canonical example is the result that all functions onthe real line are continuous [12, Section 3.3]. Our approach to constructive reasoning mustreject all such counter-classical results. We are not alone in this — the constructivism ofBishop, Bridges and others agree on this point.
Constructions are incomplete and hence should not be expected to con- struct, for every claim A, either it or its negation ∼A. Constructions havecomputational content, so a construction of A ∨ B should be a construc-tion of A or a construction of B. This jointly ensures that A ∨ ∼A oughtfail. This cannot necessarily be constructed.
What is important, here, is that for a pluralist it does not follow that A ∨ ∼A is not true, or even, not necessarily true. It is consistent to maintainthat all of the truths of classical logic hold, and that all of the argumentsof classical logic are valid with the use of constructive mathematical rea-soning, and the rejection of certain classical inferences. The crucial factwhich makes this position consistent is the shift in context. Classical in-ferences are valid, classically. They are not constructively valid. If we usea classical inference step, say the inference from x(A ∨ B) to xA ∨ xB,then we have not (we think) moved from truth to falsity, and we cannotmove from truth to falsity. It is impossible for x(A ∨ B) to be true and for xA∨xB to be false. However, such an inference can take one from a truthwhich can be constructed to one which cannot, as we have seen. So, theinference, despite being classically valid, can be rejected on the grounds ofnon-constructivity.
This pluralist account of constructive inference is not a view that will be shared by constructivists who wholeheartedly reject the use of classicalinference. However, it is a view which does justice of what constructivereasoning. When a constructivist says ‘not’, she means not. She does notmean something else, foreign to the classical mathematician.21 She differsfrom the classical reasoner only in her use of tighter canons of inference. Itis hard to see how any other view can do justice to the practice of construc-tive mathematics. It seems that classical dogmatists must either reinterpretconstructivist claims as being about something else (when she says ∼A shemeans not that A is not true, but instead that A can be proved not to betrue) or that intuitionistic logic merely a formalist game in which the rulesare syntactically restricted to allow a more limited repertoire of proof.
We have given an account of logical pluralism, and we have shown how itcontributes to our understanding of different traditions in contemporaryLogic. In this section we address a few criticisms.
ANYTHING GOES?Objection: “You say that there are many, many different consequence re-lations, and that none of these, in any objective, universal sense, is betterthan the others. Does it not follow that anything goes? On your view, thereis no disagreement about logical consequence. But that makes a mockeryof the current state of play in Logic. Stephen Read writes: Rival logical theories, such as intuitionistic logic, paraconsistentlogics, relevant logics, connexive logics, and so on, are based on 21We use the example of the mathematician merely because constructive reasoning is most developed in this tradition. It need not be restricted to mathematics. Mathematical techniqueis applied when talking about the environment. We can reason constructively not only aboutthe real line, but also about spatial and temporal distances, physical quantities, and manymore things besides.
different philosophical analyses of this basic notion. [39, page36] “According to your view, these logics are not rivals, they live in one largehappy family. Similarly, Graham Priest writes: Whether or not any of the nonstandard logics discussed here[intuitionist, many-valued and quantum, relevant and paracon-sistent, conditional and free] are correct, their presence servesto remind us that logic is not a set of received truths but a dis-cipline where competing theories concerning validity vie witheach other. [36] “On your account, these theories do not compete. You have misunderstoodcontemporary Logic.” Reply: Pluralism is not a recipe for wholesale agreement. There can be dis-agreements about logical consequence. Our pluralism holds that some for-mal logics can fruitfully be seen as different elucidations of (V), the prethe-oretic notion of logical consequence, and that (V) does not determine onelogic, but rather, a number of them. It does not follow that there are nodisagreements about notions of logical consequence. It does follow, how-ever, that in any such disagreement the ground has to be fixed to ensurethat the disputants are not talking past each other.
Here are two positions with which we disagree, while maintaining our pluralist credentials. We disagree with the dialetheism of Graham Priest,according to which contradictions may be true. According to Priest, thereare arguments of the form A, ∼A ∨ B which have true premises and an untrue conclusion [34, 35]. We dis-agree.22 So, this means that when we cash out validity as necessary truthpreservation, or anything similar, we hold that all of the inferences of clas-sical first order logic are valid. Priest does not.
