Copeland - Pure Semantics and Applied Semantics

Discussion:

Pure Semantics and Applied Semantics

A Response to Routley, Routley, Meyer, and Martin B. J. Copeland

Interpretation rules.., are sometimes left in a perilously abstract state, through failure to explicate 'valuation', 'domain' and 'world'. There is nothing objectionable about such abstract semantics as a subject of mathematical study .. . . But abstract semantics is not yet semantics at all, since metalogical formulae like 'v(I1) = 1' can only serve to interpret the connectives if the metalogical word 'valuation' and the metalogieal symbol '1' have themselves been first explicated. (Kirwan [15], p. 107.)

With the current proliferation of formal semantics it is in

my opinion essential that the distinction to which Kirwan

draws attention in this quotation be firmly borne in mind.

An abstract and uninterpreted mathematical semantics, al-

though of the utmost utility for many technical purposes,

cannot assign meanings to the logical constants, and thus is

something less than semantics in the fullest sense. As

Kirwan points out, it is only once the symbolism of the

formal semantics has itself been explained that the semantics

can be said to assign meanings to the signs of the base

language. Kirwan uses the term 'applied semantics' for

semantics in this latter state, and the term 'pure semantics'

for semantics in the purely mathematical state, where the

symbolism of the semantics has received no explanation

over and above that of its mathematical role. (op. cit., p. 100)

It seems that the terms 'pure semantics' and 'applied

semantics' were first coined by Plantinga (although, of

course, the distinction which these terms mark out goes

back a long way). In [20] Plantinga writes as follows:

Now for a crucial distinction. What is offered in the Kripke system, strictly speaking, is a formal or pure semantics. A model structure, for example, is a purely set theoretical construction that as such has no obvious connection with modal notions at all; it is just any ordered triple (G, K, R) where K is a set of which G is a member and on which R is a reflexive relation . . . . It is not to the pure semantics as such that we must look for the promised insight into our modal notions . . . . The pure semantics does not give us a meaning for ~ . . . . Instead, it simply defines 'is a valid formula' for each of the systems treated. (pp. 126-7)

To convert the pure semantics to applied semantics, Plan-

tinga goes on to say, the semanticist needs to explain that K

is the set of all possible worlds, that G is the real world,

that ~(A,H)= T is to be understood as asserting that

formula A is true in possible world H, and so on.

Dummett has marked out the pure/applied distinc-

t ion by using the terms merely algebraic notion of logical consequence and semantic notion of logical consequence properly so called. ([9], p. 204) He writes of the distinc-

tion:

Semantic notions are framed in terms of concepts which are taken to have a direct relation to the use which is made of the sentences of a language ... . It is for this reason that the semantic definition of the valuation of a formula.., is thought of as giving the meanings of the logical constants. Corresponding algebraic notions define a valuation as a purely mathematical object... which has no intrinsic connection with the uses of sentences . . . . We have examples of purely algebraic completeness. For instance, the topological interpretations of intuitionistic logic were developed before any connection was made between them and the intended meanings of the intuitionistic logical constants. Thus, intuitionistic sentential and predicate logic is complete with respect to the usual topology on the real line... [but] no one would think of this as in any sense giving the meanings of the intuitionistic logical constants .. . . Here it would be wholly in order to say that the interest of such a completeness proof, which I am calling algebraic as opposed to semantic, was purely technical. We have a mathematical characterisation of the set of valid formulas of intuitionistic sentential or predicate logic, which may serve to establish certain general results about that set. (op. cit., pp. 204-205)

The Rout ley-Meyer semantics for relevance logics are, in

Dummett 's terms, merely algebraic, and are not semantics

properly so called. As far as I am aware, Scott [27] is the

first person to have remarked in print that the R o u t l e y -

Meyer semantics have as yet no adequate philosophical

explication; and the point has subsequently been reiterated

and expanded upon by a number of authors, for example

van Benthem [29], Copeland [5], Hintikka [13], Kiel-

kopf [ 14], and Lewis [ 16]. The Rout ley-Meyer semantics

are no more than mathematical characterisations of the

valid formulae of various relevant logics. The semantics are

of purely technical interest, supplying no account of the

Topoi 2 (1983), 197-204. 0167-7411/83/0022-0197501.20. �9 1983 by D. Reidel Publishin,~ Company.

198 B.J . COPELAND

meanings of the logical constants. In short, the Rout ley- Meyer semantics are to date pure semantics.

