The logical structure of linguistic commitment I: Four systems of …individual.utoronto.ca/philipkremer/onlinepapers/lslc1.pdf · 2009-05-25 · LOGICAL STRUCTURE OF LINGUISTIC COMMITMENT

M A R K N O R R I S L A N C E A N D P H I L I P K R E M E R

T H E L O G I C A L S T R U C T U R E OF L I N G U I S T I C

C O M M I T M E N T I: F O U R SYSTEMS OF NON-

R E L E V A N T C O M M I T M E N T E N T A I L M E N T

'Entailment' can be said in many ways. In the present paper, we sketch a linguistic theory which motivates taking 'A entails B' to mean 'commitment to A is, in part, commitment to B', and we present four formal systems designed to capture this concept. These four systems turn out (somewhat surprisingly) to be the strict implicational fragments of the modal logics K, T, K4, and $4 (see Hughes and Cresswell 1984), although the motivation is not "modal" in any standard sense. (We note that, though we believe the relevance of A to B cannot be ignored in a final analysis of 'commitment to A is, in part, commitment to B', these systems are not "relevance" systems, in the sense of Anderson and Belnap 1975. We address relevance in "The Logical Structure of Linguistic Commitment II".) We proceed as follows: w 1: We discuss the appropriateness of calling a logical system (or a logic of entailment) the correct logical system (or logic of entailment).

w We briefly explain an account of assertional practice found in Brandom 1983 and 1985. A central feature of this account is the notion of an assertional commitment.

w We criticize Brandom's account of the relation between logical theory and underlying linguistic practice, arguing that the relation must be conceived of as normative rather than as descriptive. We sketch an alternative account, which coheres with a generally inferentialist account of meaning without leading to the unpleasant consequences of more straightforward approaches.

w We present Fitch-style natural deduction systems (akin to those of Fitch 1952) and simple algebraic semantics designed to capture the concept of commitment entailment. Thus, we are led to a variety of plausible explications of the notion of non-relevant commitment

entailment.

w We consider the significance, in the present context, of theoremhood.

Journal of Philosophical Logie 23: 369-400, 1994. �9 1994 Kluwer Academic Publishers. Printed in the Netherlands.

370 MARK NORRIS LANCE AND PHILIP KREMER

w We conclude.

w (Technical appendix): We prove the formal claims made

throughout the paper.

1. THE CORRECTNESS OFA LOGICAL SYSTEM

Often, debates about the correctness of logical systems takes place from an unhappily abstract perspective. The discussion of what system provides the correct account of entailment - e.g., debates between relevance and more classically inclined logicians - at times seems to have little more to go on than intuitions concerning ordinary or mathematical

usage of the term 'entailment'. We maintain that the question of what logical system - and, more

particularly, what logic of entailment - is the correct one hinges upon what its interpretation is taken to be. There are many different interpretations which might be given to a formal system fitting into the vague category of a "logic". Interest in epistemic logic, for example, could lead one to consider the inferential relation holding between two sentences if and only if the first is definitive evidence for the second. (Note that the adjective 'definitive' rules out non-monotonic systems.) Alternatively, a philosopher of language might want to explicate a feature of the content of sentences, propositions, or assertions - roughly that which follows from its meaning - in terms of an inferential relation. Or, to give a final example, one could suppose that the interpretation of a

logical system is ontological, based on relations between states of affairs or possible worlds.

These examples suggest not only that there are many different contexts fit for formal inquiry; they suggest that there is no univocal or precise pre-theoretic notion of entailment (as is argued in Lance 1993), though there is the post-theoretical notion of entailment-in-a-system. Indeed, each of the interpretations mentioned above corresponds to a use to which the notion of entailment has traditionally been put.

Now perhaps the same logic will do for all three of these interpretations, but this would not be easy to show. If formal entailment systems are to be interpreted within significant epistemic, linguistic and ontological theories, one must be confident of the correctness of these theories in order to assess the claim that entailment is univocal.

L O G I C A L S T R U C T U R E OF L I N G U I S T I C C O M M I T M E N T I 371

Rejection of, say, an accepted linguistic theory in favour of a radically different one would open the possibility of a re-interpretation of the role of entailment relations in the theory of meaning and, thus, invite us to reassess the question of whether there is really only a single entailment relation.

In the end, the question of whether entailment is univocal is not the interesting one. We would like to develop adequate theories and, wherever possible, to formalize relations defined within those theories as logical systems, some of which we would then like to call logics of entailment. The point we wish to emphasize is: it is crucial to argue for and develop a system of entailment with explicit reference to a particular theory within which the system has its life.

In what follows, we attempt to do this. We describe a particular linguistic theory and define an important relation within that theory. The understanding of this relation which we develop here is interesting in itself and raises a number of philosophical issues. Having described this theoretical relation, we begin the work of developing a range of logical systems which purport to capture it. It is not claimed that the semantic relation so defined is the only relation between semantically significant units that is theoretically important and leads to an entailment-type formal system. It is claimed that one important notion of entailment and its relation to a semantically significant feature of language use is explicated in this way.

2. A S S E R T I N G

Brandom 1983 presents an important attempt to provide an account of the act of making an assertion. He draws on the work of Sellars and, more distantly, the later Wittgenstein, by considering the act of asserting to be a kind of move within a language game. Such games are, essentially, normatively constrained practices of giving and asking for reasons.

Brandom takes the act of asserting to be fundamentally an act of undertaking a certain commitment, namely that of answering appropriate challenges to the claim in an appropriate manner. Thus, the differet~e between merely pronouncing the sentence 'There will be a collapse of the federal savings and loan system' and asserting it, is


that in the latter case one has committed oneself to providing evidence for the assertion. If one asserts this claim, one is thereby committed to answering such challenges as 'That seems implausible given that the S&L's are federally insured'. In this case, one may even have to respond to a mere demand for evidence of the form: 'How do you know?'.

Alongside the normative dimension of commitment is that of

entitlement. If one succeeds in carrying out one's justificatory responsibility for A - if one wins in the dialogical game of giving and asking for reasons - other members of the linguistic community are

bound to grant one entitlement to the claim. That is, they have to recognize as appropriate any of a range of uses of the sentence, for example, as evidence for other claims, or as a justification for non-linguistic actions.

It is possible that there be two sentences A and B such that all of the justificatory commitments undertaken in the act of asserting B are already undertaken in the act of asserting A; that is, there can be a kind of containment of commitment content among sentences such that the

commitments undertaken in asserting B are part of the commitments undertaken in asserting A. We put this in the following terms: in such a situation, commitment to A is, in part, commitment to B.

There are particular kinds of inferential m o v e s in a language game which are associated with this relation. For example, one who is challenging the claim, by another speaker, that yonder animal is a dog, may point out that commitment to this claim carries with it commitment to the claim that the animal is a mammal. This might justify challenging the original assertion by way of the asserting that the animal lays eggs. This challenge is relevant to the original assertion, indirectly, since it is directly incompatible with something that follows from the original assertion. Thus we are led to the fundamental interaction between inference and incompatibility within our linguistic practice: any claim incompatible with a given claim B, can also be used as a challenge to any assertion A which committively entails B.

