On the concept of Material Consequence

true and the conclusion false. There are, of course, those who repudiate this con-

strual (e.g., Anderson and Belnap 1975), but let us set their misgivings aside for the

present and examine what can be accomplished within classical confines. A traditional

manner of deepening the classical account appeals to truth-preservation, as evidenced

in the writings of Bernard Bolzano where deducibility is characterized in terms of truth-

preservation throughout uniform substitutions upon certain 'variable ideas' occurring

in the propositions so related.².   More recently, it has been customary to express this

analysis linguistically by speaking of sentences and distinguishing between logical and

extra-logical constants, or, to use Quine's terminology, between lexical items and

logical particles. With this, the focus is apparently shifted from talk about validityper

se to talk about logical consequence. One version of the analysis, then, is as follows:
 

(LSV)  A sentence B is a logical consequence of sentences A1, . . .An if and

only if whereA1', . . . An', B' are the results of any uniform substitutions

upon all the extra-logical constants occurring in A1, ...,An ,B, then B'

is true if A1', . . .An' are all true.³
 

Naturally, a criterion of this sort relies on a suitable characterization of 'uniform

substitution'. In particular, the substituends must be appropriate in the sense that

extra-logical constants of a given category can only be replaced by terms of the same

category; that is, singular expressions for singular expressions, predicates for predi-

cates, and so forth.⁴  The main thrust of (LSV) can be put somewhat differently. In

explicating a relationship of logical consequence it is only the logical constants, or the

operations expressed by such constants, that occur essentially or that carry implica-

tional burden in the sentences so related. The occurrence of any particular extra-

logical constant is inessential in that it, or the content it expresses, may be freely sub-

stituted for salva consequentia. Its occurrence is, in other words, idle so far as logical

consequence is concerned.

    Correlated to the substitutional approach is a charaterization that appeals directly

to the notion of logical form. Let us say that a valid logical form is one such that

every statement (sentence, proposition) having it is true. Intuitively, a valid form is

depictable by a matrix or schema containing only logical constants and having only

true substitution instances, e.g., 'p  p'. Then, exploiting the equivalence between

logical consequence and the logical truth of a conditional, we have
 

(LFV)  A statement B is a logical consequence of statements A1, . . . An if and

only if the statement "(A1 & . . . & An)  B" has a valid logical form.⁵
 

This criterion has one advantage over (LSV); if a "statement" is a non-linguistic item

possessing logical structure, then we have a construal of logical consequence that

does not seem dependent upon language in the way that (LSV) is. In this light one

might suppose that (LFV) affords a deeper analysis of logical consequence that

confers legitimacy upon the substitutional account (LSV).⁶

    The substitutional approach is often contrasted with an alternative explication in

terms of models and satisfaction. Tarski, in favoring the latter, felt that the substi-

tutional analysis supplied only a necessary condition for logical consequence if the

language in question fails to possess a 'sufficient stock of extra-logical constants'.

(Tarski 1956, 415-416). Quine, however, has argued that the two approaches are

equivalent if the language is 'reasonably rich: rich enough for elementary number

theory' (Quine 1970, 53). If the latter is correct, then (LSV) would require slight

modification, though the same need not be so as concerns (LFV) for the reasons

given. It is interesting to observe that Tarski, while rejecting the substitutional

analysis, retained the idea that formal consequence is a relation 'uniquely determined

by the form of the sentences between which it holds'.⁷
 
 
 

3. Paraphrase and the boundaries of logic

logical consequence, so construed, exhausts our intuitive notion of validity. It is

necessary to evaluate the claim by means of examples provided by natural and scien-

tific languages, examples that are not prejudged through reformulation in preferred

linguistic structures. An antecedent investigation of natural language arguments is

required, and if these cannot be brought into conformity with the criteria by means of

grammatical, though otherwise unfettered, paraphrase then it is unlikely that we can

do better with an artificial language.

    At first blush, natural languages appear to contain an overwhelming host of

counterexamples; for instance, the argument from 'he is a bachelor' to 'he is

unmarried', an argument as solidly entrenched as any modus ponens inference, is

not, at first glance, logically valid. It is commonly believed, however, that terms like

'bachelor' conceal logical complexity, that is, such lexical items are decomposible

into linguistic units that reveal underlying logical structure. Thus, 'is a bachelor', we

learn, means 'is a male and is unmarried' so that the argument actually has the valid

logical form
 

'fx & gx, therefore, gx'.
 

Similarly, 'collie' or 'is a collie' in (Al) is decomposible into a conjunction of predi-

cates one of which is 'is a dog'; hence this argument, too, can be viewed as logically

valid. An analogous account could be given of the argument from, 'this is a collie' to

'this has a pointed snout', where having a pointed snout is a differential property of

the species in question.

    Other cases, while invoking paraphrase, call into question the precise domain of

logic. A case in point is mathematics. Assume that some theory framed in set-theo-

retical terms is capable of generating a desired body of mathematical truths; are the

constants of this theory logical constants, and are its principles logical laws? This is,

as we all know, a matter of debate. Equally problematic are the modalities. Few

would question the validity of arguments taking the form 'necessarily p,therefore,

possibly p', but are such arguments logically valid? Perhaps, if the modalities are

themselves logical modalities, but suppose that they are physical or causal modali-

ties? Again, consider the argument
 

(A3) John knows that he knows that he is handsome 

        therefore, John knows that he is handsome,
 

where 'he' is used to attribute first-person reference to John.⁸  The argument is

credible enough, yet the implicational burden is carried by 'knows'; is this term, too,

a logical constant? Are the theses of epistemic or doxastic "logic" to be classified as

logical truths? The same question can be raised concerning so-called 'mereological'

predicate logic by whose means some (e.g., Massey 1976) would bring arguments like

(A2) within the scope of logic.