In [5] I presented a detailed case to the effect that the

Routley-Meyer semantics are pure semantics, not applied semantics. (In that paper I used the term 'merely formal model theory' to designate pure semantics, and the term

'illuminating semantics' to designate applied semantics. In the interests of uniformity I now use the Plantinga-Kir- wan terminology to mark out the distinction in question.) To SUlumarise, I there argue that the semantics fall to be applied semantics owing to lack of explication of three crucial items. Firstly, little explanation is given of the core function I appearing in the semantical clauses for the logical constants, save the bare set theoretic explanation of it as a function from (formulae} XK to {T,F} . Secondly, little explanation is given of the ternary relation on members of the set K. Thirdly, the * operation on members of K is assigned properties in a purely ad hoc way, the sole constraint being that the resulting semantics validates the negation axioms of whatever system is under consideration. Further, such explanatory remarks as are provided concerning * go no way at all towards converting the pure semantics to applied semantics. (For example, Routley 'explains' that a* is the reverse of world a, where "a reverse world is like the reverse side of something, e.g. a gramo- phone record"! ([22], p. 291)) 1 . Until such time as full and unequivocal explanations are given of how the formalism is to be understood at these three crucial points, the Routley-Meyer semantics remains pure semantics.

Not, of course, that there is anything wrong with pure semantics. It is hardly necessary to detail the many technical

uses and merits of pure semantics: everyone knows how freely the metatheorems begin to flow once a formal modelling of a system has been constructed. But the important point is that there are certain sorts of philosophical assertion which can be established by reference to an applied semantics, a semantics properly so-called, but not by reference to a pure semantics. Yet Routley and Meyer do misguidedly claim to have established several such asser- tions by reference to their pure semantics. For example one cannot, as Routley and Meyer have tried to do, use pure semantics to "validate... some of the most controversial dogmas.., of Marxism-Leninism" ([24], p. 13), nor, indeed, to show that "Aristotle made a mistake when he concluded there are no contradictions in things" (ibid.). These resounding philosophical claims are not the sort of thing which can be established by constructing a mathematical characterisation of the set of valid formulae of some system or other. Another such claim is to be found in

Routley and Routley [26], where the authors attempt to use their pure semantics in philosophical polemic against Disjunctive Syllogism (henceforth DS), seeking to show classical logicians that it is not correct to infer B from the formula (A v B)& ~A. Yet a pure semantics cannot be used to substantiate such a thesis. As I point out in [5]

(p. 403) a classical logician is going to be convinced by a semantical argument to the effect that his inference of B from (A v B) & ~A is erroneous only if the semantics in

question assigns the classical meanings to the connectives &, V and ~. (For if the classical logician and his critic differ in their assignments of meaning to any of these connectives

they will simply be talking past each other in their dispute over the correctness of DS: the principle of inference that the critic has putatively shown to be unsound will not be the principle which the classical logician is upholding as correct.) The Routley-Meyer semantics does not satisfy the condition just italicised, for pure semantics assign no meanings whatever, classical or otherwise, to the logical constants.

It is now time to meet head on the attack on my paper recently presented by Routley, Routley, Meyer, and Martin. 2 RRM & M have grossly misinterpreted my paper,

attributing to me claims which I have never made whilst passing over in silence the real challenges offered by the paper. My original case stands unanswered. Nor have RRM & M presented me with a case to answer. The prob- lem before me is not to reply to a collection of philosophical arguments, but rather is to dispel the illusion that I said what RRM & M would have it that I said.

Let me commence by stating in outline the overall argument of [5]. The major part of the paper is taken up with establishing the claim that the Routley-Meyer semantics is a pure semantics, not an applied semantics (in the older terminology of [5] this is written: the Rout- ley-Meyer semantics is merely formal model theory, not an illuminating semantics). The arguments used to this effect were summarised earlier in the present paper. This claim established, I am able to state that "it is totally

unclear what account of the meanings of the logical constants is given in the Routley-Meyer 'semantics' " ([5] , p. 406). From this I move to my overall conclusion that "the 'semantics' yields no semantical argument for the fallacious- ness of the entailment DS" (p. 406). This move is licensed by the fact that "such an argument must show how the invalidity of DS arises from a classical account of the meanings of the propositional connectives together with a relevance account of the meaning of entailment" (p. 406). This is a fact already established on page 403 by the argu-

PURE SEMANTICS AND APPLIED SEMANTICS 199

ment rehearsed above, namely that a demonstration which fails to assign the classical meanings to the propositional connectives is hardly going to convince a classical logician that he makes a mistake when he infers B from (A V B) & ~A.