3. ENTAILMENT CLAIMS

Brandom 1985 addresses the relation between logical theory and linguistic practice. For a Sellarsian, the content of a sentence is primarily

LOGICAL STRUCTURE OF LINGUISTIC COMMITMENT I 373

a function of the inferential role of that sentence in the linguistic practice of the relevant community. It is such facts as that commitment to

something's being a dog carries with it commitment to its being a mammal that determine the meaning of 'dog'. But if meaning is

inferential role, then a perspicuous and explicit means of determining a meaning for a bit of vocabulary is to provide it with a proof theory,

a set of rules determining the acceptable inferential moves involving

sentences containing the bit of vocabulary. Logical systems provide proof theories for certain logical concepts,

notably that of entailment. So a proof theory for sentences involving

the constant '--+' read as expressing (committive) entailment, should serve to give meaning to a bit of vocabulary; in particular, it should give an inferential content to a bit of language whose role is to make possible the formulation of sentences which express the very inferential relations that gives content to the underlying language. When we say that A entails B, we are making explicit a relation which was previously implicit in linguistic practice and whose existence there served to

confer content on the sentences A and B. It is primarily this increase in the expressive resources of our language rather than the determination

of the set of logical truths that is the valuable contribution of logical theory.

Now this picture is correct in many respects, but it has a particular consequence that calls for its emendation. We must ask what grounds are to be given for one proof theory over another. If Brandom is correct in claiming that sentences like 'A ~ B' purport to "codify" underlying linguistic appropriatenesses - to make inferential relations, previously implicit in a linguistic practice, explicit - then the correct system is the one that most accurately describes these underlying social criteria of inferential appropriateness. However, to adopt such a line is to endorse a form of cultural relativism. It is to say that we ought to enshrine in logical theory whatever inferential moves are endorsed by our linguistic community. Indeed, it is to build into the very concept of entailment that one's linguistic community is correct about it.

This cannot be. It is conceptually possible for a community to be mistaken about inferential matters. To say 'A ---+ B' is not to say that we antecedently take it that commitment to A is, in part, commitment to B, but to say that commitment to A really is, in part, commitment

374 M A R K N O R R I S L A N C E AND P H I L I P K R E M E R

to B. To say that, however, is to say that one really ought to assign commitments to speakers in such a way, whether society endorses this or not. Thus, entailment claims are normative in a culturally transcendent sense that is akin to the way in which some argue that

ethical claims are. How are we to make sense of this on our account of

meaning, however? We said that the inferential role of entailment claims would be

spelled out by way of a proof theory. The problem then arose of the epistemic methodology of proof theory formation; what grounds could justifiably by given for one proof theory over another? This led us to consider the precise relation between such a proof theory and the underlying practice of inferring. What has been ruled out, on the ground that it entails relativism, is the possibility of taking this relation to be

one of description. We can see how to cash out the metaphor of genuine culture

transcendent normativity by paying more attention to the structure of a Sellarsian language game. It was a simplification to suggest that such games only consider inferential moves between sentences. In fact there are three broad categories of linguistic moves according to Sellars: language-language moves which include the various sorts of inferential moves; language entrance moves, for example the move from observing a duck to saying '1o, a duck'; and language

exit moves such as the move from saying that one is leaving to the

act of leaving. Another type of language exit move is one that involves more than one agent. In certain cases a linguistic move by one agent may bring about a commitment on the part of others

to perform a certain sort of act. For example, when a speaker says "the end" or otherwise indicates that her talk is finished, this linguistic act carries with it a commitment on the part of the members of an audience to indicate a reaction, usually by applauding.

This sort of propriety is most important for our purposes and indicates that linguistic practice is not an isolated institution, but one which is connected with other social practices in ways that are crucial to its own nature as well to that of the practices with which it interacts. It is such a cross-practice connection that is characteristic of normative assertions generally. In the case of commitment entailment claims, the


connection is between two distinct linguistic practices rather than a

linguistic and a non-linguistic one, however. The first, the underlying practice, is an ordinary language practice

which lacks logical vocabulary. Meaning is conferred on the assertions of this underlying practice by the appropriateness of moves such as that from 'x is a dog' to 'x is a mammal' . The second practice, the meta- practice, is the one whose structure is determined by a proof theory for sentences involving logical vocabulary.

The feature of this higher level practice which makes it one definitive of a normative content is a normative connection to the underlying practice. Specifically, we require that entitlement to an assertion of

an entailment claim within the higher level practice carry with it - be, in part - entitlement to the corresponding move in the underlying practice. Thus, if one justifies the claim 'A ~ B, by deriving it from accepted claims, then this act will confer entitlement to make committive inferential moves from A to B.

Thus, we play two inter-related linguistic games. In one we assert claims about dogs and the like, challenge these, and infer things from them according to certain rules which are taken by everyone to be prima facie binding. Occasionally a dispute will arise concerning some of these rules, perhaps an inferential one. In this case we play a game centering

around the assertion, challenge, and defense of a claim of the form 'A ~ B'. There are (primafacie) accepted rules for playing this game also. (One example might be that one can infer 'A --+ B' from 'A -+ C' together with 'C ~ B'.)

Since it is perfectly' possible to justify, by the standards of a reasonable proof theory, an entailment claim not corresponding to an antecedently recognized inferential propriety, this relation between the practices does not presuppose relativism as a descriptive one would. Whereas Brandom's discussion invited us to see logical theory as a codification of ordinary linguistic practice, we should, rather, see it as a mechanism or set of rules for the emendation of that practice. We ascend to the level of logical theory primarily so as to resolve disputes concerning inferential proprieties operating at the basic level.

In terms of our methodology for the construction of an explicit p roof theory for commitment entailment, then, the present account implies that we should look for proof rules which are such that we ought to

376 M A R K N O R R I S L A N C E A N D P H I L I P K R E M E R

endorse committive inferences from A to B just in case 'A --, B' is derivable from previously acceptable entailments according to these rules. (For more discussion see ~5, below.)

4. P R O O F T H E O R Y F O R C O M M I T M E N T E N T A I L M E N T

Our suggestion is to read 'A --+ B' as a commitment to A is, in part, commitment to B'. Now there are two sorts of force which can be identified in such a claim. On the one hand, it suggests

"universality of agent". That is, we are tacitly quantifying over assertors, meaning that commitment to A by anyone is, in part, commitment to B by that person. This universality of the asserted content of a conditional implies that if A --+ B and if c~ is committed to A, then c~ is committed to B.

However, this is not all that is suggested by the intuitive locution. To say that commitment to A is, in part, commitment to B is to suggest, as Anderson and Belnap might urge, that commitment to A is relevant to commitment to B. The point is that c~ might be committed to B, on grounds independent of c~'s commitment to A. Thus, even though

everyone committed to A happens to be committed to B, this latter commitment might not be a part of their commitment to A. Nonetheless, in the present paper we consider the universality side of commitment entailment in isolation from considerations of relevance, putting these considerations off until Part II. Thus, for the remainder of Part I we look to the explication of an entailment relation, A ~ B, which can be read 'anyone committed to A is committed to B'. This strategy leads to a very elegant proof theory. For simplicity we consider, in what follows, a language whose only connectives are the conditional, --+, and conjunction &. (We are primarily interested in -+. The inclusion of conjunction simplifies the axiomatizations of our systems.)