    The ensuing controversy is widespread and, becoming one with the issue of distin-

guishing logical from extra-logical constants, difficult to resolve.g Where the line is

drawn is not crucial for present purposes, but it is important that the matter not be

taken lightly. Each adjustment of the boundaries of logic theatens the autonomy of

the discipline; as symbolic techniques develop there is a growing threat that what is

called logic' will usurp metaphysics, epistemology, etc., and encroach upon the

overlapping disciplines. The point, of course, is not to disparage symbolism, but only

to stress that its mere introduction into a given field of study does not imply that its

users are suddenly engaged in logical investigations. Caution must be urged in the

application of the terms logic', logical' and logically' if the domain of logic is not

to become too diffuse.

    Though the boundaries of logic may be unsettled, it is illegitimate to include

within them the subject-matter of the physical or social sciences. To be sure, much of

the reasoning that occurs within the latter is rightly classified as purely logical or

mathematical, but occasionally a given formula or law seems to relate to a given

inference as a rule, or as a rule-legitimating principle, than as a premise. The latter is

a common practice in pure mathematics, but it also occurs in the physical sciences as

when a theoretician concludes E = T + m₀c²  from mc²   = T + m₀c²  by  using the

well-known 'E = mc²' as a rule-legitimating principle and not as a premise. The said

"rule" may never be articulated, though if it were it might read something like
 

from f(E) = y  infer  f(mc² )=y,
 

where 'f'  is a schema for any functional expression. Lack of articulation, of course,

does not mean that the rule is not operative in the inference. To be sure, we cannot be

certain of what goes on in the minds of mathematicians and physicists when they

draw conclusions, but if expressed inferences are any guide then it appears that we

have unearthed a class of arguments that are not logically valid as they stand and, yet,

are plausible in their own right.

    Similar remarks pertain to arguments involving predicates which do not seem

decomposible, that is, those that express apparently simple properties, as in,
 

(A4) this is scarlet 

        therefore, this is extended,
 

though, admittedly, it is no easy matter to determine which properties are simple and

which are complex. Additionally, while the devices of decomposition and paraphrase

might bring inferences that move from determinates of species to differentia or deter-

minables, e.g., (Al), into accord with (LSV) and (LFV), there are also those that go

from genus or determinable to a disjunction of species or determinates, for instance,

from 'this is a human' to 'this is a male or this is a female'. The latter becomes logic-

ally valid if determinable properties simply are disjunctions of determinates, but

there is no way, short of metaphysical perversity, that we can simultaneously analyze

determinables into disjunctions of determinates and determinates into conjunctions

of their differentia and determinables. So, if logical validity is conferred upon (Al) it

must be withheld from the argument that moves from 'this is a dog' to the disjunction

'this is a collie or this is a terrier or this is a dachshund or . . . '--provided that such a

disjunction can exist. While some technique of regimentation is suitable to each type

ofdeterminable-determinate-differentia argument, there is no one technique suitable

for all.

    Neither the appeal to paraphrase nor the practice of extending the boundaries of

logic is sufficient to render all credible arguments logically valid by means of the

criteria (LSV) or (LFV). A third approach insists that the aberrant cases are not valid

at all, and to this we now turn.
 
 

4. Enthymemes

ing it; an enthymeme, so goes the story, is, by that very fact, invalid as it stands, even

though a 'complete' valid argument simmers underneath. Perhaps a word or two

about enthymemes will shake the confidence of those who take refuge in this strategy. It

must be asked; what is an enthymeme? Traditionally, the notion has been relativized to

syllogistic logic; an enthymeme is a syllogism with a missing or suppressed premise. As

logic outstriped the syllogism, however, this definition became too restrictive. An alter-

native is to characterize an enthymeme as any argument with a suppressed premise

(Haack 1978, 245). But this definition is too broad for the approach in question.

Consider someone who expresses the following inference:

 
(A5) I am hungry

        therefore, either I am hungry or my brother is hungry,

and yet who believes that the conclusion follows only if the premise is supplemented

by the unexpressed conditional 'if I am hungry then either I am hungry or my brother

is hungry'. The latter qualifies as a suppressed premise of the argument, thereby

rendering the argument (A5) enthymematic, at least for that person, but those famil-

iar with truth-functions would be unwilling to conclude that the argument is, thus,

invalid as it stands (assuming that the disjunction is inclusive). To salvage the strategy

the definition can again be relativized, not to syllogisms, but to any reliable system of

logical deduction. Quine, for example, writes that an enthymeme is ' . . .a logical

inference in which one or more of the premises are omitted from mention on the

ground that their truth is common knowledge and goes without saying' (Quine 1974,

169).¹⁰  Presumably, as a logical inference, that is, a logically valid argument (if this is

the correct reading of Quine), is not an enthymeme if the expressed premises logically

imply the conclusion, no matter what other premises lurk underground; with this

characterization the strategy is resurrected.

    The notion of a 'suppressed premise' is less innocent than might first appear.

Speaking loosely, a suppressed premise would seem to be "in the mind", unex-

pressed, though in some way operative in the psychological process constituting the

inference. It should not be assumed, however, that every inference is actually

expressed or that each so-called 'enthymeme' is a truncated expression of some

complete argument in the mind. Perhaps a given episode of reasoning is itself

enthymematic, in which case the idea of a suppressed premise becomes more prob-

lematic. Every inference requires some rule or leading principle' (to use Peirce's

terminology) which is not included among the premises of the involved argument.

