Bergmann Logic and Sortal Incorrectness

Logic and Sortal IncorrectnessAuthor(s): Merrie BergmannSource: The Review of Metaphysics, Vol. 31, No. 1 (Sep., 1977), pp. 61-79Published by: Philosophy Education Society Inc.Stable URL: http://www.jstor.org/stable/20127017 .

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LOGIC AND SORTAL INCORRECTNESS*

MERRIE BERGMANN

A sortally incorrect statement (non-significant statement, category mistake), in its simplest form, may be characterized as a statement

in which the subject and predicate are mismatched. The phe nomenon of sortal incorrectness arises as follows. Any predicate in natural language may be assigned both an extension and a sortal

range. The extension of a predicate is the collection of all those entities of which the predicate is true, while the sortal range con

sists of all those entities to which the predicate is significantly ap

plicable. When the entity designated by the subject-expression of an atomic statement (a predication) falls outside of the predicate's sortal range, the statement is sortally incorrect. The following are typical examples of sortally incorrect statements.

(1) The theory of relativity is blue (Drange);

and "Socrates is a prime number" (Pap); "Quadruplicity drinks pro crastination" (Russell); "This stone is thinking about Vienna"

(Carnap). (1), for instance, is sortally incorrect because the pred icate "is blue" is not significantly applicable to abstract entities.

The theory of relativity is not the sort of thing that could be blue. A wealth of literature has been devoted to the problem of de

veloping an adequate0 semantic analysis of sortally incorrect state

ments.1 The motivation is clear, albeit controversial: sortally in

correct statements appear to exhibit an unusual behavior when

coupled with other statements in logically complex statements.

For instance, two senses of negation are distinguishable when the

operation is applied to sortally incorrect statements. In this paper,

* This is the prize-winning essay of the 1976 Dissertation Essay Competition sponsored by The Philosophy Education Society, Inc., publisher of The Review of Metaphysics. 1

Three recent articles, with extensive bibliographies, are: J. N.

Martin, "A Many-Valued Semantics for Category Mistakes," Synthese, 31 (1975), pp. 63-83; R. Routley, "The Need for Nonsense," Australasian Journal of Philosophy, 47 (1969), pp. 367-384; and R. H. Thomason, "A Semantic Theory of Sortal Incorrectness," Journal of Philosophical Logic, 1 (1972), pp. 209-258.



62 MERRIE BERGMANN

I shall reopen the question of the "correct" semantic analysis of

sortally incorrect statements.

I

A few preliminary remarks are in order. First, under the rubric

"sortally incorrect statements" I intend to exclude a closely related

variety of statements which may be called "type-errors." Type errors are statements which, if treated as being true or false, may

lead to semantic paradoxes. Examples include: "What I am now

saying is false"; "The term 'heterological' is heterological" (Grelling). Unlike type-errors, the statements which I call "sortally incorrect" do not lead to semantic paradoxes. Sortally incorrect state

ments may be odd or deviant, but not paradoxical. For this reason I

choose to examine sortal incorrectness as a phenomenon distinct

from type incorrectness.

Second, I am treating sortally incorrect statements as seman

tically deviant. R. H. Thomason has shown that treating these statements as syntactically deviant leads to the following prob lem.2 Let us suppose that sortally incorrect statements are syn

tactically deviant?that is, the expressions within the statements have been coupled in violation of syntactic rules. Then we may

say that statement (1) combines an "abstract" subject-expression

("the theory of relativity") with a "concrete" predicate ("is blue"), in violation of the implicit grammatical rule that "concrete" predi cates can be meaningfully combined only with "concrete" subject

expressions. Now consider the following statement:

(2) What I am thinking of is blue.

If I am thinking of the sky, then the statement is sortally correct

and true. But consider the occasion on which I utter (2) while

thinking of the theory of relativity. On this occasion, (2) would

appear to be sortally incorrect. But then sortal incorrectness cannot

be a purely syntactical matter; the reference of the expression "what I am thinking of" is the crucial factor in evaluating the sortal status

2 Thomason, p. 212. The grammatical rule suggested below in the

text is an example of a linguist's "subcategorization" rule. See N. Chomsky, Aspects of the Theory of Syntax (Cambridge, Mass.: M.I.T. Press, 1965), pp. 148-160.