RRM & M commence their attack by citing a claim of mine which they label TC, and which is in fact the claim just discussed to the effect that their semantics needs to preserve the classical meanings of the propositional connectives. However, RRM & M fail to comprehend my reasons - which I have just repeated - for making this claim. On the contrary, they tell the reader that TC is supposed to be the

touchstone for distinguishing illuminating semantics from merely formal model theory: the feature which distinguishes applied semantics, semantics properly so-called, from pure semantics is supposed to be that the former assigns the classical meanings to the propositional connectives (p. 72). I need hardly say that this claim is nonsense. The difference between a pure semantics and an applied semantics resides in the fact that the latter provides an account of the meanings of the logical constants, whereas the former is unable to provide such an account. There is absolutely no restric- tion on which meanings an applied semantics may assign to the logical constants: it may assign the classical meanings, or the intuitionistic meanings, or whatever other meanings the semanticist may be concerning himself with. This non- sensical view of the distinction between pure and applied semantics does not, of course, appear in my paper. Indeed, had I been advocating this view it is hard to fathom how I could have cited as an example of applied semantics the work of Urquhart (p. 407), whilst remarking that his

semantics seems to me to assign a meaning other than that of classical disjunction to the connective V (p. 408); and it is equally hard to fathom how I could have cited the Beth

tree semantics for intuitionistic logic as an example of applied semantics (p. 405).

This basic misunderstanding of my position vitiates RRM & M's reply to my paper. For this fictitious claim about the distinction between pure and applied semantics figures (under the label Cl) as the main premiss of the argument upon which they undertake to focus their attack (pp. 72-3) . Since - as readers may verify for themselves - this argument is nowhere to be found in [5], the attack is totally misdirected. RRM & M quaintly describe the argument as distilled from my paper (p. 72), yet they produce no quotations or other textual evidence to show that the argument may be fairly attributed to me. Indeed, the one quotation which they do provide in the vicinity is in fact a misquotation, the word "classical" having been inserted

where it does not, and should not, appear in the original (their page 72, quoting from page 406 of my paper). And the misrepresentation goes on. Having saddled me with C1, RRM &M proceed to "distil" (p. 73), "reconstruct" (p. 74) and "extract" (p. 75) arguments for it. One catches one's breath in astonishment to read that one's paper contains "three brief and fallacious arguments" for a claim which one has never made. The three arguments to which RRM &M devote pages 73, 74, 75, 76 and 77 of their paper may indeed be brief and fallacious - but that is no concern of mine. (The authors are certainly in earnest when, on page 71, they warn the reader that they will be "proceeding beyond what is directly stated" in Copeland's paper.)

I find the argument attributed to me on page 76 partic- ularly wearisome. Here RRM & M insist that I believe the meaning of natural language negation to be the classical meaning. Yet as a result of discussion and correspondence the authors have in fact been aware for over two years that I eschew all talk of the meaning of natural language negation, preferring to have to hand several negation concepts from which to select the best match for any given occurrence of colloquial or mathematical denial. However, the authors state in a number of places that a major assump- tion of the case I have presented against their semantics is the claim that the meaning of natural language negation is the classical meaning (pp. 76, 77, 83). As readers may verify for themselves, there is absolutely no mention of natural language negation in [5].

In footnote 3 RRM & M actually confess to knowing that my requirement that their semantics assign the classical meanings to the propositional connectives is not intended as some absolute requirement on illuminating semantics,

but rather is presented as a requirement which their semantics needs to satisfy if it is to be used in an attempt to show a classical logician that he makes a mistake when he infers B from (A V B) & ~A. (They assign my claim to this effect the label E2 ~ Since this footnote is the only point in their

paper where RRM & M are in some form of contact with the real case of [5], it is worth discussing in detail the remarks they make there. (For reasons best known to themselves RRM & M insist that E2 ~ is not to be found in [5] (op. cit.). But readers may find it for themselves on lines 3 to 9 of page 403.)

RRM &M outline two objections to E2 ~ The first proceeds by counterexample: E2 ~ is false, the argument goes, because there is at least one classical logician who has been persuaded by the semantics that DS is fallacious. This argument is a howler: whether any classical logician has

200 B.J . COPELAND

ever actually been so persuaded is a matter quite separate

from the question of whether classical logicians ought to be so persuaded - quite separate from the question of whether the semantics can properly be used in polemic against the

Disjunctive Syllogism of classical logic. Moreover, RRM &

M in fact fail to provide an example of a classical logician who has been so persuaded; they cite C. F. Kielkopf, yet

it is hardly fair to describe Kielkopf as a classical logician.

(In the preface to his [14] Kielkopf remarks that before

his conversion (as he puts it) to Anderson-Belnap style

relevance logic he believed that the various classical implication and entailment systems were "all defective in some

way or another".)