How does one know that a suppressed statement--for instance, in the case of (Al),

'every collie is a dog' --is functioning as a premise rather than as a rule, or, more

exactly, as a rule-legitimating principle? Undoubtedly, the issue merits close

psychological scrutiny, but it is significant that some investigators have concluded

that most everyday inferences cannot be modeled on extant logical calculi, and not all

inferential rules are appropriately characterized as "logical" in any standard sense,

and that content, in addition to logical structure, is crucial in accounting for de facto

human reasoning.¹¹  Their findings suggest that the principles relating material

contents enter into inferences as rule-legitimating principles and not as premises. As a

consequence, it is presumptuous to hold that a person who sets forth what seems to be

a plausible argument, but not logically valid as expressed, has some completed logical

argument in mind.

    It can be claimed that such psychological considerations are irrelevant to the

objective matter of argument validity. To some extent this is correct, but it must be

remembered that an appeal to suppressed premises, ipso facto, lifts the discussion

onto the psychological plane. It is best, perhaps, to drop the term 'suppressed' and

define an enthymerne as an incomplete argument which when supplemented by an

additional premise(s) becomes logically valid. Of course, such talk is elliptical; the

original argument does not "become" valid; instead, one locates a further, though

correlated, argument and recognizes it to be valid. The definition is peculiar,

however, for now any invalid argument is an enthymeme no matter how outrageous.

Inferences like (Al) and (A2) are no more and no less enthymematic than the

reasoning of the hillbilly who concludes that his neighbor Ivan is a communist upon

learning that Ivan hails from Russia. Insisting that the added premise be true

improves matters little since truths which when combined with premises logically

imply a conclusion are plentiful. Any bad argument with a false premise, for

instance, 'London is in France, therefore, Paris is in Mexico', can be transformed

into a valid argument upon supplementation with a material conditional whose ante-

cedent is a conjunction of premises and whose consequent is the conclusion. To leave

matters at this point is to fail to do justice to an important aspect of many so-called

enthymemes, namely, their apparent credibility as theystand. Arguments like (Al)

and (A2) will not be objected to as readily as those offered by the hillbilly or by the

student who flunked both geography and logic, at least not by any reasonably in-

formed and intelligent person. The weakness of the strategy of resorting to enthy-

memes is obvious; it simply fails to account for the initial credibility that arguments

like (Al) and (A2) have and that flagrant non sequiturs lack.

    To acknowledge that some inferences are not logical is to admit that reasoning

occasionally proceeds by way of material inference rules, that is, by what Carnap

once called 'P-rules' (Carnap 1937). One attempt to discredit extra-logical or

material validity reasons that since inferences can be carried out by means of logical

rules alone, together with an increased stock of premises, material rules are dispensible

and, consequently, there is no need to countenance any brand of argument credibility

beyond that of logical validity. Though such reform of the ways we actually do reason is

possible it should be observed that logical rules can similarly be dispensed with by

restricting all inferences to a system characterized by material rules alone. Logically

valid arguments will be enthymematic with respect to such a system, illustrating that

being enthymematic is a relational property of virtually every argument vis-a-vis some

inferential system or other. It would be extreme, naturally, to conclude that we have

thereby disposed of or reduced logical validity. In sum, dispensability is no more a guar-

antee of reducibility than being enthymematic is of invalidity.

    Related to these positions is a strategy of Carnap's that appeals to 'meaning

postulates', but as this approach has been effectively criticized elsewhere discussion

of it will be omitted here.¹²  Without doubt, the understanding of any expressed infer-

ence depends upon knowing the meanings of certain terms used in its formulation.

But this should not be taken to imply that material inferences, as expressed, pre-

suppose 'definitions' of extra-logical constants any morethan logically valid infer-

ences, as expressed, presuppose definitions of logical constants. In this regard,

material and logical inferences are on equal footing.

    There is a long-standing correlation between validity and implication, and also

between implication and the necessity of a conditional. The attempts to dispense with

material validity and to withhold legitimacy from the arguments mentioned above

effectively deny that there is such a thing as extra-logical implication or necessity.

This is a controversial position, to be sure, but it has the tenebrous result of coralling

many of the common inferences of scientific and everyday thinking into the bin of

invalidity. The important distinctions that would be lost through such a move are

considerable, and this provides some rationale for examining approaches that take

seriously an irreducible relation of material consequence.
 
 

5. Material validity

credibility upon content. Unfortunately, the second conjunct of this claim is rarely

developed, and to leave things at this stage is inadequate for the concerns of logical

theory; principles characterizing material consequence are highly desired, if not

demanded, as is an explanation why there should be this curious hiatus in the first

place. If validity, whether logical or material, just is the impossibility of a false

conclusion given true premises, as the classical account would have us believe, and if

the appeal to substitution and form is a way of explicating this conception for logical

validity, then why should not an analogous device work for material validity?

    It is significant that the substitutional approach admits of a natural extension;

while certain extra-logical constants are idle, so far as implication is concerned, and

may be freely replaced, others bear implicational burden and must be regarded as

occurring essentially. One criterion that nicely parallels (LSV) is as follows:
 

(MSV.1) A sentence B is a material consequence of sentences A1,. . . , An  if and

only if   (1) B is not a logical consequence of A1, . . . , An and (2) where

A1', . . . , An', B' are the results of any uniform substitution upon some

of the extra-logical constants occurring in A1, . . . An, B, then B'  is

true if A1', . . . , An' are all true.