In a similar vein, we also disagree with intuitionists who hold that there are arguments from A to B ∨ ∼B with a true premise and untrue conclu-sion [12]. We disagree; we take every instance of B ∨ ∼B to be true.23 In these cases disagreement is possible. It is possible once we have set the terms of the debate. In both cases, with paraconsistent and intuitionistlogic, we find a place for these non-classical logics — for both are elucida-tions of the pretheoretic notion (V) of logical consequence.
There might be other logics for which we can find no place in our cat- alogue of True Logics, as much as we admire their technical subtlety.24 22We respect the dialethic tradition in logic, according to which there are true contradic- tions. However, it is not the view we take in this paper.
23There is nothing essential to pluralism to take the classical side of the debate in these disagreements. We could just as well be pluralist dialetheists or intuitionists. For example, wemight hold that some claims of the form were true, and still accept both a “classical” paraconsistent logic and a constructivist one, to model constructive reasoning.
24Chief among those left out of our catalogue involve any systems for which transitivity or identity of consequence fails. For example, the Martin and Meyer system S-for-Syllogism,which rejects on grounds of circularity, is ruled out given the lack of reflexivity [25, 29].
Moreover, Tennant’s “relevant logic” which rejects transitivity likewise fail to fall under thebanner of logical consequence given in (V) [51].
There are too many modal logics to hold each of them as the logic ofbroad metaphysical necessity. So, given a particular interpretation of eachof the symbols in our formalism (including consequence) we can admit thatthere is a great deal of scope for rivalry. For the propositional modal logicof necessary truth preservation, we think that a logic somewhere betweenS4 and S5 is a candidate for getting things right. Anything else gets it wrongwhen it comes to metaphysical necessity. There is scope for rivalry anddisagreement, when the meaning of the basic lexicon is settled. The moralof our pluralism goes as follows: Once you are specific about what yourlogic is meant to do, there is scope for genuine disagreement.
This raises a general question: What is it to disagree with an account of consequence? What kinds of disagreement are possible? There are at leastfour different ways in which disagreement and difference between formallogics can be understood. Here is a rough spectrum of what one mightthink about a logical system L.25 Abstract Geometries: L is a logic because it is formally similar to other logics.
It models a consequence relation. It is to logical systems what a finiteprojective geometry is to Euclidean geometries. Euclidean geometries andtheir close neighbours are used to model physical space. A finite projectiveplane is not going to be used to model physical space, but it may be used tomodel something analogous to physical space. Similarly, system L mightbe used to study something analogous to consequence relations. And so,it is called a logic for reasons of structural similarity.
Applied Geometries: Take two geometries, a three-dimensional Euclideanspace, and a particular non-Euclidean three-dimensional space. These twospaces might be competing models for the physical space in our region.
Here the geometries are applied, for there is a notion of what it is to whichthe theoretical entities must correspond. Once rules of application of themodel are settled, there is scope for a genuine disagreement between the twotheories. Similarly, once applied, there is a scope for genuine disagreementbetween logical systems. However, this disagreement comes about simplyby applying the logic to model the validity of real argument. Differentformal systems can be equally appropriately used to model the validity ofarguments. The analogy with applied geometry becomes appropriate onlyonce the pretheoretic account (V) is fleshed out. Once you have a specificaccount of what kind of cases are in use (be they, worlds, constructions,situations) then there is scope for disagreement.
Different Subject Matter: We do not know how to label this position. Xthinks what Y is doing is attempting to get at the same kind of thing aswhat X is trying to get at, but Y is going about it in completely the wrongway, and is actually either doing gibberish or talking about something else.
The intuitionist view of the classicalist, or vice versa, can be seen like this,but need not be. A debate between the two which hinges upon whetherthe proper analysis of meanings ought proceed by way of truth conditionsor in terms of provability or evidence conditions can be seen in this way.
Pluralism: Finally, you can hold that two different logics L and L are bothaccurate and systematic accounts of (different specialisations of) the one 25Thanks to Daniel Nolan for discussion on this point.
notion of logical consequence. We hold that this position is the appropri-ate one in each of the cases we have discussed.
All points on this spectrum are inhabited in debates between rival log- ics. Furthermore, we think that useful things can be said about the dif-ferent ways in which plurality can arise. Pluralism comes in different axes.