For their second objection to E2 ~ RRM & M merely

refer the reader back to remarks made earlier in the paper

concerning a pair of propositions E2 and D2. The chief

burden of these remarks seems to be to stress that the

authors believe the consideration of inconsistent and in-

complete situations to be important for the study of deduc-

tive reasoning. But this has no bearing on the truth of E2 ~

A logician may agree that inconsistent and incomplete situa-

tions should somehow be counted in; but if, as he consis- tently may, he remains wedded to the classical meanings of

the propositional connectives then he will not be (or ought not to be) convinced by just any semantical 'demonstra-

tion' concerning entailment - he will find acceptable only those demonstrations in which the classical meanings are

assigned to the propositional connectives of his language.

This stricture will not of course apply if our logician is wedded to some non classical set of meanings for the connectives, or if he is so formally minded that he does not care what if any meanings are assigned to the symbols.

But then he cannot be described as a classical logician, and so poses no threat to the truth of E2 ~

RRM & M accuse me of having lapsed into incoherence

in my remarks on inconsistent and incomplete situations (p. 80). In [5] I asserted that the following three conditions

may be simultaneously satisfied with respect to an intensional functor 1 which assigns the value T to a pair (A ,a )

just in case sentence A is believed by individual a: (i) I (A , a) = T and I ( ~ A , a) = T may both hold; (ii) I (A , a) = T

and I ( ~ A , a)= T are not contradictories; (iii) these first

two properties continue to hold when ~A is taken as the classical negation of A (p. 404). P,P,M & M assert that "(i)-(iii) form an inconsistent triad" (p. 80). However, their argument for this depends on the claim that an account of classical negation is given by the semantical rule

I ( ~ A , a ) = T iff I (A ,a ) ~ T(p. 78).

This claim is false. 3 For, to reiterate, a formal rule such as

this gives by itself no account of meaning: an explanation of how the formalism is to be understood is additionally required. Moreover, there will be ways of understanding the formalism such that the rule does not assign the meaning

of classical negation to the symbol ~ . For example, if the rule is understood as saying that HA is believed by indi-

vidual a if and only if A is not believed by a then some

meaning other than that of classical negation is assigned to by the rule (for nothing in the nature of human belief

rules out the possibility that individual a should believe both a sentence and its classical negation). 4 Yet RRM &M's

argument that (i)-(iii) are inconsistent precisely depends on the claim that the above rule does assign the meaning of

classical negation to ~ when understood in this way. In

short, RRM & M have no good argument for their assertion that inconsistent and incomplete situations can be accom-

modated semantically only if the classical meaning of ~ is relinquished.

My critique of the Routley-Meyer semantics has two

prongs to it. The first consists of the argument that the

semantics is pure, not applied; and the second consists of the argument that since the semantics is pure it cannot

properly be used to engage with the classical logician in polemic against DS. If it were true, as in footnote 3 RRM &

M aver it to be, that the authors have not attempted to so use the semantics then this second prong would be ineffective. But the authors' averment is simply false. In [26]

Routley and Routley mount a sustained attack on DS (see, for example, the very first paragraph of that paper). Indeed one has only to turn to pages 88, 89 and 90 of the RRM & M paper itself to find one of the authors presenting a "philosophical case against DS" (p. 90). DS "must be

rejected" we are told, "[asl can be shown.., from the semantical conditions for its validation" (pp. 88-89) . "A host of writers" (Copeland included) are berated for be- lieving DS to be a correct principle of inference (Note 19).

We are admonished that "there is nothing intuitive or even

slightly appealing about" DS (p. 89); italics in the original).

Whilst one does not expect the authors of a joint work to have no conflicting opinions, one does expect there to be some modicum of consistency within the paper.

In [5] I wrote:

I have often heard voiced the thought that this fact [that DS is invalid in the Routley-Meyer semantics] provides a semantical argument for the truth of the claim that DS embodies a fallacy of relevance. The thought is committed to print by the Routleys ([26], see especially pp. 343,349) (op. cit., p. 400.)

I wish to defend this passage against RRM &M's charge


that I have erroneously attributed to the Routleys something they did not say: the claim that DS commits a fallacy of relevance is, state RRM & M, nowhere to be found in the paper in question (p. 88, and again on p. 94). Fortunately, defence is simple. Here is what the Routleys wrote.

It is perhaps because Disjunctive Syllogism involves negative and not positive suppression that its suppression features have been overlooked .... What is wrong with suppression? First, it is responsible for relevance-violating paradoxes, positive suppression for positive paradoces and negative suppression for negative paradoxes. ([26], pp. 342-343)

Meyer has indeed stated ([19], Note 20) that traditionally minded logicians require remedial training in reading, but I really do not think that the fault is with the reader in the present instance.