To prove that commitment to A by anyone implies commitment to B by that person, it is natural to proceed by assuming that an arbitrary person is committed to A and then showing that she is committed to B. Let us write C~A for "c~ is committed to A" (where c~ is any positive integer) and call the operator (or string of operators) preceding the sentence A "the prefix of A". Then, the above assumption is that our rule of conditional proof (CP) should be of


the form:

From a proof of CaB on hypothesis C~A to infer A --* B.

Our structural Fitch rule of Hypothesis (Hyp) would then allow that we can introduce a new subproof with hypothesis C~A provided that a has

not occurred earlier in the proof. This formulation of the structural rules, however, ignores the

possibility of nested conditionals. For if the use of a reiterated assumption within a subproof requires the incorporation o f the

reiterated assumption's prefix into the prefix of the conclusion, then such a conclusion can never figure into an application of CP as we

have formulated it. Consider the intuitive manner in which we would prove a nested

conditional, 'A --+ (B --+ C)', given the reading of 'A -+ B' as "anyone

committed to A is committed to B". We would begin by assuming that an arbitrary person, c~, is committed to A and setting out to show

that she was committed to the claim that anyone committed to B is committed to C. How can we show this? Well, we assume - under the condition of having already assumed that c~ is committed to A - that a is committed to an arbitrary person/3's being committed to B and then show that it follows that a is committed to/3's being committed to C. In line with this strategy, we suggest the following as a reasonable Fitch-style proof:

C1A --~ B

CtC2B-+C

C1C2C3A

C1A--+B

CIC2C3B

CIC2B--~C

CIC2C3C

C 1 C 2 A ~ C

C1 (B ~ C) --+ (A ~ C)

(A---~B)---+((B---~C)--+(A---+C))


Several points in this proof call for comment. The first three lines are

in accord with the strategy suggested above. At the next line we reiterate. (In the absence of any motivation for restrictions, we allow reiteration of any sentence, retaining the prefix, of course; all of the work of

avoiding fallacies will be done by natural restrictions on the proof rules governing such reiterated sentences.) At line 5, we apply a form of modus ponens (MP) which is natural on the interpretation. At line 8 we have our first application of CP. It is of the general form given in w below.

A natural formulation of the rules MP, and CP for sentences with commitment prefixes already avoids what some consider the worst of the fallacies of material implication. Thus, to prove A --+ (B --, A) would

require us to allow that a conclusion of CIA under hypothesis C1C2B warrant the conclusion C1B ~ A and there is no plausibility to such a move. The proof would proceed (fallaciously) as follows:

CxA

C 1 C 2 B

C1A

C1B ~ A (fallacious step)

A ~ (B -~ A)

4.1. The Fitch-Style Natural Deduction Rules

We now lay out the Fitch-style rules implicitly used in the first deduction above. The rules for conjunction, '&', are natural, and facilitate axiomatization. First, note that for any line of a Fitch-style natural deduction proof there is a number of vertical lines to the left of the sentence written at that line. We call this number the " rank" of the line.

The rules are as follows:

Hyp: A step may be introduced as the hypothesis of a new subproof and each new hypothesis receives a prefix C I . . . Ck, where k is the rank

of the subproof. Rep: A sentence occurring at an earlier line may be repeated, retaining

the prefix.


Reit: A sentence occurring earlier may be reiterated into hypothetical

subproofs, retaining the prefix.

CP: From a p roof of C1 �9 .. Cn+ 1B on hypothesis C1 . . . Cn + lA to infer

C x . . . C n A ~ B, n>~0.

MP: From C1 . . . CnA -+ B and C1 . . . Cn . . . Cn+mA to infer C I . . . Cn+mB, where n/>0, m ~> 0. (Below, we consider restricting the range of m.)

&I: From CI . . . C~A and C1 . . .C~B to infer C1 . . .CnA&B

&E: From CI . . .C~A&B to infer either C1 . . . C , A or C1 . . .CnB.

We call the natural deduction system fC0+, and the set of sentences

provable from it CO+ ( " f " is for "Fi tch" and " C " for "commitment" ;

the use o f " 0 + " becomes clear). Delightful fact: CO+ is the --+& fragment

of $4 (where --+ is strict implication).

4.2. Restrict ions on the Use o f M P

There are two natural restrictions on the use of MP; in particular, there are reasons to disallow cases in which m = 0, and reasons to disallow

cases in which m > 1. And so, we define four natural deduction systems

which purport to capture the notion of non-relevant commitment

entailment, depending on whether we insist on one restriction, the other

restriction, both or neither. F rom a formal point of view, the generation

of these related systems is quite elegant, their differences depending upon

a slight change in the usability of MP; furthermore, they correspond to antecedently interesting and studied formal systems. From a

philosophical point of view the different possibilities highlight interesting issues as to the scope of the commitment one undertakes in making an entailment claim.

Presently, we advance considerations in favour of the two restrictions.

We do not take these considerations to be decisive, and so we do not

come down in favour of one or another system. These considerations are

only meant to show that all four systems are plausible candidates for a logic of commitment entailment, and all four are worthy of further study.

4.2.1. Why not allow m = O? commitments concerning one's own

commitments . Not allowing MP in cases in which m = 0 amounts to

380 MARK NORRIS LANCE AND PHILIPKREMER

denying the validity of such inferences as from C1 (A --* B) and CIA to C1B. Why not allow such inferences? ,

The fundamental feature of our interpretation of ' -* ' is to take it as

determining the inferential articulation of linguistic commitments. So we

certainly want it to be the case that if it is true that A ~ B and that c~ is

committed to A then a is committed to B as well. But this only explains

the upshot of the truth o f A ~ B; it does not solve the question of what it

is that is involved in commitment to A -~ B.

Since the truth of A -~ B is a normative fact to the effect that

commitment to A on the part of anyone carries with it commitment to B,

it is reasonable to suppose that commitment to A ~ B should involve

commitment to the claim that commitment to A on the par t of anyone

carries with it commitment to B. Thus, one natural normative construal of the act of undertaking a commitment to A ~ B is to take it that a

person, c~, who undertakes such a commitment becomes such that, for

any fl, commitment by ~ to 'fl is committed to A' is, in part, commitment

by oe to 'fl is committed to B'.

Now this construal does not, at least pr ima facie, imply that if ~ is

committed to A and to A ~ B, then o~ is committed to B as well. This

construal does imply that if ~ is committed to A -~ B and to '~ is

committed to A' (rather than simply to A), then c~ is committed to 'c~

is committed to B'; but there is no clear reason why commitment to A need carry with it commitment to the claim that one is committed to A, nor that commitment to the claim that one is committed to B need carry

with it commitment to B. Thus, we can see a debate over whether to allow MP when m --- 0, as a debate over whether the inferences f rom

C1A to C1C1A and vice-versa are valid - the C - C thesis. In the Appendix, w we advance certain technical considerations concerning

the C - C thesis. These considerations must wait for the semantics and the

definition of canonical models. This is not the only way to understand the difference between

allowing and not allowing MP when m = 0. One might reject the C - C thesis, but nonetheless try to argue on independent grounds that there

is reason to take a person committed to both A and A ~ B to be committed to B as well. Nonetheless, it can at least plausibly be argued (as in the following two paragraphs) that we do not want such an inference to go through.