The occurrence of 'some' in the second clause must be interpreted as having largest

scope otherwise we fail to obtain a sufficient condition for material consequence,

that is, (2) is to be read as saying that some of the extra-logical constants occurring in

AI,  .  . . ,  An, B are such that any uniform substitution upon those constants is truth-

preserving. It is clear how this criterion can then handle cases like (A1)-(A4); in both

(Al) and (A4) the indicator 'this' is idle and can be freely substituted for salva conse-

quentia, while in (A2) and (A3) the occurrences of proper names are inessential. In

(A3), however, the first occurrence of 'he' must also be replaced by pronouns appro-

priate to whatever replaces 'John', though not necessarily the second, since 'he is

handsome' is itself, as a whole, replaceable.

    Unfortunately, (MSV. 1) proves to be too liberal; if we choose our subset of extra-

logical constants in such a way that none occurs in the conclusion then any argument

will be valid so long as the conclusion is true. Some adjustment is needed, and it is

helpful to look at a definition offered in George 1972 of what he calls 'enthymematic

consequence', where some sharing of constants by premises and conclusion is

required. Paraphrasing George, the criterion is:
 

(MSV.2) A sentence B is a material consequence of sentences A1, . . . , An if and

only if   (1) B is not a logical consequence of A1, . . . ,An and (2) there is a

set S of extra-logical constants occurring in A1, . . ., An, B some

members of which occur both in B and in one or more of A1, . . . , An

such that if A1', . . . An', B'  are the results of any uniform substitution

upon the members of S then B'  is true if  A1', . . ., An' are all true.¹³
 

and (MSV.2) is able to account for the validity of each while circumventing the prob-

lem affecting (MSV.1). Since many plausible arguments from natural language do

not exhibit such constant-sharing, however, there is still a need for regimentation.

There are no shared extra-logical constants in the arguments 'the tallest man is a

bachelor, therefore, some mammal is unmarried', and no constants appear idle, but

the alleged decomposibility of 'bachelor' suffices to bring the argument into line with

the new criterion. George cites the example of 'it rains, therefore, the streets are wet',

and suggests that it is to be paraphrased as 'it rains at t, therefore, the streets are wet

at t ', where 't'  is a schema for a replaceable temporal (and spatial) constant (George

1972, 115). Any substitution upon instances of this schema is truth-preserving. A

similar procedure may be followed in accommodating an inference taken from rela-

tivistic mechanics where the mathematical formalism conceals implicit constants (or

schemata):
 

(A6) E = mc²

Therefore, T + E₀ =₀^s Fds+ m₀  ×   ½600,000k/sec.

This inference is materially valid given E = T+E₀ and
 

mc²  =₀^s Fds+ m₀   ×  ½ 600,000k/sec,

assumed within the theory, as principles legitimating material rules of inference. On

the face of it, this argument exhibits no shared constants, but this is remedied when it

becomes evident that expressions like 'E' and 'm₀' are elliptical for 'the energy of x'

and 'the initial mass of x', where 'x' is a schema for constants designating physical

objects. Paraphrase frequently brings such implicit constants into the open,

illustrating that a wide variety of arguments can, with some plausibility, be brought

into conformity with (MSV.2).¹⁴

    An interpretation of material validity in terms of form might seem unlikely; after

all, does not the essential occurrence of extra-logical constants indicate that material

consequence is determined by content and not form? Does the parallel between

substitutivity and form break down at this point? The answer to these questions

depends upon what exactly & form of a statement is, and this is an issue that has

received comparatively little philosophical attention. Traditionally, the primary

linguistic access to logical forms has been by way of open sentences or schemata in

which some variables occur freely and the only constants are logical constants. One

extension of this representational device is to allow the occurrence of extra-logical

constants in form-specifying matrices, retaining free occurrences of certain variables,

so that 'x is extended', say, specifies a material form just as 'x is f' represents a logical

form. Accordingly, it would be the presence of some free variables or other rather than

any particular constants that indicates form, and it is worth observing that some writers

appear to be operating with just some such assumption.¹⁵  Stipulations can be intro-

duced that would forbid the employment of 'form 'in this fashion, but the real question

is whether the very grounds upon which logical forms are posited, namely, to deepen the

classical account of validity or to provide it with additional content, might not also lend

credence to the view that the class of forms is broader than is usually thought. Any

mystery surrounding this idea is dispelled if forms, in general, are construed as

statement types relevant to the analysis of implicational relationships and to the

classification of statements into various categories. The advantage is a uniform treat-

ment of validity and the sustention of the correlation between form and substitutivity.

    This in mind, an analogue of (LFV) can be proposed:
 

(MFV) A statement B is a material consequence of statements A1, . . . , An  if and

only if (1) B is not a logical consequence of  A1, . . . , An and (2) the statement

'(A1 & . . . & An)   B'  has a valid material form.

With the broadened notion of "form" it is easy enough to locate validating forms for

arguments like (A1)-(A4); that of (A4), for instance, is represented by the

conditional matrix
 

x is scarlet  x is extended,

boundaries of logic, also submit to a ready analysis; trivially, the argument
 

7>5,5>3, therefore, 7>3

is validated by the form specified by
 

(x>y & y>z)   x>z.

A validating form can even be found for physical inferences like (A6), specifiable by

a matrix in which a free variable replaces the implicit constant.¹⁶

    Many questions arise at this point. What sorts of constants can occur in matrices

or open sentences that are allowed to specify material forms? Does every open

sentence determine some such form? What becomes of the traditional form-content

distinction once a criterion like (MFV) is permitted?¹⁷   What is a form anyway? Does

the posit of material form prove to be of any value in the treatment of material conse-

quence? This latter question, in particular, must be answered affirmatively if the

notion of material form can be taken seriously by logical theory.
 