SORTAL INCORRECTNESS 63

of (2). In the interest of a unified approach, I shall treat all sortal incorrectness as a semantic phenomenon.

II

The by now classic defense of bivalent logic?indeed, classical bivalent logic?as being fully equipped to handle sortally incorrect statements is given by Quine:

. . . there has been a concern among philosophers to declare mean

ingless, rather than trivially false, such predications as 'This stone is

thinking about Vienna' (Carnap) and 'Quadruplicity drinks procrastina tion' (Russell). Here we witness sometimes just a spontaneous revul sion against silly sentences and sometimes a remote project of cutting

meaningful language down to something like empirical size. But since the philosophers who would build such categorial fences are not generally resolved to banish from language all falsehoods of mathe matics and like absurdities, I fail to see much benefit in the partial exclusions that they do undertake; for the forms concerned would remain still quite under control if admitted rather, like self-contradic

tions, as false (and false by meaning, if one likes). Tolerance of the don't cares ... is a major source of simplicity in theory; and in the

present instance it counts double, sparing us as it does both the

settling of categories and the respecting of them.3

The suggestion seems simple enough: once the deviant statements

have been admitted into the language (as syntactically well-formed), simply assign to these statements the value false.4 I shall assume

for the moment that it is atomic sortally incorrect statements which should be assigned the value false. Once we have made this assign

ment to the sortally incorrect statements, they are no more obnoxious

than contradictions that have likewise been admitted into the lan

guage. The expressions will play the same role in deductions as

any inconsistent sentence or set of sentences and no more. More

over, truth-functions over atomic sortally incorrect statements are

then simply defined?because exactly the classical truth-functions

may be used.

3 W. V. O. Quine, Word and Object (Cambridge, Mass.: M.I.T. Press, 1960), p. 229. The "meaninglessness" to which Quine aludes here is taken to be a result of syntactic non-well-formedness.

4 This is an extension of the Fregean method of allowing the signifi

cance-range, or sortal range, of a predicate to include everything that can count as an object. See especially "Function and Concept," in P. T. Geach and M. Black (trans, and eds.), Translations from the Philosophical

Writings of Gottlob Frege (London: Basil Blackwell, 1970), pp. 33ff.



64 MERRIE BERGMANN

One objection against any bivalent theory dealing with sortal incorrectness is that sortally incorrect statements aren't false; they are just meaningless or non-significant. If these statements are to

be admitted into a semantically interpreted system they should be

assigned either wo truth-value (e.g., in supervaluational systems) or a

truth-value other than those designated as the truest and the falsest

(e.g., in many-valued systems).

But the mere claim that sortally incorrect statements are "mean

ingless" is not sufficient to warrant a departure from the classical

semantic approach. Along Quinean lines, we may simply stipulate that the classical value false applies to those statements which we call sortally incorrect, which are at any rate not true, as well as

to contradictions and empirically false statements. There may be a

sense in which such statements are "meaningless," but that need

not preclude us from developing a semantic theory in which they are simply treated as being false.

Brushing this objection aside, then, it seems that the conse

quences of any bivalent approach should be the decisive factors in

accepting or rejecting it. R. Routley offers the following problem: ... a theory which simply reclassified non-significant sentences as

false would be quickly discredited as inconsistent: since the classical

negation of a nonsignificant sentence is, by definition, also non-sig nificant, for some sentence p both p and ~p would be false in the

theory.5

That is, if "Quadruplicity drinks procrastination" is sortally incorrect then so is "Quadruplicity does not drink procrastination." On the

assumption that the latter is the classical negation of the former, and that sortally incorrect statements are to be designated as false,

the classical system is rendered inconsistent.