Entailment, as we understand it here, is primarily a relation that determines implicit commitment on the basis of explicit commitment. That is, the set of implicit commitments of an agent is just the set of inferential consequences of the set of explicit commitments of that agent.

The question concerning the rule of assertion then comes down to the question of whether it is better to take the determination of the set of

implicit commitments to be determined by true entailments only, or to also consider those entailments to which the agent is committed. This is not, of course, to say that there is a question of whether a person can be committed to false entailments; the question is only

whether such commitments should be taken as determining the inferential content of the agent's other commitments.

Suppose that some person incautiously claims that material implication is the correct logic of commitment entailment. Such a person is, then, committed to (A&NA)~B. Should we take that person to be committed to everything if, in his theory of class struggle, there turns out to be a hidden contradiction of which he is unaware? It seems

plausible to deny that a contradictory commitment on the part of an arbitrary person carries with it commitment to everything; so, for the same reasons, we might not want to endorse this implication for a person who happens to adopt a false logical theory. What follows from this person's theory of class struggle is a function of what the theory explicitly says and what really follows from what. To treat his logical theory as determining what follows from his theory of class struggle is akin to supposing that, in the realm of inference, whatever he says goes (or, to suppose, in allegiance with Humpty Dumpty, that when he uses words they mean precisely what his logical theory says they do).

4.2.2. Why not allow m > 1? commitments concerning other's

commitments. Why not allow the case m > 1? Suppose that we know that ~ is committed to A -+ B and that c~ is committed to/3 's being committed to 7's being committed to A. (In symbols: C~(A ~ B) and C~C~C~A). For simplicity, let us assume that o?s commitments are explicit and conscious; that is, a thinks that A ~ B and thinks that/3 thinks (or is committed to the claim) that 7 is committed to A. Now it certainly does not follow that ~ thinks (or is committed to the claim)


that fl believes A ~ B; c~ is committed to attributing commitments to /3 on the basis of A ~ B, but that does not imply that/3 himself agrees with c~'s inferential view.

Let us suppose, for concreteness, that c~ believes quite explicitly that

fl does not believe A --* B. Now in this situation, we know that believes that fl is committed to 3`'s being committed to A. The question that arises is whether fl's implicit attributions of commitments (according to c~) should follow from fl's other attributed commitments according to (what ~ takes to be) true implicational facts or according to (what ~ takes to be) the implicational claims that/3 is committed to.

We argued above, and it has been implicit throughout, that it is actual implicational facts that determine what follows from one's commitments. But when we consider not another 's actual commitments, but rather the commitments she is committed to attributing to others, it

seems quite natural to suppose that we should not take the implicit attributions to be determined by what is really the case inferentially, but by what we suppose is taken to be the case by the attributor.

After all, suppose that c~ is a logician who believes that material implication is the correct theory of commitment entailment, but o~ (correctly) takes it that/3 is a relevantist. Suppose further that A is P&~P, which is a claim that c~ believes that/3 takes to follow from 3`'s theory of class struggle. Then c~ will believe A ~ B for arbitrary B since this is a theorem of strict implication and also that fl is committed to 3`'s being committed to A. Should o~ then be committed to taking/3 to be committed to 3`'s being committed to B?

Maybe not. c~ does have to defend the claim that 3' really is committed to B, ifc~ agrees with/3 that 3 ̀is committed to A; but in determining what fl must attribute to 3`, c~ should give/3 the benefit of/3 's own logical theory, o~ may not agree with/3's logical views, but o~ should agree that since/3 holds them,/3 has the advantage of not having to attribute commitment B to 3'. That is, after all, one of the points of holding /3's theory, whether or not this motivation is sufficient to justify the theory.

4.3. The Four Systems

So we are considering four ranges for m in the MP rule. Each range

L O G I C A L S T R U C T U R E O F L I N G U I S T I C C O M M I T M E N T I 383

TABLE I.

Range of m Fitch-style Commitment Modal Axiomatization System Logic Logic

m = 1 fC1 C1 K~& 1-5 and 8 -1 I m = 0 or 1 fC01 C01 T~& C1 + 6 m >_ 1 f C l + C I + K4~a C1 + 7 m _> 0 fCO+ CO+ S4~& C1 + 6 + 7

Axioms and rules for axiomatizations:

Basic Axioms: 1. A ~ A 2. A & B ~ A 3. A & B ~ B 4. (A --+ B) & (A --* C) --4 .A --+ (B 84 C) 5. (A --+ B) & (B --+ C) --+ (A --+ C)

Additional Axioms: 6 . ( A & ( A ~ B ) ) ~ B 7 . ( B - + C ) ~ . ( A ~ B ) ~ ( A ~ C )

Rules: 8. Modus Ponens (MP): 9. Conjunction:

10. Prefixing: 11. Relevance destroyer:

Comments:

from A and A ~ B to infer B from A and B to infer A & B from A -~ B to infer (C --~ A) --+ (C -* B) from A to infer B --~ A

(1) CO+ is Anderson and Belnap 1975's relevance logic E ~ a + rule 11. Part II shows that E ~ a results when we combine relevance indices with commitment prefixes, and when we allow the same scope, regarding commitment prefixes, in the use of MP as we do for CO+. Weaker relevant commitment systems result when we restrict MP as restricted here. In the present non-relevant context, the relevant destroyer makes perfect sense: if B is a theorem, then everyone is committed to it, so that anyone committed to A is committed to B.

(2) Rule 10 is redundant in CO+ and C I+ , and axiom 5 is redundant in CO+ (though either one o f axiom 5 or rule 10 is needed to formulate CO+). A referee tells us that rule 10 is redundant in all of our systems, but we do not know how to derive it. (More precisely, the referee tells us that rule 10 can be derived from axioms 1, 4 and 5 and rules 8, 9 and 12, together with the transitivity rule for entailment; but the transitivity rule can itself be derived from rules 8 and 9 and axiom 5).

(3) Al though MP is a rule of our axiomatic systems, we do not here endorse it in the broadest sense: we do not in general endorse the inference from A and 'anyone committed to A is committed to B' to B. It may, for example, be the case that A is true, B is false, and that everyone committed to A happens to be committed to B. What we do endorse is the inference from 'A is a theorem' and 'A ~ B is a theorem' to 'B is a theorem'. (See w on the distinction between truth and theoremhood.)


is associated with a Fitch-style natural deduction system. We identify the corresponding commitment logic with the set of theorems of the Fitch-style system. Each of these logics turns out to be the -*& fragment of a modal logic, and can be axiomatized in a natural way. Table I summarizes this information. The list of axioms and rules occurs after Table I. (The first column should explain the names chosen in the second and third columns.)

4.4. An Algebraic "Commitment Semantics"

Our informal "semantics" has relied on notions of persons' commitments to certain claims, including claims involving other persons' commitments. So we need to formalize not only the notion that c~

is committed to P, but that c~1 is committed to c~2 is committed to . . . c~n is committed to P. In such a case, we take the sequence

(O~1, O~2,... , OZn) to be committed to P, in symbols (oq, ~2, . . - O~n) ~ P. We also want to formalize the interaction between the semantic notion of commitment (~) and the syntactic notion of commitment entailment (4).