 

6. Some difficult cases

for which no intuitive paraphrase is of any help. The first of these are arguments in

which every extra-logical constant appears to occur essentially so far as the validity of

the arguments is concerned, while the second involves cases in which, though not

every constant occurs essentially, truth is not preserved throughout all uniform

substitutions. Let us examine each type of case in turn.

    From 'Nixon is angry* it follows logically that some individual is angry, and it

follows materially that Nixon is an emotive state, assuming that 'Nixon' is a singular

expression designating some individual person. It would seem that 'some individual

is in an emotive state' is a material consequence of 'Nixon is angry'. Perhaps by

introducing a shared temporal parameter, a la George, this argument can be

adequately handled by (MSV.2). Consider, however, the following:
 

(A7) Nixon is angry

        therefore, some individual is, at some time or other, in an emotive state.
 

sense that each extra-logical constant in the premise expresses an element that is

generalized with respect to in the conclusion and no such constant is shared. The

generalization with respect to time suggests that the introduction of a temporal para-

meter is of little aid. On the other hand, it could be argued that a proper name like

'Nixon' conveys content, e.g., being an individual, that must be brought out in

canonical paraphrase. In a given context, for example, the description 'the individual

who was President of the U.S.A. in 1970' might denote exactly what 'Nixon' does,

and in this case 'individual' is itself a replaceable constant.

    There is a sizeable controversy over this treatment of proper names, however, and

its introduction brings with it a theoretical price. It is even less obvious that the first-

person indicator 1' can be replaced by descriptions in such a fashion. Substituting 1'

for 'Nixon' and generalizing to 'person' rather than to 'individual' yields an

argument that constitutes an even stronger challenge to (MSV.2):
 

(A8) I am angry

        therefore, some person is, at some time or other, in an emotive state.

But nothing should be thought to turn on the presence of singular expressions,

whether proper names, descriptions or indexicals, as evidenced by the following

argument:
 

(A9) some collies are brown

        therefore, some dogs are colored,

where both premise and conclusion may be understood as implicitly involving a

generalization with respect to time as in the conclusions of (A7) and (A8).¹⁸   Calls for

lexical decomposition will, no doubt, arise immediately as regards (A9), but the dis-

cussion of determinable-determinate-differentia arguments in section II above

should be recalled to forestall hasty conclusions. In addition, attention should also be

directed towards cases in which decomposition is of dubious value, as with,
 

(A10)  some blue object is a cube

            therefore, something extended is not a sphere,

where 'object' is used in a wider sense than 'physical object'.

    Arguments like (A7)-(AIO) can be used to show that material consequence, a la

(MSV.2), fails to be transitive. The premises of (A9), for instance, implies a propo-

sition expressed by 'some collies are colored', and the latter implies the conclusion of

(A9). Both implications are sanctioned by (MSV.2) and the shared constants are

replaceable (within limits). Since the premise does not imply the conclusion by

(MSV.2), however, then this criterion fails to support transitivity of material conse-

quence and this seems sufficient to render it a failure as an explication of any relation

of consequence.

    An interesting class of arguments involving implicit constants provide counter-

examples of the second sort. These arguments contain principles that have different

but overlapping ranges of applicability or domains of discourse. For instance, given

current physical theory, certain laws which hold of photons, say, do not hold of all

physical entities, e.g., E = hv and, hence, hv = mc² ,   where h is Planck's constant.

Let us use E = mc² as well as
 

mc² = ₀^s Fds + m₀   ×   ½600,000k/sec.

as rule-legitimating principles, and suppose that the description 'the first photon

emitted in our next experiment with black body radiation' is an implicit constant in

the equational expressions. Consider the following argument:
 

(A11) E = mc²

Therefore, hv = ₀^s Fds + m₀   ×  ½600,000k/sec.

This argument is on the same footing as (A6) with respect to validity, that is, if (A6) is

valid by virtue of the fact that its rule-legitimating principles are true then so is (A11).

However, there is an important difference between the two; any object that satisfies

E = mc²  will satisfy the conclusion of (A6), but the same does not hold for the con-

clusion of (A11) since E = hv, while a law governing the behaviour of all photons,

fails for other sorts of objects. So, it is not the case that every substitution upon the

implicit constant, i.e., 'the first photon emitted in our next experiment with black

body radiation', is truth-preserving. Still, there is a sense in which the occurrence of

this constant is idle and inessential; any other constant designating a photon can be

substituted for it with truth being preserved. (MSV.2), however, cannot be rescued by

this fact nor can it accommodate the credibility of (A11).

    No doubt there are simpler cases to illustrate the point concerning different

domains. But (Al1) has the advantage of being set forth in the relatively clear langu-

age of mathematical physics while bearing continuity with the earlier example (A6).

Together with (A7)-(A10), it indicates the limitations to the applicability of

(MSV.2). More strongly, these five arguments are counter-examples to this principle,

and if  (MFV) is equivalent to (MSV.2), then to (MFV) as well. A further criterion

which would drop the demand that any extra-logical constant be shared yet require

that both premises and conclusion contain substitutable constants would fare no

better with respect to (A7)-(A11). Attempts to defend (MSV.2) that appeal to the

concepts of enthymemes and implicit definitions must be measured against the

arguments of section 4 above. The crucial fact that virtually any lawlike generali-

zation can be related to an argument as a rule-legitimating principle cannot be

overlooked. It is advisable, hence, to examine an amended account of material conse-

quence that can handle cases like (A7)-(A11).
 