Note that in this quote, Routley makes an important assump

tion: that the classical negation of a sortally incorrect statement

is itself sortally incorrect. But if the Quinean theory treats sortally incorrect atomic statements as false, thus, making falsehood a neces

sary condition of sortal incorrectness, classical negation may simply

function as usual, and will produce from a false statement a true

one. No reference is made in the definition of classical negation to any semantic aspect of the statement operated upon other than

its truth-value. Thus, any distinction that may be drawn between

5 Routley, p. 368.




atomic sortally incorrect statements and other non-true statements

is obscured; Quine has indicated that this was his intention. The negation which Routley has in mind is not classical nega

tion, but internal negation.6 The distinction between internal and external negation can be seen most clearly by an examination of

predicates with negating prefixes such as "un-" or "dis-." For ex

ample, from the nontruth of

(3) The theory of relativity is interested in classical music,

it can be inferred that

(4) It is not the case that the theory of relativity is interested in classical music,

but not that

(5) The theory of relativity is uninterested in classical music.

(4) and (5) may be said to express, respectively, the external nega tion and the internal negation of (3). In statement (4), the prefix "it is not the case that" is interpreted as denying the truth of the

subsequent statement, while in the case (5) something more seems to be at stake. The assertion appears to entail that the theory of

relatively can have dispositions?a clear absurdity. It is this sort

of negation, then, that takes a sortally incorrect statement into

another sortally incorrect statement.

The distinction between these two modes of negation is tied to the concept of a predicate's sortal range (or range of applicability).

The semantic deviance of sortally incorrect statements can be ex

plained with reference to the claim that (at least) certain predicates of the language have restricted sortal ranges; they are not sig

nificantly applicable to everything which can be picked out (referred to) by the subject-expressions of the language. The sortal range of

a predicate includes both those things of which the predicate is true?that is, the predicate's extension?and those things to which

the predicate could otherwise be significantly applied. The problem with negation, then, is that on one reading (inter

nal negation) the negation of a predication can be taken as indicat

ing that the denotation of the subject-expression falls outside of the

6 The distinction between internal and external negation may be traced to Aristotle, Prior Analytics, I, 46.



66 MERRIE BERGMANN

predicate's extension but within the predicate's sortal range. On this

reading, the internal negation of a sortally incorrect statement can

not be true. But if bivalent logics only admit of classical negation, then internal negation is not expressible in those languages. Classical

negation yields true statements from sortally incorrect statements

when the latter are treated as being false, and for this reason Routley discredits semantic approaches which evaluate sortally incorrect

statements as false.

But it is not clear that Quine's position cannot be saved. Note

that the examples of sortally incorrect statements given by Quine are atomic; these statements are to be admitted into the theory as falsehoods. The question of the truth-values of non-atomic sortally

incorrect statements may still be left open. Consider the following statement, from Quine's discussion of non-denoting terms in Word

and Object: Even such truth-value gaps can be admitted and coped with, per

haps best by something like a logic of three truth values. But they remain an irksome complication, as complications are that promise no

gain in understanding. Let it not be supposed that these various perplexities issue merely

from a pedantic distinction between what is false and what is neither true nor false. Nothing would be gained by pooling those two cate

gories under the head of the false; for they are distinguished, under whatever names, in that the one category contains the negations of all its members while the other contains the negations of none of its

members.7

The point to be noted is that the negation which Quine is considering here, internal negation, would not be truth-functional over a bivalent

semantics.

It is the apparent non-truth-functionality of internal negation

that has led to the most serious criticism of bivalent approaches; for, it is claimed, at least three values are needed to make the dis

tinction between internal and external negation. Routley claims that

. . . the addition of the limited [internal ] negation . . . forces in

terpretations of the theories up to at least 3-valued models. Con

sequently the non-significance discerned is not just a sort of falsity;

non-significance falsity?nonsense for short?can be split off as a

third value.8

Once this third value is admitted, internal negation can operate

truth-functionally (taking the third value into the third value). The

7 Quine, p. 177.

8 Routley, p. 378.




problem is not in deciding whether sortally incorrect statements

should be called false, but rather in deciding how many values the

underlying logic, if truth-functional, must have. (It seems that

Quine and Routley would not be in disagreement on this point.)