D E F I N I T I O N 1. Let CL be C1, C01, C I + or CO+. A CL-commitment-

model is an ordered pair M = (S, ~) where (1) S is a non-empty set (of persons, or other entities to whom we assign commitments); (2) pc_ S <~ x Form, where S. <~ is the set of finite sequences of members of S (including the empty sequence, t3) and where Form is the set of formulas; and (3) ~ satisfies the (i) and (ii) below for b in S <~ and A and B in Form: Here, bc = b concatenated with c, and ScL c S <~ depends on CL as in the following table. (We identify a E S with (a) E S <~, and we identify S with {(a) : a 6 S}.)

D E F I N I T I O N 2. Given an CL-commitment-model, M = (S, ~}, and a formula A, we say that M validates A(M ~ A) just in case

~ A. We say that a formula is CL-valid if it is validated by every CL-commitment-model.

Comments: (1) We have suggested a "commitment" reading of

'(Oq, O~2,... , O~n) ~ A' for non empty sequences (c~1, c~2,..., c~,),


CL: C1 CO1 CI+ CO+ so,: s s u {~) s <~ \{0} s <~

(i) b ~ (A & B) iff b ~ A and b ~ B; and (ii) b ~ (A ~ B) ifl" (Vc E ScL)(be ~ A ~ bc ~ B). Note: Given CL, ~ is uniquely determined by the values it takes on atomic formulas. So, given CL and given whether or not b ~ p for every atomic formula p, we can use (i) and (ii) as recursive definitions of validity for non-atomic formulas.

namely 'a t is committed to 'a2 is committed to . . . an is committed to

A" . . . ' . Possibly, 0 ~ A should not be given a commitment reading,

but rather that 0 ~ A be interpreted as asserting the truth of A. So

we get a point in S <~ which plays a special role. I f one thinks, on the other hand, that semantics is purely a matter of commitment and

entitlement - say with truth claims understood disquotationally - then

the entire model can be seen as determining the commitments that

a person must undertake. On this reading, 0 ~ A asserts that the

special person must, in virtue of the logical rules, be committed to A. (2) In C I + and C0+-commitment-models , for sequences b, c,

and formulas A, B, if b ~ (A --* B) then bc ~ (A ~ B). So according to both C I + and CO+, commitment to an entailment claim (though

not to every claim) carries with it commitment to the claim that everyone

is committed to that entailment claim. This observation brings into

sharper focus some of the discussion of w concerning allowing m > 1

in MP. In particular, allowing m > 1 in MP commits one to this: commitment to an entailment claim is, in part, commitment to the claim

that everyone is committed to that entailment claim (whether she knows it or not). This is not completely unreasonable: commitment to an

entailment claim is commitment to a claim concerning the underlying linguistic practices of the community; so it is plausible to take it that

members of the same linguistic community are committed to the same

such claims. Reconsider the example of w a believes that material

implication provides the correct account of committive entailment; a correctly believes that /3 is a relevantist about such matters; and a believes that /3 believes that 3' believes P &uP. Is a committed to "/3 is

committed to '3' is committed to B ' " for arbitrary B? Though we previously argued not, it is open for a to say that /3 is committed to

entailment claims which/3 fails to recognize, and, furthermore that/3 's


implicit commitments (even those regarding others' commitments) are determined by his explicit commitments and all true entailment claims (which a thinks are captured by material implication), whether or not/3 wishes to accept those claims. (In the end, one must do more work in spelling out the details of the entire account of language, and only in the context of such a theory can these issues of the content of a normative logic be settled.)

4.5. A classical logician's worry: have we correctly parsed 'anyone committed to A is committed to B'?

A classical logician might note that our Fitch-style natural deduction systems result from adding commitment prefixes to the Fitch-style system already in place for the ~ & fragment of intuitionism; that is, if we restate the rules without the commitment prefixes, we have a Fitch-style system from which we can derive all and only intuitionistic theorems. (See Anderson and Belnap 1975, p. 10.) So 'commitment to A is, in part, commitment to B', which is first parsed as 'anyone committed to A is committed to B', is ultimately parsed as '(Va)((a is committed to A) D (a is committed to B))' where D is intuitionistic. A classical logician might object that a better reading of 'anyone committed to A is committed to B' is as '(Va)((a is committed to A) D (a is committed to B))' where D is classical. How can we capture this classical intuition?

We specify an object-language with:

- propositional variables, P, Q, R, etc.; - D, & and parentheses;

- a universal quantifier, V; - variables, a,/3, -y, etc., ranging over persons; - and a commitment operator, com.

"D" is read classically. "(acomA)" is read as " a is committed to A". (A ~ B) is an abbreviation of (Va)((acomA) D (acomB)). The idea is this: if the "base" logic is to be classical rather than intuitionistic, then we ought to get reasonable logics of commitment entailment by taking our axioms to be the traditional axioms of classical logic, as well as


some special axioms for the corn operator. (Different logics would arise from different sets of corn axioms.) The resulting classically based commitment entailment logics would then be the ~ & fragments of the systems thus defined.

One's choice of corn axioms depends upon one's attitude concerning the upshot of a commitment claim. In w we considered four different such attitudes, and we reflected these differences in the scope allowed for m in the statement of MP. Here we attempt to reflect just the same attitudes and differences in attitude in slightly varying sets of corn

axioms. And so we produce four classically based systems, analogous to the four systems of w

One might expect these classically based systems to be strictly stronger than intuitionism. If they turn out not to be, the classical logician's objection loses its punch, and our rhetorical point is made.

We now specify four classically based systems. The axioms and rules are as follows:

Rules: 1. MP: from A and (A D B) to infer B 2. &I: from A and B to infer (A & B) 3. Generalization: from A to infer (Vo~)A 4. Commitment to theorems: from A to infer c~comA

Axioms;

0. any standard list of axioms for the D & V fragment of classical logic.

1. [(c~comA) & (c~com(a D B))] D c~comB;

2. (V/3)(o~comA) D (~com(V/3)A), where c~ and/3 are distinct variables;

3. ( (~comA) & (oecomB)(c~com(A & B)); 4. (c~com(A&B)) D ((c~comA) & (c~comB));

5. [(a --~ B) & (cqcom(~2eom. . . (CemeomA))...)] D (oq com(o~zcom . . . ((XmcomB) ) . . .).

The four systems are distinguished by the range allowed for m in axiom 5 : m = l ; m = 0 o r l ; m _ > l ; m > 0 . ( W h e r e m = 0 ,


(~lcom(o~zcom... (~mcomA))...) is just A.) Much to our delight, we have the following:

THEOREM: the ~ & fragments of these four systems are C1, C01, CI+ , and CO+ respectively.

Rhetorical point: we can read (A ~ B) in C1, C01, CI+ , and CO+, as ' (Va)((a is committed to A) D (a is committed to B))', remaining indifferent as to whether "D" is taken to be classical or intuitionistic. Note 1: Axioms 1 and 2 and Rule 4 may seem too strong, and one can imagine plausible weakenings. However, the stronger these axioms and rules, the stronger (given our theorem) our rhetorical point.

Note 2: It would be interesting to see the result of taking D to be something other than classical or intuitionistic implication - e.g.

relevant implication.