The distinction between pure and applied semantics is crucial to my critique of the Routley-Meyer semantics. Yet there is plentiful evidence in their attempt to reply to the critique that RRM & M have little knowledge of this well-known distinction. The natural consequences of this are that their comprehension of the critique is limited, and their response to it is ineffectual. I have already discussed their gaffe concerning TC (as they call it): the gaffe of thinking that the distinction turns on whether or not a semantics assigns the classical meanings to the propositional connectives. I will proceed to discuss three further crucial points at which RRM & M exhibit lack of understanding of the distinction. Firstly, there is their erroneous belief that citing any of the following facts shows me mistaken in my claim that the Routley-Meyer semantics is pure semantics: (a) the semantics "answers to a well developed axiomatic and algebraic structure" (p. 9 2 ) - i t "reflects [a] previously axiomatised system" (p. 93); (b) completeness theorems are forthcoming concerning the Routley-Meyer semantics- this is "illegitimately overlooked in the indict- ment of relevant semantics as doing nothing to clarify meaning" (p. 92); (c) the proof of outstanding conjectures concerning various syntactical systems has been enabled by the semantics (p. 93); (d) the semantics has enabled im- proved proofs of already established results (p. 93). Consid- eration of the early semantics for intuitionistic l o g i c - .cited by Dummett ([11], p. 167) as an example par excel- lence of pure semantics- will serve to counter each of these points. The valuation systems in question were, of course, developed to characterise pre-existing axiomatic structures but are no less pure for that, nor for the fact that

appropriate completeness proofs were forthcoming. Whether or not a semantics is pure has nothing to do with whether or not it "reflects a previously axiomatised system". The modelling of an axiomatic system can be achieved without concern for the interpretation of the mathematical formalism employed. But the price to be paid is that the semantics give no account of the meanings of the symbols of the axiomatic theory modelled. It is a mistake to suppose that the existence of a completeness proof establishes that a semantics "clarifies meaning". (c) and (d) are equally easily disposed of by the above counterexample. As Dum- mett records (op. cir.) the valuation systems in question were originally introduced for the purpose of establishing proof theoretic results; and no doubt the technique did enable outstanding conjectures to be proved and existing results to be established more elegantly. The whole point of working with pure semantics is to achieve such things.

The second point that I want to discuss in this connection concerns my claim (made in [5] and reiterated above) that little explanation is given by Meyer and the Routleys of the central function I of their semantics over and above the bare set theoretic description of it as a function from {formulae} XK to (T, F} (this, it will be recalled, is one of my three reasons for concluding that the semantics is pure, not applied). RRM & M seek to rebut this charge (p. 83) by pointing out that in past work they have used the following English rendition of the context I(A, a) = T: formula A holds in situation a. But this reply misses the point. An English rendition is not an explanation. For all the understanding of I which their rendition offers it might as well be replaced by: situation a holds in formula A. What is it for a formula to hold in a situation? Indeed, what is a situation? A world? Or a set of sentences? And if the latter, how are such sets to be understood? Pending complete and unequivocal answers to these questions, we have been supplied just with a way of pronouncing that which we do not understand, and not with an explanation at all. (My [8] contains further remarks on this issue, together with some suggestions as to how the function I may be understood.)

Perhaps it is worth remarking at this point that one could not correctly claim that the recursion by which I is introduced itself constitutes a complete account of the relation holds in. RRM & M preclude this possibility by their intention to use the recursion to give an account of the meanings of the logical constants (see, for example, their page 82). The point here is similar to the familiar one that the recursion by which Tarski gave a definition of truth cannot be used both as an account of the truth

202 B. J. COPELAND

predicate and as an account of the meanings of the logical constants: given an independent understanding of the meanings of the logical constants Tarski's recursion gives an account of the truth predicate, and given an independent understanding of the truth predicate the recursion gives an account of the meanings of the logical constants, but one cannot have it both ways. (This point is made by Dummett ([10], p. 460), and by Strawson ([28], p. 180).) Similarly, the recursion by which I is introduced may be used to provide an account of the relation hoMs in, given an independent hnderstanding of the meanings of ~, V, & and (and of situations, *, and the ternary relation); or alterna- tively the recursion may be used to provide an account of the meanings of ~, V, & and -+ given - w h a t is thus far lacking- an independent understanding of the relation holds in (and of * etc.).