Ill

Within a three-valued (truth-functional) semantics, the matrices

for internal and external negation are:

~A A

T F N

F T N

T F N

F T T

internal negation external negation

The two matrices may be related by the use of the Bochvar-Frege horizontal h:9

hA

T F N

T F F

-A is defined as ~hA. With respect to the matrices for negation, the third value may indifferently be interpreted as signifying non

truth or as signifying semantic deviance (in the present case, sortal

incorrectness). The internal negation of a sortally incorrect state

ment is both nontrue and sortally incorrect. The horizontal may be read: "it is true that. ..." Then the external negation of any

statement A may be read: "It is not true that A." Where A is

sortally incorrect, -A is both true and non-deviant?for unlike

?A, -A involves no derivative affirmation concerning sortal

applicability. But adopting a three-valued semantics has an additional con

sequence: the binary logical connectives must be redefined over

9 For a discussion of this connective, see H. G. Herzberger, "Truth and Modality in Semantically Closed Languages," in R. L. Martin (ed.), The

Paradox of the Liar (New Haven: Yale University Press, 1970), pp. 25-46.



68 MERRIE BERGMANN

the three values.10 Many alternatives are possible; of these I shall

consider matrices for two normal varieties of disjunction. A con

nective is normal just in case whenever a compound statement gov

erned by that connective has immediate components which have

been assigned classical values (T or F), the compound statement itself is assigned the value it would be assigned by the corresponding classical matrix. Normality is a minimal concession to classical logic; it means in the present case that although semantic recognition has been given to sortally incorrect statements (by means of assigning the third value to these), the admission of these statements will not affect the operations on the non-deviant statements of the lan

guage. Note that in any three-valued logic with normal connectives, all tautologies (with value T designated) are classical tautologies.

As far as the remaining assignments of values to normal con

nectives are concerned, other principles may be invoked. The prin

ciple of purity yields Bochvar's "internal" disjunction11; the principle is that any compound statement one of whose components is assigned the value N will itself be assigned the value N:

AvB I T F N

T T T N F T F N N N N N

Letting the assignment of A7 to a statement mean that the statement

is semantically deviant, the acceptance of the principle of purity

amounts to saying that if any part of a compound statement is de

viant (nonsignificant, meaningless), the compound as a whole is

thereby rendered deviant. This is the most natural interpretation

of A7 in the present case.

Against the principle of purity, it may be objected that even

where the components of a statement are such that only one is as

10 F. Sommers avoids this problem by making internal negation an

operation on predicates, and retaining a bivalent semantics. See his "Pre

dicability," in M. Black (ed.), Philosophy in America (Ithaca: Cornell

University Press, 1967), pp. 262-281. The alternative semantics I develop in Section IV is likewise bivalent, but internal negation is defined over all statements of the language. 11 D. A. Bochvar, "On a 3-Valued Logical Calculus and its Application to the Analysis of Contradictions," Matematicheskii Sbornik n.s., 4 (1938), pp. 287-308.




signed a classical value, this one assignment may be sufficient to

warrant the assignment of a classical value to the compound. Take

as an example a disjunction, one of whose immediate components is

assigned the value T. In classical logic, the compound statement

would be assigned the value T regardless of the value assigned to

the other disjunct. The principle of uniformity states that connec

tives should have uniform matrices: whenever a given row (or column)

of a matrix has the same entry for both the value T and the value

F, the value N will also have that entry. Kleene's "strong" dis

junction conforms to this principle:12

AvB I T F N

T T T T F T F N N I T N N

The most natural interpretation of the value N in this case is non

truth, rather than deviance. In fact the latter reading appears

anomalous, as shown by the following example.13

(6) Either today is Friday or today is less than ten.

When the disjunction is interpreted as Kleene's strong connective, (6) will be assigned the value T on Friday and A7 on every other day of the

week. If N signifies deviance, this will mean that the statement is

deviant on every day of the week but Friday. Each of the two interpretations of the third value is plausible

as applied to sortally incorrect atomic statements. When the matrix

for disjunction is chosen, however, it seems that a decision between

the two interpretations is forced. A recent alternative to the three

valued approach is R. H. Thomason's supervaluational system for

sortal incorrectness.14 The approach diverges from truth-functional

12 S. C. Kleene, Introduction to Metamathematics (Toronto: van

Nostrand, 1952), pp. 332-340. 13 The example, together with a different but related point, is from