4.6. Non-relevant commitment entailment and modality

Anderson and Belnap 1975 propose to divide the force of an entailment claim into a model component and a relevance one. On the modal side, they take it that to claim that A entails B is partly to claim that B follows

from A necessarily, i.e. according to a system of alethic strict implication, in particular, $4. This is then combined, in a certain sense, with a system expressing relevance, in order to achieve a system of entailment.

Alethic necessity is beside the point when it comes to commitment entailment. Whether an additional assumption can reasonably be imported into a proof from a given premise has nothing to do with whether it is a necessary truth (say in $4). Nonetheless there is a sense in which the conceptual division of entailment into modality and relevance is vindicated by the present approach, since all of the systems of non-relevant commitment entailment tried so far are systems of strict implication.

This is a surprising and (at least aesthetically) pleasing result. There has, for some years, been a split in the community of logicians attracted to relevance logic. One the one hand there are those champions of the relatively strong systems such as E and R. These are also the systems admitting of the conceptual division into a system of strict


implication ($4) and a system of relevant implication (R). On the

other hand, there are the defenders of much weaker systems of relevance logic. These philosophers, largely the Australians, have investigated and sought to apply systems weaker than the relevance logic T; in particular much attention has been paid to the relevance logic DL, whose --+& fragment can be axiomatized just as our C1, with the

"relevance destroyer" rule removed. It is now possible to show that one can adopt the weaker Australian systems and still make use of the American distinction between a modal force and a relevant force.

(See Part II.)

5. T H E O R E M H O O D

In comment (3) of w above, we relied on the distinction between a sentence being true and a sentence being a theorem. Now, in some sense,

it is a perfectly straightforward matter to specify which, among the many true sentences, are theorems: the ones which can be derived using the correct commitment entailment system (whichever it is). But theoremhood has often been taken to be more significant that derivability in a formal system. Theoremhood has been associated with necessity, a priority, indubitability, and other philosophically significant

notions. Can we attach special significance to theoremhood in the present context?

One attempt is to read 'A is a theorem' as 'A is true (and everyone is committed to it?) by virtue of the meaning of its logical vocabulary (in particular '---')'. But consider 'Fido is a dog ~ Fido is a carnivore'. It can be argued that, although this isn't a theorem, it is true partly by virtue of the meaning of 7 . After all, the meaning of '--+' is intimately tied to its role; and its role is to make explicit the language.-language moves to which we are entitled. To make explicit our entitlement to make the move from 'Fido is a dog' to 'Fido is a mammal' is, then, to partially specify the meaning of ' ~ ' .

Here we are going beyond a purely inferentialist account of meaning; indeed, the moment we pointed out the significance of language-entrance and language-exit moves we went beyond a purely inferentialist account of meaning, since the meanings of terms are bound up with such moves as well as inferential (language-language)


moves. The previous paragraph suggests that the meaning of ' 4 ' transcends, to some extent, its inferential role, without suggesting that the meaning of '--+' transcends its role in the language game.

Consider the following four "moves":

1. from asserting 'Fido is a dog' to asserting 'Fido is a mammal';

2. from looking at a dog to asserting 'that is a dog'; 3. f r o m asserting '(A --- B) & (B ~ C)' to asserting

' (A --+ C)'; 4. from noting the appropriateness of move 1, to asserting

'Fido is a dog -~ Fido is a mammal'.

While the role of 'dog' in the language game is bound up with the appropriateness of the inferential language-language move 1, it is also bound up with the appropriateness of the non-inferential language- entrance mo;ce 2. Similarly, while the role of ' ~ ' is bound up with the appropriateness of the inferential language-language move 3, it is also bound up with the appropriateness of move 4. Though move 4 is, in some sense, a language-language move, it has much in common with move 2. In the same (vague and problematic) sense that move 2 "ties the term 'dog' to the world" and grounds its semantic content; move 4 "ties the term ' ~ ' to lower level practice" (as discussed in w and grounds its semantic content.

Theorems are not, then, sentences which are true by virtue of the meaning of their logical vocabulary. Rather they are sentences which are true by virtue of the inferential role of their logical vocabulary, a role that contributes to, but does not exhaust, their linguistic role. This is not meant to downplay the importance of formalizing this inferential role. This third-level practice allows us critically to approach the second level (and thus the first); it allows us to see what assertions are beyond first and second level reproach, and yet to critically consider them; and so it adds to an overall understanding of "the relationship between language and the world".

6. C O N C L U D I N G R E M A R K S

A proof theory is a formal theory of the inferential content of some


bit of vocabulary. A proof theory for entailment will specify the

inferential content of an entailment connective, but the content such a connective ought to be assigned depends on the theoretical role of that connective in some substantive theory. A proof theory for a conditional expressing commitment entailment is a theory which expresses the inferential content of a bit of vocabulary whose role is

to allow us normatively to appraise the semantic content of other

sen t ences .

Since this content is one that arises within the context of a linguistic

practice of giving and asking for reasons, it is essential to be clear on the role that inferential moves play in that practice. Since the theory is a normative one - the upshot of the justification of claims made using the inferential locution is a license to make certain inferential moves - our methodology must be to consider what proof theory we ought to endorse as governing practice in the underlying language game. It would be epistemically irresponsible to consider only the question of whether our proof theory captured the ways in which we do infer. Thus, our discussion has had two goals. Specifically, we have tried to make progress on the question of the inferential role of commitment

entailment claims as one important clarification of the preanalytic notion of entailment. More generally, we have exhibited arguments for and against systems, arguments which are of a certain type, namely, that which produces a content for sentences the role of which is to impose normative constraints upon underlying practices. Hopefully we can come to understand and appreciate the methodology of such theorizing more clearly by seeing how it works in this case. It is a

methodology that is applicable not only in logic, but, we would claim, in translation (see Lance and Hawthorne 1990), epistemology, political science, sociology, and - the genus that covers them all - normative theory.

7. T E C H N I C A L A P P E N D I X

For the purposes of this appendix, L ranges over K, T, K4, and $4, and where L is K(T, K4, S4)CL is CI(C01, CI+ , CO+). Except where otherwise noted, formulas A, B, etc., are taken to be in a language whose only connectives are ~ and &. Also, we abbreviate C1 �9 �9 C~A


as n:A. The results of the first three sections are:

w CL C_ L_~& w A ~ CL ~ A is not CL-valid w A is not CL-valid ~ A ~ L__+~

These results suffice to establish the claims of column 4 of Table I in w and the theorem of w In w we show that the logics can indeed be axiomatised as in column 5 of Table I in w In w we establish the theorem of w

7.1. CL C L~&.

Before we continue, we remind the reader of the Kripke semantics for modal logics. (See Hughes and Cresswell 1984.) A possible worm model structure (pwms) is an ordered triple W --- (W, w0, R) where W is a set (of "possible worlds"), w0 E W, and R is a relation (called an 'accessibility relation') on W. A possible worm model (pwm) is a four-tuple M = (W, w0, R, V), where (W, w0, R) is a model structure, and where V assigns to every atomic formula p a subset V(p) of W. For an object language whose only connectives are strict implication, 4 , and conjunction, &, the validation relation, ~, between possible worlds and formulas is defined as follows:

For atomic formulas, p, w ~ p iff w E V(p);

w ~ (A&B) iffw ~ A a n d w ~ B;

w ~ (A ~ B)iff (Vw' E W)((wRw' andw' ~ A) =~ w' ~ B).