Thirdly, and lastly, the failure of RRM & M to grasp that there is a difference between pure semantics and applied semantics is shown by their supposition that since their semantics contains a "quite precise semantical rule" for the symbol ~, the semantics therefore gives an account of the meaning of that symbol (p. 82). This judgement reveals a lack of awareness of the fact that a semantics gives an account of meaning only once the mathematical formalism of the semantics itself has been explained in terms of concepts relating to the actual or intended use of the sentences of the language for which the semantics is given. Precisely what is lacking in the authors' work is a complete and unequivocal explanation of how we are to understand the mathematical formalism of their semantical rule for ~; and until RRM & M provide such an explanation it remains completely obscure what meaning is given to ~ in their writings. RRM & M are quite mistaken when they think it clear that their ~ differs in meaning from the negation of classical logic (p. 78), and, indeed, when they think their

to be closer in meaning to natural language negation than is classical negation (ibM.): judgements of this sort can be made sensibly only once the conversion of the pure semantics to an applied semantics stands completed, s

Thus far in this paper I believe I have established three things. Firstly, RRM & M have largely failed to comprehend the case I addressed to them in [5]. Secondly, the arguments which in their reply they have attributed to me have nothing to do with what was actually said in [5]. Thkdly, and consequently, their response to [5] is wholly ineffective: the case I presented there stands unanswered. For the remainder of this paper I wish to turn to an issue of second- ary importance. In the course of their response RRM & M

allege that [5] contains various misattributions, misre- presentations, and errors. I shall proceed to show that these allegations are without foundation. 6

Firstly, RRM &M allege that my attribution of the system R to Anderson and Belnap is "inaccurate" (Note 9). If it is then Meyer himself is guilty of the same inaccuracy, having written in [17] "Anderson and Belnap have constructed a system R" (p. 472). But in fact there is no inaccuracy. It is part of the very folklore of relevance logic that Anderson and Belnap constructed the system they christened R by adding an axiom A -~ NA to their system E. This is recorded by Belnap in [3]. And as is equally well known there is an alternative axiomatisation of R, again due to Anderson and Belnap, obtained by adding the negation, conjunction and disjunction axioms of E to Church's weak theory of implication (vide [2], pp. 20, 339-341).

Equally strange is RRM & M's reaction to my use of the phrase 'relevant entailment', which they claim "is not

included in the terminology of Anderson and Belnap [2], but is drawn from a rival conceptual framework" (Note 9). Whilst it is perhaps true that the precise sequence of symbols 'relevant entailment' never happens to occur in [2], it is

simply absurd to claim that talk of relevant entailment is foreign to the conceptual framework of [2]. The very title of the book is Entailment: the Logic of Relevance and Necessity.

In [5] I reported that at the time of writing (namely 1977) research in relevance logic centred on systems neighbouring the system II' due to Ackermann, in particular the systems R and NR (for lists of such neighbours consult Anderson and Belnap [2], pp. 339,348,349) . RRM & M allege that this report is inaccurate (Note 9), but once again their allegation depends upon the falsification of well known facts. Fortunately, one can here appeal to a contri- bution to the Relevance Logic Newsletter (May 1977, pp. 103-104) entitled 'Current Research in Relevant Logic' and written by Meyer himself. We are told "a new stage in the development of relevant logics has been reached . . . . Briefly, there now exist a class of logics.., which are called here relevant logics." Meyer then lists six members of this class: one of these is the Anderson-Belnap system (E which is equivalent, as Meyer notes, to the Ackermann system II'), whilst four of the remaining five are deemed neighbours of II' by Anderson and Belnap (op. cit.). The system R, which is of course one of these neighbours, is placed "in the middle" of the class by Meyer. A description of the then-seductive features of R and her sister NR may be found in a paper by Ken Collier in Vol. 1 (1976) of


The Relevance Logic Newsletter: (Incidentally, Meyer 's

Current Research article further confirms my previously

discussed at t r ibut ion o f R to Anderson and Belnap. He

writes "E, R, and T are due, of course, to Anderson and Belnap" (op. cit., p. 104).)8

RRM & M further allege that "Copeland's discussion

seriously misrepresents the historical situation of work on

relevant logic", and by way o f corrective they insist that

"in fact the formal breakthroughs arose out of and were

built upon preceding philosophical work" (p. 86). I t would

have been only proper to have supported this allegation

with quotations from [5], but none are given. Indeed, none

could be: there is no discussion whatever in [5] of the

history or genesis of the Rou t t ey -Meyer semantics. More-

over, the extent to which Meyer and the Routleys were led

to their formal semantics by motivational ideas is quite

irrelevant to my point that crucial elements of the mathe-

matical formalism of the semantics have never been fully

and unequivocally explained by its creators. 9

RRM & M allege that my discussion in [5] of Urquhart 's

semantics contains a "major technical error", reporting me

to have made the false claim that the Urquhart semantics

validates the same theses as the positive part of the Rout-

i cy -Meye r semantics for the system R (p. 85 and Note 13).