 

7. Further proposals and problems

of the substitutional approach and the allied analysis in terms of form. While

retaining the idea that each valid argument contains factors that are idle and replace-

able it introduces restrictions upon the class of available substitution instances;

instead of holding that validity requires truth-preservation throughout all substitu-

tions per se it insists merely that truth be preserved throughout all substitutions of a

specified sort. To explain, let us recall that the notion of an "appropriate"

substituend is central to the substitutional analysis of validity, logical or material, as

pointed out in section 2. Previously, this notion was introduced in terms of linguistic

types, but it can also be presented in relation to logical categories; an expression E1 is

an appropriate substituend of an expression E2 just in case E1 and E2 designate or

signify items falling under the same logical category, e.g., logical subjects,

predicables, etc . . . It seems fair to construe logical categories as very general sortals

or kinds--i.e., as sortal or kind properties or concepts expressible through common

nouns or common noun phrases. This in mind, a further relativization is in order;

given any sortal or kind K, expression E1 is a K-appropriate substituend of expression

E2 just in case E1 and E2 designate or signify items or entities of kind K.  For example,

'red' is an appropriate substituend of 'angry' with respect to the higher-order logical

category of being a predicable. The singular term 'Reagan' is an appropriate substi-

tuend of 'Carter' with respect to the category of being a logical subject as well as with

respect to the kind (property) of being a person. With this, (MSV.2) can be forfeited

in favor of the following:
 

(MSV.3) A sentence B is a material consequence of sentences A1, . . . , An  if and

only if  (1)  B is not a logical consequence of A1, . . ., An and (2) there is a

set S of extra-logical constants occurring in A1, . . ., An, B, some of which

occur in A1, . . . ,  An and some in B, such that for some kinds K1, . . . , Kj,

where each c_i in S signifies an item of kind K_i, 1  i  j, and if

A1',  . . . , An', B'  are the results of any uniform replacement of each c_i, by a

K_i -appropriate substituend then B'   is true if A1', . . . ,An' are all true.
 

unlike (MSV.2), does not require that any constant be shared by premises and

conclusion, and (ii) (MSV.3) allows for restrictions to be placed upon the admissable

substituends of a constant with respect to any kind of entity signified by that

constant, whereas (MSV.2) requires that only logical kinds, i.e., logical categories,

can determine the class of admissible substituends. In connection with this last point

it is important to note that (MSV.3) states that an argument is valid just in case there

is some set of constants substitutions upon which, with respect to some kinds, are

truth-preserving. The occurrences of 'some' are crucial, for not every substitution

upon a given constant will be truth-preserving--even if the constant and the substi-

tuend both designate entities of a given kind K.

    (MSV.3) is obviously a complex and, perhaps, unintuitive account of material

consequence. Its merit must be shown through a careful examination of particular

examples. First of all, given that logical categories are included among the kinds

quantified over in the second clause of (MSV.3), it is clear that the criterion is capable

of accommodating all the arguments that (MSV.2) can. But how does (MSV.3) help

in explaining arguments (A7)-(A11)? Concerning (A11), it is sufficient to restrict the

admissible substituends to constants designating photons so that the transition from

the premise to the conclusion will never involve a shift from truth to falsity given

replacements for the implicit constant. In this case, the kind property in question is

the sortal "being a photon" while the replaceable constant is, as before, 'the first

photon emitted in our next experiment with black body radiation'. Cases (A7)-(A10)

require a slightly different move, namely, an acknowledgment that predicate and

sortal expressions may themselves signify attributes (concepts) falling under given

kinds. For instance, 'angry' expresses not only a specific emotional state, it also signi-

fies a property that can be classified as a state, a mental state, an emotive state, and so

forth, while 'individual' expresses a kind of property that is had by, among other

things, persons. For (A7), then, the set S of replaceable constants might be the set
 

{'Nixon', 'individual', 'angry'}

where 'Nixon ' can be replaced by any constant designating a person, 'individual ' by any

constant designating a property of each and every person (e.g., being a living thing), and

'angry' by any constant designating an emotive state; any substitution upon these

constants with respect to those kinds of truth-preserving. By means of similar selections

and substitutions (A8)-(A10) can also be handled. In addition, because (MSV.3)

harbors no requirement of constant sharing it escapes the criticism of (MSV.2)

concerning transitivity. Seemingly, (MSV.3) succeeds where (MSV.2) fails.

    How does the move to (MSV.3) affect the analysis of material validity in terms of

material form? Theobjections to (MSV.2) apply equally to (MFV) if  material forms can

only be specified by matrices whose freely occurring variables range over everything,

or, for that matter, over all and only elements of given logical categories. If, on the other

hand, form-specifying matrices are permitted to contain freely occurring restricted

variables, i.e., variables whose ranges, with respect to any domain, are determined by

specific sortal concepts other than those determining the logical categories, then (MFV)

can be retained. Allowing restricted variables in this way is the counterpart to paring

down the class of appropriate substituends for constants. To illustrate, 'x_p is angry'

expresses a material form j ust as 'x is angry' does, where 'x_p', unlike 'x', ranges over all

and only persons. Letting  'f_E '  be a predicate variable ranging over all and only kind

properties of each and every person (with common noun expressions as substituends),

and 'fg ' a predicate variable ranging over emotive states, then a validating form of (A7)

is represented by the conditional matrix
 

x_p is f_E    some f_p is, at some time or other, in an emotive state.

abbreviates the implicit constant noted earlier now made explicit, and similarly for

the conclusion. A validating form of this argument is specified by
 

E(y) =m(y)c²  hv(y) = ₀^sFds(y) + m₀(y) ×   ½600,000k/sec

--where 'y' is a variable ranging over all and only photons. Arguments (A8)--(A10)

receive a similar treatment.