Given a pwm M, we say M ~ A (M validates A) iff w0 ~ A. We say that every pwm is a K-pwm. We say that a pwm is a T-pwm

(K4-pwm; S4-pwm) iff the accessibility relation is reflexive (transitive; reflexive and transitive). We say that a formula A is L-valid iff it is validated by every L-pwm. We note that, for any formula A whose connectives are ~ and &, A is L-valid iff A E L--+8~.

We now show that CL C L_~. It is a corollary of the theorem below.


LEMMA. Every line of every Fitch-style deduction in CL is of the form n:A, for some formula A (where 0:A is just A). Also, a Fitch-style deduction ending in n:A must have at least n undischarged hypotheses.

T H E O R E M . Suppose

(1) there is a Fitch-style deduction in CL whose last line is n:A and whose first n undischarged hypotheses are 1 :B1, . . . , n:B~; and

(2) M = (W, w0, R, V) is an L-model.

Then (VWI,. . . , Wn E W){('7'i = l , . . . , n)[(wi_ 1Rwi) and (Wi ~ Bi)] ==~ w, ~ A } .

Proof By induction on the length of the Fitch-style deduction. I f the

last line was introduced by Hyp, Rep or Reit, the result follows almost

immediately. Similarly for &E and &I.

Suppose that the last line was introduced by CP. Then A is of

the form (C ~ D), and there is a Fitch-style deduction in CL

whose last line is n + 1 :D and whose first n + 1 undischarged

hypotheses are I :B1, . . . n:B~, n + I:C. We want to show

(VW1,..., Wn E W){Vi = l , . . . ,n)[(wi_lRwi)(wi ~ Bi)] ==~ w. (c D)}.

Suppose, then, that Wl , . . . , w, ~ W and (Vi = 1 , . . . ,n)[(wi-lRwi) and (wi ~ Bi)]. To show wn ~ (C ~ D), we suppose that w, Rw and that w ~ C, and we note that w ~ D, by the inductive hypothesis.

Suppose that the last line was introduced by MP. Then there are

two previous lines, m:(C ~ A) and n:C, where the first n undischarged

hypotheses are still I :B1, . . . n:B~, and where

(1) i f L = K then n = m + 1;

(2) i f L = T then n = m or m + 1;

(3) if L = K4 then n _> m + 1; (4) if L = $4} then n _> m.

Now, suppose that w l , . . . , w~ E W and (Vi = 1 , . . .n)[ (wi_lRwi) and (wi ~ Bi)]. Then, by the inductive hypothesis, Wm ~ (C ~ A) and w~ ~ C. To show that wn ~ A it suffices to show that wmRwn. I f n = m + 1, then wmRw~ follows from (Vi = 1 , . . . , n)[(wi_ 1Rwi). I f

n = m, then L = T or $4, and wmRwn follows from the reflexivity of R.


I f n > m + 1, then L = K4 or $4, and wmRwn follows from the transivity

of R and the fact that wmRwm+ 1 , . . . , wn-1Rwn �9 QED

7.2. A r CL ~ A is not CL-valid

We show this by constructing (for each CL) a canonical commitment-

model which validates all and only the theorems of CL.

D E F I N I T I O N . MCL = (S, ~) where S is the set of formulas and ~ is

defined on S as follows:

(A1 , . . . , An} ~ A iff A1 ---+ .A2. . . ~ .An --+ A E CL.

T H E O R E M . MCL & a CL-commitment-model.

Proof We must show that

(i) a ~ (A&B) iff a D A and a ~ B; and (ii) a ~ (A --+ B) iff (Vb E ScL)(ab ~ A =~ ab ~ B),

where SCL C_ S <~ depends on CL as in the following table. (We identify

a E S with (a} E S <~, and we identify S with {(a): a E S}.)

CL: C1 CO1 CI+ CO+ ScL: s s u {~} s<~\{~} s <~

So we must show that (i) A1 --~ .A2. . . ~ A~ ~ (A&B) E C L i f f

A1 -+ .A2 . . . --~ .An --~ A, and A1 --~ . A 2 . . . -~ .An --~ B E CL; and (ii) A1 ~ .A2. �9 ~ .An ~ (A -~ B) E C L i f f (VB1,. . . , B m, where m satisfies the restrictions below) (A1 --~ . A 2 . . . --+ .An ---+ B1 --~ . B 2 . . . -~

.Bin --+ A E CL ~ A1 -~ .A2. �9 -+ .An -~ B1 ~ .BE... --+ .Bin --~ B E CL). Restrictions on m:

CL: C1 C01 CI+ CO+ m: =1 = O o r l >1 _>0


First we note that the following rule is admissible into all four Fitch-style

natural deduction systems: From 0:(A1 ~ .A2. . . ~ .An --+ A), 1 :A1, 2:A2 . . . , and n:An to n:A.

(i) is easy to show. (ii) ( 3 ) is shown by appeal to MP. (ii) ( ~ ) is shown by letting k = 1

and B1 = A, and then by appeal to CP.

Comment: In w we promised some technical considerations concerning the C - C thesis. First, observe the following for any canonical

model, MCL. (Here, we assume the commitment reading of a ~ A, and we will write c~comA for ' a is committed to A'.)

(1) if acomA then acomacomA; (2) if acomacomA then acomA, if CL is either C01 or CO+; (3) if/3comacomacomA then 13comacomA, if CL is either C01

or CO+.

However the following fails, if CL is CO1 or CI:

(4) if f3comacomA then/3comacomo~comA.

So what? (1) seems to tell us that one direction of the C - C thesis is validated by every model, and (2) seems to tell us that the entire C - C thesis is validated by just the right canonical models: those associated with allowing m = 0 in the MP rule. So it seems that the C - C thesis is

supported by these canonical models. However, if the C - C thesis is sufficiently fundamental to motivate one logic rather than another, one should expect (3) and (4) to hold for models corresponding to logics allowing m = 0. But (4) fails, as noted, for C01. And so, if we take the canonical models philosophically seriously, C01 seems to undercut one

of its primary philosophical motivations. The reason that (4) fails for C01's canonical model is that the following is not a theorem of C01: (A ~ B) ~ (A --+ (A ~ B)). So we might consider adding it, to get yet another logic of non-relevant commitment entailment.

7.3. A is not CL-valid ~ A r L ~ a

We show this by defining, for every Cl-commitment-model, M, an L-pwm, M ~, which validates exactly the same sentences as M.


DEFINITION. Suppose that M = {S, D) is a Cl-commitment-model.

M' = (W, w0, R, V) where (i) W = S<~;

(ii) w0 = the empty sequence, 9;

(iii) R = {(a, ab): a E S <~ and b E ScL}; and (iv) V(p) = {a E W: a ~p} .

7.4. Axiomatisations

So far we have shown that CL = L~e . For the purpose of this section, we define aCL to be the set of sentences containing the axioms given for CL and closed under the rules given for CL in w We now show that CL = aCL.

It is straightforward to show that each axiom of aCL is in CL or alternatively, L~&, simply by providing appropriate Fitch-style deductions. To show that L ~ e admits all of the rules of aCL proceed as follows:

(1) note that a formula is validated by every L-pwm just in case it is validated by every worm of every L-pwm;

(2) show that the rules preserve validity at every point in every

L-pwm. This shows that aCL c_ CL. That CL c_ aCL is a corollary to Theorem 1.