This claim is in fact nowhere to be found in [5]. The point

made there is simply that ideas developed by Urquhart may

be used to supply an understanding, of the kind appropriate

to app!ied semantics, of the mathematical formalism of the

positive part of the Rou t l ey -Meye r semantics. This no

more involves the claim RRM & M impute to me than the

suggestion that numbers can be understood as sets involves

the claim that under translation number theory and set

theory have the same theorems. Moreover, there should

be no presumption that the process of interpreting a pure

semantics must respect the theses of some given axiomatic

system. Once an account is constructed of how the formal-

ism of a hi therto pure semantics is to be understood, one

may decide that the postulate set of the pure semantics

(or indeed some other part of the pure semantics) is not

quite foursquare relative to that understanding, and add or

delete formal constraints as necessary, thus altering the set

of sentences validated. It is precisely this feature of applied

semantics which renders it capable o f adjudicating between

alternative axiomatic systems - a welcome abili ty in a field

such as relevance logic, where axiomatic systems are a dime a dozen.~ o

RRM & M accuse me of having vacillated in [5] between

issuing a challenge and making a condemnation (p. 84 and

Note 12). In reality there was no vacillation: I condemned

the a t tempt to use pure semantics to establish that the

classical logician makes a mistake when he asserts that

(A V B ) & ~A entails B, for this is a task to which only

applied semantics is suited; and I challenged Meyer and the

Routleys to provide the explanation o f the formalism

which would serve to convert their semantics to an applied

semantics. Even though their response to my paper is now

on hand, this challenge remains unanswered.

Let me end with a quotation from Kielkopf, who in

his text book on relevance logic writes o f the R o u t l e y -

Meyer semantics:

No secret has been kept; we do not have any significant interpretation of model structures . . . . Interpretation of these model structures is a field wide open for discovery and invention ([ 14], p. 369).

The Queen's University o f Belfast, Belfast, Northern Ireland

Notes

Meyer has in fact evinced a certain pessimism concerning the possibility of giving an adequate account of �9 ([18], p. 80). He even descries the semantical clause for negation in which he and Routley use �9 as "screwy" (ibM.). 2 R. and V. Routley, R.K. Meyer and E. P. Martin, On the Philo- sophical Bases of Relevant Logic Semantics', Journal of Non- Classical Logic 1 (1982), 71-105. 3 This false claim gets used again in an argument on p. 89; and also in the argument on pp. 78-79 to the effect that it is impossible to give an account of relevant er[tailment whilst retaining the classical meaning of negation. In fact there are at least three methods on the market for explicating relevant entailment whilst retaining classical negation: vMe Copeland [4] and [7], and Epstein [12]. 4 My [8] contains a broader discussion of tiffs point. s Kielkopf, an exponent of the Routley-Meyer semantics, has also stressed that the semantics assign no meanings to the logical constants. In a chapter of his text book on relevance logic devoted to the Routley-Meyer semantics, he writes:

[The Routley-Meyer semantics] teach us forcefully that in regard to interpreting a system work with model structure semantics is as formal as work with matrix semantics. They make it clear that interpretation of a language on a model structure does not give meaning to the signs. ([ 14], p. 370).

6 I must confess to the presence of one mistake in [5]. I incorrect- ly stated that in the version of their semantics which Routley and Meyer present in [23], the base world 0 is consistent. I neglected to notice that the special postulate which secures the consistency of 0 is in fact omitted from that presentation of the semantics. (This special postulate pops in and out of the semantics from paper

204 B . J . COPELAND

to paper: it appears in the presentation given in [21], and also in that given in the important [26] .) Fortunately, this oversight does not affect the central arguments of [5]. 7 In [6] I showed the seductive features of the sistersE, R andNR to be entirely cosmetic. s A further point which should be made in this connection concerns RRM & M's remark in the footnote under discussion that "it might be simply said that while some of [Copeland's] points tell against R and NR they are ineffective against deeper relevant systems such as DK". (Note 9) This position is untenable. My critique applies to the Routley-Meyer semantics per se: the mathematical formalism of the semantics has not been adequately ex- piained. Whether the postulates of the semantics are manipulated to achieve completeness relative to R or completeness relative to DK has no bearing on this matter. 9 If RRM & M believe that all "the formal breakthroughs" in their subject "arose out of and were built upon preceding philosophical work", then they are sorely mistaken. Anderson once wrote: "Can- dour compels me to admit, again in the interests of historical accuracy, that one of the principal reasons for dropping this rule (which I shall hereafter refer to as "the disjunctive syUogism") was that in the presence of this primitive rule, almost none of the arguments in the papers of Belnap and myself, cited above, can be carried through; or so it seems". ([1], p. 10) 10 RRM & M state that there exists an "enlargement" of Urquhart's theory, to appear in [25], which "should supply the sought interpretation for the full Routley-Meyer theory" (p. 85). Judgement as to whether this enlargement really does serve to convert the Rout- ley-Meyer semantics to an applied semantics must be suspended, since unfortunately all details are omitted. But RRM & M's remarks that the notion of a piece of information is to become "semi-technical" (pp. 84, 86) perhaps sound a warning that the material they are referring to is yet more pure semantics, and not applied semantics at all.