    Though they retain vestiges of traditional approaches, (MSV.3) and (MFV)

constitute an unwieldy account of material validity. Whatever defense they can be

given must be judged in terms of adequacy and comprehensiveness, not elegance.

Regrettably, however, (MSV.3), like its predecessor (MSV.I), proves too much;

unless some restrictions are placed upon the substitution kinds or upon the

replaceable constants arguments will be valid which would not intuitively be so. For

example, let A be any true statement and let B be any statement, true or false; it

follows that the statement (sentence) B  A is true. There seem to be cases, however,

where A is not a consequence of B even if this holds, viz., where A is not a necessary

truth and B is not a necessary falsehood, e.g., where A is lam now reading this

paper' and B  is 'I am now 75 miles away from Sophia Loren'. However, given B A,

B is a member of the class of statements determined by the following kind property

'being a statement p such that p  A', and A is a member of a class of statements

determined by the kind 'being a statement such that B  p'. Any substitution upon

B and A with respect to those kinds will be truth-preserving, so, by (MSV.3), it

follows that A is a (material) consequence of B. This result runs contrary to a widely

held belief that there are true statements which are implicationally independent of

each other, regardless if the implication or consequence relation in question is logical

or material.

    The same phenomenon emerges with respect to substitutions upon individual and

predicate constants. Consider a village in which the police chief is, albeit

contingently, the mayor, and suppose that both 'the police chief is angry' and 'some

mayor is, at some time or other, in an emotive state' are true. Does the former imply

the latter? A negative response seems in order; surely the latter could be false even if

the former is true. In fact, a valid argument appears only when the sentence 'the

police chief is the mayor' is added as a premise, and it is important to notice that this

sentence does not express a principle capable of legitimating an inference rule. As

such, the argument from 'the police chief is angry' to 'some mayor is, at some time or

other, in an emotive state' is not on a par with arguments (A1)-(A11). On (MSV.3),

however, this argument turns out valid since all substitutions upon 'angry' with

respect to the kind "being an emotive state" and upon 'mayor' with respect to the

kind "being an attribute of the police chief" are truth-preserving. (MSV.3) is, there-

fore, an embarrassment. Analogous considerations apply to (MFV) if this criterion is

joined by the liberal policy of representing material forms by means of matrices

containing free variables of arbitrary restriciton.

    There are ways of salvaging (MSV.3) and (MFV). One might allow that every true

proposition is, in some sense, necessary, so that there is no oddity in saying that B

implies A whenever B  A is true. But embracing this Spinozistic perspective is a

drastic move, too drastic for many that are otherwise sympathetic to material

validity. An alternative is to modify (MSV.3) by requiring that the substitution kinds

be essential to the items designated by the replaceable constants. Similarly,

concerning (MFV), it can be mandated that the kinds or concepts to which variables

are restricted be essential to the values of those variables. With these strictures the

counterexamples could be blocked; as to the first, B is not essentially a statement p

such that p  A, while, regarding the second, 'mayor' does not express a kind

property that is, essentially, an attribute of the police chief. But while moves of this

sort might be welcome to some, it must be observed that they generate the monu-

mental task of accommodating not only essentialism but also essential attributes

(kinds) of higher order entities like statements and properties as well.

    For those who spurn essentialism or abstract entities while remaining partial to

the idea that some arguments are materially valid, there are other alternatives.¹⁹

The failure of the three criteria (MSV.I)-(MSV.3) might indicate the futility of

attempting to account for material consequence by appeal to form and substitutivity,

and perhaps there is more than meets the eye to the contention that some implica-

tional relationships are dependent upon content rather thanform. Or, again, maybe

the real culprit is the classical analysis of validity; examination of materially valid

arguments merely reveals this fact in a novel manner. These hypotheses are,

obviously, important contrasts to the traditional approaches that have been pursued

here and whether the latter can be salvaged by further refinements remains to be seen.

In the meantime, we face the bleak result that our understanding of material

consequence and, hence, of consequence simpliciter, is still at a rudimentary and

unarticulated stage.
 
 
 
 

Acknowledgments

While writing this paper the author was a fellow at a National Endowment for the

Humanities Seminar offered at Indiana University during the academic year 1980--81

under the direction of Professor Hector-Neri Castaneda. Many thanks are due to him

for advice and encouragement; and also to a referee, for his helpful remarks

concerning the inability of (MSV.2) to secure the transitivity of material

consequence.
 
 
 
 
 

Notes

1. Church 1956, 1; compare Henkin 1967, 61-62.
 

2.  Bolzano 1973, 204-126; and see Kneale and Kneale 1962, who write that Bolzano interpreted deduci-

bility in terms of propositional structure and went on to apply this idea to the analytic-synthetic contrast

as well.
 

3 See, for example, Tarski 1956, 415; Quine 1970, 47-51; and Harman 1973, 75. It is common to char-

acterize logical truth in terms of substitution and to then define other logical notions, specifically, logical

consequence, in terms of logical truth, as in, for instance, Carnap 1937passim. With a criterion like

(LSV) it must be allowed that each sentence "occurs in"  itself despite the oddity of this mode of speech.
 

4.  Compare Tarski 1956, 415, whose statement of the substitutional criterion embodies a reference to

like signs.
 

5.  See, for example, Wittgenstein 1961, 5.131; Tarski 1956, 417; Strawson 1952, 50-52; Lewis and Lang-

ford 1959, 340; and Castaneda 1975,71. Quine 1966,37 also defines a 'valid form' in this way, and it is

important to realize that the occurrence of 'every', in the definition given, must be understood as

expressing a nomological or lawlike generalization--however, the notions of "nomological" and

"lawlike" are themselves to be interpreted. It should be noted, finally, that the practice of repre-

senting logical forms through open sentences in which the only constants are logical constants is

adopted throughout.
 