T H E O R E M 1. I f there is a Fitch-style deduction in CL whose last line is n:A and whose first n undischarged hypotheses are I :B1,. . . n:Bn, then the following is in aCL:

B1 ---+ .B2 --+ . . . - + .Bn ---+ A .

Theorem 1 is a corollary to this Lemma.

LEMMA. The following rule is admissible in aCL:

1. from A to infer (B1 ~ .B2 --+ . . . --+ .Bn --+ A) and the following are theorems ofaCL:


.

.

4.

5.

[(B1 --~ .B2

(B1 -4 .B2 -4

(BI -4 .B2 --+

(B1 -4 .B2 --~

(B1 --+ .B2 -4

(B1 -4 .B2 --4

�9 . -4 .Bn ~ A) &

�9 . -4 .Bn -4 B)]

. . - 4 .Bn ~ (A & B))

. . -4 .B n -4 ( a & B)) -4

�9 . - 4 .Bn -4 A )

. . - 4 .B n -4 (A & B)) -4

(B~ -~ .B2 -4 .. -4 .Bn ~ B). [(As -4 .A2 - 4 . . . - 4 .An -4 (A ~ B)) &

( A l - + . A 2 - 4 . . . -4 .An--+.B1 --+ .B2 -4 . . . -4 .Bin -4 A)]

(A1 -4 .A2 -4 . . . - 4 .,A n ~ .BI -4 B2 -4 . . . ---, .Bin -4 B),

where m is restricted as in the following table:

aCL: aC1 aC01 aCl+ aC0+ m: =1 = 0 o r l >1 >0.

7.5. Classically based systems (as in w

Le t us n a m e the f o u r sys tems de f ined in w c o r r e s p o n d i n g to the f o u r

res t r i c t ions on the r ange o f m in a x i o m 4 as in the f o l l o w i n g tab le (el is fo r

' c lass ical ' ) :

m: = l = 0 o r l >1 > I system: CIC1 CIC01 clC1 + clC0+

W e n o w es tab l i sh the t h e o r e m o f w by s h o w i n g tha t c l C L ~ , = C L for

C L = C1, C01, C I + or CO+. T h a t C L C elCL_+8, can be s h o w n by

de r iv ing each a x i o m o f C L in e l C L , a n d by s h o w i n g each rule to be

admis s ib l e in e lCL. T h a t cCICL_~8, _c C L is a c o r o l l a r y to T h e o r e m 2,

be low.

D E F I N I T I O N I. A cl-commitment-model is an o r d e r e d pa i r M = (S, I)

w h e r e (1) S is a n o n - e m p t y set ( o f pe r sons , o r o t h e r ent i t ies to w h o m we


assign commitments); (2) I is a function which assigns a member of S to each variable, and a subset of S <~ to each atomic sentence.

DEFINITION 2. Given a model M = (S, I) a variable c~ and an s c S, I[s/c~] is the function just like i except that is assigns s to a; and

= ( S ,

DEFINITION 3. Given a model M = (S, I) and a logic elCL, the relation ~M,cLCL between members of S <~: and formulas is defined recursively as follows: (Here, for b and c in S <~, bc = b concatenated with c)

(i) b ~M,clcL(A&B) iff b ~M,clcLA and b ~M,clCLB; (ii) b ~M,clCL (A D B) iff (b ~M,elCL A ~ b ~M,elCL B); (iii) b ~M,dCL (Vc~)A iff (Vc E Sctci~)(bc ~Mts/~1,aCL A, where s is the

last member of the sequence bc); (iv) b ~M,clCL (c~cornA) iff b(I(a)) ~M,clCL A.

Here SdCL C_ S <~ depends on elCL as in the following table:

clCL: C1 C01 CI+ CO+ So~CL: S S U {0} S\{O} S

Comment (1): Since, for the purposes of the present subsection, (A ~ B) is an abbreviation of (Vo~)((aeomA) D (o~comB)), we do not have to provide a separate clause telling us how to evaluate (A ~ B) in the models. We can, however, derive the following clause:

b ~4,~iCL (A ~ B) iff (Vc E SclCL ) (bc ~M[s/a],clCL A ~ bc ~M[s/a],aCL A, where s is the last member of the sequence bc)

Comment (2): Clause (iii) in the above definition seems wrong, since we quantify over all c in SclCL even though only one "person" a is involved in (Vc~)A. While this does present a problem of philosophical interpretation, our purpose in giving the definitions and theorems of w is to establish the main theorem of w And so the definitions of w can be seen as purely technical devices. It would be nice to have a more philosophically interesting semantics for dCL, and we may be in its neighbourhood.


DEFINITION 4. Given an el-commitment-model, M = (S, I), a logic elCL and a formula A, we say that M elCL-validates A (M ~clCL A) just in case ~ ~M,dCn A. We say that a formula is elCL-valid if it is elCL-validated by every el-commitment-model.

THEOREM: 1. I rA E elCL then A is elCL-valid.

THEOREM 2. Suppose that M = (S, ~) is a CL-commitment-model, and that M'(S, I}where, for atomic formulas P, I(P) = {b E S: b ~ P}. Then, for all formulas A whose only connectives are ~ and &, and all b E S,

b ~ A / f f b ~M';dCL A.

A C K N O W L E D G E M E N T

We would like to thank Stephen Kuhn and Nuel Belnap for helpful comments on an earlier version of this paper. A referee at the JPL corrected an error in an even earlier version.

R E F E R E N C E S

Anderson, Alan Ross, and Nuet Dinsmore Belnap Jr. 1975. Entailment: The Logic of Relevance and Necessity, Volume I. Princeton University Press, Princeton NJ.

Brandom, Robert 1983. "Asserting". Nous 17, 637-650. Brandom, Robert 1985. "Varieties of Understanding", in Rescher 1985, 27-51. Fitch, Frederic Brenton 1952. Symbolic Logic: An Introduction; Ronald Press Co., New

York. Hughes G.E., and M.J. Cresswell 1984. A Companion to Modal Logic. Methuen, London. Lance, Mark 1993. "Two Concepts of Entailment". Unpublished, lead symposium paper,

American Philosophical Association Eastern meetings. Lance, Mark and John Hawthorne 1990. "From a Normative Point of View". Pacific

Philosophical Quarterly 71, 28-47. Lance, Mark and Philip Kremer 1994. "The Logical Structure of Linguistic Commitment

II: Systems of Relevant Commitment Entailment". Forthcoming, this Journal. Rescher, Nicholas 1985 (ed). Reason and Rationality in Natural Science: A Group of Essays.

University Press of America, Lanham MD. Routley, Richard and Val Plumwood, and Robert K. Meyer, and Ross T. Brady 1982.

Relevant Logics and Their Rivals, Part I. Ridgeview Publishing Co., Atascadero CA. Sellars, Wilfrid 1963. Science, Perception and Reality. Humanities Press, New York.

4 0 0 M A R K N O R R I S L A N C E A N D P H I L I P K R E M E R

Mark Lance, Georgetown University, Washington, DC 20057, U.S.A.

and

Philip Kremer, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.

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