One is the difference between models and what is modelled. Logic can dealwith both models (say, Tarski’s) and what is modelled (say, possible worlds,or situations). You might have a preferred site on this axis, yet still allowa degree of plurality. For example, you might allow variance over the sizeof the domain of quantification (empty domains, proper classes), or youmight allow plurality over the kinds of situations (or models) considered.
These three kinds of pluralism are independent of each other. We haveadvanced each variety here, but one is enough to justify logical pluralism.
Objection: “Another potential problem with pluralism comes from theother direction. You have shown that there is a number of different waysthat ‘case’ can be interpreted in (V). But (V) has a universal quantifier inthe front. It says that an argument is valid if and only if in all cases inwhich the premises are true, so is the conclusion. Is not real validity thenpreservation of truth across all cases? Will this not mean that the true logicis the intersection of all logical systems given by (V)? You have one truelogic after all.” Reply: Firstly, classical first order logic is logic after all. If the premises of aclassically valid argument are true, so is the conclusion. Those argumentsare valid. They are not all constructively valid, or relevantly valid, but thisdoes not stop them being valid, in an important and useful sense. Theclass of all Tarskian models is an important and natural class of cases, andit is appropriate to restrict our quantifiers in (V) to those cases.
Secondly, we see no place to stop the process of generalisation and broad- ening of accounts of cases. For all we know the only inference left in theintersection of all logics might be the identity inference A it would be say that identity is the only valid argument. It seems a muchmore appropriate use of the term to call each of these systems logic.
Thirdly, each formal system is used to regulate inference, each falls un- der our original pretheoretical banner for logical inference. So, each ofthem are logics.
Question: “What is the status of your investigation? Are you engaging ina conceptual analysis of the concept of logical consequence?” Reply: The nature of conceptual analysis is contested, so our remarks onmust be tentative. As many have noted, Tarski aimed to give an analysis,of the “intuitive” notion of consequence. Etchemendy repeats the story: Tarski begins his article by emphasizing the importance of theintuitive notion of consequence to the discipline of logic. Hedryly notes that the introduction of this concept into the field“was not a matter of arbitrary decision on the part of this or that investigator.” The point is that when we give a preciseaccount of this notion, we are not arbitrarily defining a newconcept whose properties we then set out to study — as we arewhen we introduce, say, the concept of a group, or that of a realclosed field. It is for this reason that Tarski takes as his goal anaccount of consequence that remains faithful to the ordinary,intuitive concept from which we borrow the name. It is for thisreason that the task becomes, in large part, one of conceptualanalysis. [17, page 2] Insofar as Tarski was doing conceptual analysis in his “On The Concept ofLogical Consequence”, we are too.26 We are not introducing a new con-cept and recommending that people study it. (V) captures the pretheoreticnotion to which Tarski held his own account accountable. (V) is the mostimportant guide to logical theory, and it does not constrain the field downto one candidate. Instead, it leaves the field open for a great deal of “play”.
Logic is a matter of truth preservation in all cases. Different logics are givenby different explications of these cases. This account of the nature of log-ical consequence sheds light on debates about different logics. Once thisrealisation is made apparent disagreements between some formal logics areshown to be just that: merely apparent. A number of different formal log-ics, in particular, classical logics, relevant logics and intuitionistic logics,have their place in formalising and regulating inference. Each is an eluci-dation of our pretheoretic, intuitive notion of logical consequence. Suchis our pluralism, which we have here tried to clarify. Two tasks remain:Showing that pluralism is superior to monism, and defending pluralismagainst objections. These are tasks we take up elsewhere [4].27 References
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27Thanks to Jon Barwise, John Bishop, Phillip Bricker, Stewart Candlish, James Chase, Colin Cheyne, Peter Clark, Mark Colyvan, Mike Dunn, John Etchemendy, Jay Garfield, Fred Kroon,Gary Hardegree, Daniel Nolan, Graham Priest, Stephen Read, Mike Resnik, Tim Williamson,Crispin Wright, Ed Zalta, and to audiences at the 1998 AAP(NZ) Conference, and at the ANU,Indiana, Massachusetts, Stanford, Edinburgh, Glasgow, St. Andrews and Leeds, for commentson earlier versions of this paper. Thanks also for comments from two anonymous refereeswhich helped greatly with the presentation of the paper.
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