6 There are several philosophers who speak of the logical forms of non-linguistic items, viz.,

propositions; for instance, Bolzano 1973, who wrote of the forms of Satz (propositions); Strawson

1952, ch.2; Castaneda 1975, ch.3 and 1977, 298-299 and 327-329; and, most recently, McCawley

1981, 2-5. One who takes seriously the idea that logical forms belong to propositions might hold that

sentences have their forms only derivatively and, similarly, that logical relationships among sentences

are parasitic upon logical relationships among the propositions expressed by sentences.
 

7.  Tarski 1956, 414: and see p.417 where, after presenting his analysis of consequence in terms of mod

and satisfaction, he writes that 'the consequence relation which holds between given sentences

completely independent of the sense of the extralogical constants which occur in these sentences.'

Compare Ouine 1970, 58.
 

8.  The expression 'he' in this context is, in Castaneda's terms, a 'quasi-indicator', i.e., an expression

used to attribute indexical reference to agents (see Castaneda 1967 and 1980).
 

9.  The distinction between "logical" and "extra-logical" has long been recognized as problematic.

Recent discussions of the issue can be found in Peacocke 1976, Haack 1978 and Hacking 1979.
 

10.  This definition of 'enthymeme' raises interesting problems if an inference or argument is construed as

an ordering of sentences. If a given argument lacks a premise then it is not obvious that the

(completed) "logical" inference even exists, unless we are willing to say that the latter is a possible

ordering of sentences or an ordering of possible sentences pace Quine. The notion of a "suppressed"

premise, in fact, suggests a construal of arguments in terms of orderings of propositions.
 

11.  See Wason and Johnson-Laird 1972, 93-95 and 245, who write that no existing formal calculus

correctly models ordinary inferences and that 'it is clear that no conventional notion of logical form is

viable for the analysis of ordinary deduction' (p.93). More strongly, they conclude that no purely

formal calculus could succeed in modelling ordinary inference.
 

12.  See Carnap's 'Meaning postulates', in 1958, Supplement B, and criticisms ofCarnap's views in Pap

1958, 150-156, 407-416; Quine 1976, 107-132; and Chisholm 1966, 89.
 

13.  This criterion bears some similarity to W.T. Parry's notion of 'analytic implication' (see Parry 1933.

in that a statement A implies a statement B only if there is some sharing of content between A and B

More strongly, Parry's system requires that the conclusion of a valid argument contain no more than

what is contained in the premises, i.e., that the conclusion simply "unpacks" what is already present

in the premises. A metatheorem of this system is that if B is analytically implied by A then all the

variables occurring in B also occur in A', as a consequence, the system does not contain the law of

addition p  (p v q), where ' ' symbolizes analytic implication. See Anderson and Belnap 1975,

430-432, for a more precise exposition of Parry's system. It should be noted that (MSV.2) does not

place such stringent demands upon material validity since it does not imply that whatever is present in

the conclusion is already contained in the premises.
 

14.  George 1972 reports, with respect to his version of(MSV.2), that he has ' . . .not been able to find any

intuitive enthymemes which are not warranted under the definition, though certain cases yielded only

after some analysis' (p. 115). His treatment of time involves taking times as individuals open to quanti-

fication, a feature of his view that will be exploited in section 6 below.
 

15.  Certain philosophers have used the terms 'form' and 'propositional form' in just this broader fashion,

for example, Russell 1964, 85-88 and 1919, 158; Lewis and Langford 1959, 264; and Castaneda 1975,

ch.3. On such views one could hold that constants suffice to differentiate among types of propo-

sitional forms.
 

16.  It is difficult, of course, to know which scientific inferences are materially valid--this is a matter for

specific investigations. Every example will presuppose the correctness of some measure of current

scientific theory, and nothing should be thought to depend upon the particular cases cited since each is

representative of a given type of argument.
 

17.  Some might argue that the posit of material forms defaces the time-honored distinction between form

and content since the so-called 'material forms' effectively smuggle in content and, so, illicitly parade

under the banner of 'form'. Perhaps such dissenters are correct, but they face the challenge of

explaining why it is that only logical constants can occur in form-specifying matrices. Given the usage

of 'form' in ordinary English, this stricture is arbitrary; and if there are materially valid arguments,

then such constants are not alone in expressing implicationally relevant content. How else is their

restriction to be justified? In the absence of such justification another alternative is to insist that all

claims about 'form' and 'content' be reiafivizeclto given prepositional or statement types and to ack-

nowledge with F.H. Bradley that there is ' . . .no absolute divorce of matter from form, but there

remains after all a relative distinction' (1883, 524).
 

18.  In cases (A7)-(A9) times are construed as individuals open to quanitification after the manner of

George 1972 (see footnote 14 above). This is, however, an accidental feature of these arguments since

the same point can be made even if time is handled differently, e.g., through sentential operators or

through tensed copulae or predicates. If, on the other hand, predications are timeless then George's

strategy of introducing temporal constants to save (MSV.2) is to no avail.
 

19.  There is an alternative explication of material consequence that appeals to the notions of satisfaction

and models, analogous to Tarski's preferred account of logical consequence. The actual definition of

material consequence would be identical to that of logical consequence, viz., truth-preservation in all

models, save for the fact that the conditions placed upon the models would be far more stringent and

numerous in order to anchor the acknowledged implications in the sets constituting the models. However,

this development would be an uninteresting and tiresome extension of the model-theoretic analysis.
 
 
 
 

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