Thomason, p. 238. 14 Thomason. I should mention here that Martin contains an alterna

tive semantic analysis based on H. G. Herzberger's "two-dimensional" methods. (See H. G. Herzberger, "Dimensions of Truth," Journal of Philosophical Logic, 2 [1973], pp. 535-556.) The methodology is

invaluable; I have given a similar analysis of my own semantics (Sect. IV) in my Ph.D. dissertation and which will be presented in a forth



70 MERRIE BERGMANN

systems in the rejection of truth-functionality with respect to admis

sible valuations. The scheme is roughly as follows. Let L be a

propositional language with disjunction and negation taken as primi tive connectives. Define a partial valuation p: p is a partial function

with the atomic statements of L in its domain, and the set of classical values in its range. The partial function is defined for all and

only members of the class of sortally correct atomic statements of the

language. Let a complete valuation v generated by p be any classical

valuation over the statements of L, subject to the constraint that where A is a sortally correct atomic statement, p(A)

= v(A). Let

V(p) be the class of all complete valuations generated by p. Then the super valuation s(p) generated by p is defined as follows:

s(p)(A) = T iff v(A) = T for all v e V(p)

s(p)(A) = F iff v(A) = F for allv G V(p)

s(p)(A) is undefined otherwise.

Hence no sortally incorrect atomic statement is assigned a truth

value.

Let us say that a logical connective is truth-functional just in case

the truth-values (or lack thereof) of the immediate components of a

compound governed by that connective wholly determine the value (or lack thereof) of the compound. A logical system is truth-functional

exactly when all of its connectives are truth-functional with respect to the admissible valuations of the system. It can be seen by examina

tion that a supervaluational system where the supervaluations are

the admissible valuations is non-truth-functional.15 The divorce from

truth-functionality is important, for the method of generating super

valuations described above guarantees that exactly the set of classi

cal tautologies will be tautologies over the supervaluations (with T

designated), and entailment coincides with the classical relation.

(The proof is straightforward). The virtue of the supervaluational approach lies in this fact.

Neither of these results will, in general, hold for three-valued sys

tems. For example: coupling the three-valued matrices for internal

coming article, "Presupposition and Two-Dimensional Logic." But I

will not consider Martin's work here, for his analysis is based on certain intuitions against which I am arguing in this section.

15 E.g., if C is sortally incorrect, s(p)(C) and s(p)(~C) will both be

undefined. But while s(p)(CvC) is undefined, s(p)(Cv~ C) = T.




negation and Bochvar's "internal" disjunction we obtain Bochvar's

"internal" system (call it Mw); and Kleene's "strong" disjunction

together with internal negation comprise Kleene's "strong" system

(call it Ms). Taking T as designated, the class of tautologies in Mw is a proper subset of the class of classical tautologies, and the entail

ment relation is a proper subrelation of the classical. Taking both T and N as designated, the sets of tautologies coincide, but the entail ment relation of Mw is still a proper subrelation of classical entailment.

The same results hold for Ms.16

Yet the supervaluational concession to classical logic seems to be

too strong. Note that the results for statement (6) in M s hold also for the supervaluational system. (The statement is true on Friday and receives no truth-value on other days of the week.) So here absence

of truth-value should be interpreted as signifying nontruth, rather

than deviance. But why should the statement

(7) Either the theory of relativity is interested in classical music or the theory of relativity is uninterested in classical music

be true, if neither disjunct is true? Because the statement has the form of a classical tautology, it will be true on the supervaluational approach even when the negation is interpreted as internal negation.

I fail to see the virtue in retaining the set of classical tautologies, if this is the result. Thomason has shown how to define external

negation over the supervaluations; and it seems correct that the

statement

(8) Either the theory of relativity is interested in classical music or it is not the case that the theory of relativity is interested in classical music

should be considered true. But this because the second disjunct (which involves external negation) is itself true.