References

[1] Anderson, A.R.: 'Some open problems concerning the system E of entailment', Acta Philosophica Fennica Fasc. 16 (1963), 7-18.

[2] Anderson, A.R. and BeInap, N.D.: Entailment: TheLogic o f Relevance and Necessity, Vol. 1, Princeton University Press, 1975.

[3] Belnap, N.D.: 'Intensional models for first degree formulas', Journal o f Symbolic Logic 32 (1967), 1-22.

[4] Copeland, B. J.: Entailment, Doctoral dissertation, University of Oxford, 1978.

[5] Copeiand, B.J.: 'On when a semantics is not a semantics', Journal of Philosophical Logic 8 (1979), 399-413.

[6] Copeland, B.J. : ~rhe trouble Anderson and BeInap have with relevance', Philosophical Studies 37 (1980), 325-334.

[7] Copeland, B.J.: 'Horseshoe, hook and relevance', forthcoming in Theoria.

[8] Copeiand, B.J.: 'What is a semantics for classical negation?', Typescript, 1982; to appear.

[9] Dummett, M.A.E. : q?he justification of deduction', Pro- ceedings of the British Academy 59 (1973), 201-232.

[10] Dummett, M. A. E.: Frege: Philosophy o f Language, Duck- worth, 1973.

[11] Dummett, M.A.E. : Elements o f Intuitionism, Oxford University Press, 1977.

[12] Epstein, R.L.: 'Relatedness and implication', Philosophical Studies 36 (1979), 137-173.

[13] I-lintikka, J.: 'Intuitions and philosophical method', Revue Internationale de Philosophie 35 (1981), 74-90.

[14] Kielkopf, C.F.: Formal Sentential Entailment, University Press of America, 1977.

[ 15 ] Kirwan, C.: Logic and Argument, Duckworth, 1978. [16] Lewis, D.K.: 'Logic for equivocators', No~s 16 (1982), 431-

441. [ 17] Meyer, R.K.: 'Entailment and relevant implication', Logique

etAnalyse 11 (1968), 472-479. [18] Meyer, R.K.: 'New axiomatics for relevant logics', Journal

o f Philosophical Logic 3 (1974), 53-86. [19] Meyer, R.K.: Why I am not a relevantist, Research Paper

No. 1, Logic Group, Australian National University, 1978. [20] Plantinga, A.: The Nature o f Necessity, Oxford University

Press, 1974. [21 ] Routley, R.: 'A semantical analysis of implicational system

I and of the first degree of entailment', Mathematische Annalen 196 (1972), 58-84.

[22] Routley, R.: Exploring Meinong 's Jungle and Beyond, Depart- mental Monograph No. 3, Philosophy Department, Research School of Social Sciences, Australian National University, 1980.

[23] Routley, R. and Meyer, R.K.: 'The semantics of entailment l ' , in Truth, Syntax and Modality, ed. by Leblanc, H.~ North Holland, 1973.

[24] Routley, R. and Meyer, R.K.: 'Dialectical logic, classical logic, and the consistency of the world', Studies in Soviet Thought 16 (1976), 1-25.

[25] Routley, R., Meyer, R. K. et al.: Relevant Logics and Their Rivals, Unpublished at time of writing.

[26] Routley, R. and Routley, V.: 'The semantics of first degree entailment', Noas 6 (1972), 335 -359.

[27] Scott, D. S.: 'Rules and derived rules', in Logical Theory and Semantic Analysis, ed. by Stenlund, S., Reidel, 1974.

[28] Strawson, P. F.: Logico-linguistic Papers, Metimen, 1971. [29] van Benthem, J .F .A.K. : 'What is dialectical logic?', Erkennt-

nis 14 (1979), 333-347.

Documents

Copeland - Pure Semantics and Applied Semantics