What we have here is a conflict of intuitions. Three different

criteria may be used for choosing the non-bivalent dyadic connectives,

and each has some intuitive plausibility. First, sortal deviance may be said to be contagious: statements such as (6) would be judged odd on

sortal grounds by many native English speakers, and odd on any day of the week. Second, uniformity is a plausible characteristic; al

though (6) is odd it also appears to be true by virtue of the first disjunct

16 For proofs, consult Chapter Two of my Ph.D. dissertation.



72 MERRIE BERGMANN

(on Fridays). Third, classical tautologies expressible by statements such as (8) seem true?despite the apparent nontruth of their counter

parts containing internal negation?in this case, statement (7). Note

once again that the interpretation of the third value as indicating deviance in the case of Mw is the most plausible interpretation, while in the case of Ms and the supervaluational approach, the third value and truth-value gaps are most plausibly interpreted as indicating nontruth.

But here is the key to the conflict: in all three systems, the sortal incorrectness of a statement has been held semantically identi fiable through the statement's truth-value. When this identification is made, problems squeeze out somewhere: either a statement which

seems to be true is rendered nontrue (Mw), or a statement's deviance

is obscured semantically through a passion for truth. In the next sec

tion, I shall present a logical system in which the definitions of truth value and sortal incorrectness are separated, but both dealt with in the

semantic theory and reflected in the accompanying language.

IV

My proposal is to build a logical system in which R. H. Thomason's refinement of B. C. van Fraassen's "logical space" serves as a semantic

basis, and which has only two truth-values.17 Because of the under

lying semantic structure, not all connectives will be truth-functional

in the ordinary sense.

Let the language Lc have the following primitive vocabulary:

For each integer n, a finite number m of n-adic predicates Pn p n p n

i , r 2 y . . . , r m

(I would prefer to restrict the number n to something smallish, but leave it open for generality. At any rate, for any number

n, the number m of n-adic predicates may be zero.)

Individual constants ax,a<?, . . .

Logical connectives and punctuation: -, D, y, (, ).

I will later introduce defined connectives. The statements of Lc are

defined recursively:

If for some n, tu . . . , tn are individual constants and P is an

n-adic predicate, then Ptxt2 . . . tn is a statement.

17 Thomason; and B. C. van Fraassen, "Meaning Relations Among

Predicates," Nous, 1 (1967), pp. 161-179.




If A and B are statements, then -A, y A, and A D B are state

ments.

That's all.

The basis for the semantics will be a logical space S. S is a non

empty set, which we may assume to be the collection of all entities which occur in at least one possible world. (I assume that the domains of all possible worlds are disjoint. However, non-disjoint domains

may be assumed, provided that each possible entity is included in the extension/sortal range of a given predicate in one world if and only if it is included in that extension/sortal range in every possible world of which it is a member.) The meaning, or intension, of a predicate determines a class of possible entities to which the predicate could

truly apply; or the meaning is that class of possible entities.18 I

say "could," because the idea here is that if those possible entities were actual existent entities, the predicate would be true of them.19

Likewise, the sortal range of a predicate includes those possible indi viduals of which the predicate is significant, or sortally correct. Any predicate's intension will be included in its sortal range; the possible entities of which P could be true must be part of P's sortal range.

Letting S be the logical space for Lc, define a sortal specification R on S: R is a function defined over the predicates of Lc. For each

n-ary predicate P, R (P) is an ordered pair of subsets of Sn, such that the first member of the pair is included in the second. The first

member of R(P), P's intension, will be denoted by [RKP)]; the

second, P's sortal range, by [R2(P) J. Let refbe a function (reference function) defined over the individual constants of Lc: for arbitrary constant a, ref(a) E S. We may think of ref(a) as the individual denoted by a. Let M =

(R, ref) be a model relative to the sortal

specification R on S.

18 Van Fraassen's theory does not ostensibly have this ontological com mitment. But I believe that ontological commitment does squeeze out some

where; so rather than leaving intents as nebulous entities only being represented in the theory, I prefer simply to think of logical space in such a

way that its regions can be identified with the intents of the predicates. Van Fraassen's alternative account of logical space is, of course, com

patible with the formal apparatus developed below. 19 I am here taking the sense of predicates, as we ordinarily use them,

to be existence-entailing. It is because of this existence-entailment that I say that possible entities in the logical space would be such-and-such

if they existed (as members of the actual world). See also H. S. Leonard, "Essences, Attributes, and Predicates," Proceedings of the American Philo

sophical Association, 37 (1964), pp. 25-51.



74 MERRIE BERGMANN

Sortal specifications on logical space provide a reasonable?be

cause explanatory?means of distinguishing between internal and ex

ternal negation within a bivalent language. The internal negation of a

predication involves a derivative affirmation that the denotation of the subject-expression of the predication lies within the predicate's sortal range. "The theory of relativity is uninterested in classical

music" is true if: it is not true that the theory of relativity is interested in classical music, and it is true that the theory of relativity is the sort

of thing that could be interested in classical music. External nega tion does not involve this derivative affirmation. So where Pa is a

statement of Lc, and "~" stands for internal negation (I will later intro

duce "~" as a defined connective), ?Pa will be true exactly when

refia) E ?2(P) -

R\P). That is, the referent of a is within P's sortal range but excluded from P's intension. Thus, if ref(a)

E R2(P), exactly one of Pa, ?Pa will be true and the other false. On

the other hand, if ref(a) ? R2(P), then both statements will be false. The external negation of a predication is defined only with respect to the predicate's intension?viz.,

- Pa is true if and only if Pa is

false (iff ref(a) & RKP)). Given the previous comments, a necessary condition for the truth

of Pa or ?Pa is that ref(a) E R2(P). This means that ?Pa is true iff Pa is false and Pa is sortally correct. Carrying this idea a bit further, if we had in the language for each statement B a statement B '

which

says that B is sortally correct, then we could extend the above

definition to range over all statements: ?B is true iff B is false andB'

is true. This will be the function of the operation y?yB will mean: B is sortally correct. Now we may turn to a definition of the valua

tions for Lc.

A bivalent valuation for Lc with fixed S is defined as follows.

For each model M on S, v is the bivalent valuation generated by M

iff for every statement A of Lc, v(A) E {T,F}; and the following con

ditions are met:

(9) v(P?% . . . td = T iff (refitj, . . . , ref(td) E R*(Pfl

(10) v(A DB) = T iff v(A) = F or v(B) = T

(11) v(-A) = Tiffv(A) = F

(12) viyPfa . . . ii) = T iff (ref(tt), . . . , refit,)) E R2(P2(P,

(13) viyiA 3B)) = T iff v(yA) = v(yB) = T




(14) v(y-A) = T

(15) v(yyA) = T.

Note that "y" is an unusual connective, in the following sense (noted

by van Fraassen). If we let | A |R =

{ref: A is true in (R,ref)}, we

find that | -A \R and \A D B |R are functions, respectively, of |A |R, and | J5 |R and | A |R. There is no corresponding function for | y A |R. The operation is non-truth-functional, and allows us to develop a

bivalent theory. The connectives definable in Lc may be divided into four groups,

as follows.20

(16a) A composite limited connective is a connective which forms a statement whose truth-value depends on both the truth

value and sortal status of its immediate substatement(s), and whose sortal status depends on the sortal status of its immediate substatement(s).

(b) A simple limited connective is a connective which forms a statement whose truth-value depends on the truth-value of

its immediate substatement(s) and whose sortal status

depends on the sortal status of its immediate substate

ment(s).

(c) A composite unlimited connective is a connective which forms a statement whose truth-value depends on the sortal

status of its immediate substatement(s), and which is always sortally correct.

(d) A simple unlimited connective is a connective which forms

a statement whose truth-value depends on the truth-value

of its immediate substatement(s), and which is always sortally correct.

Let me now classify some of the connectives of Lc. The negation "-" is a simple unlimited connective. It forms a statement which is

always sortally correct (14), and which is true iff the statement

negated is false (11). "It is not the case that the theory of relativity

20 The terminology "limited" and "unlimited" comes from A. Pap, "Types and Meaninglessness," Mind n.s., 59 (1960), p. 54. Pap uses these terms to distinguish between modes of negation, where "limited"

negation is internal negation, and "unlimited" negation is external nega tion.



76 MERRIE BERGMANN

is blue" is true and sortally correct, despite the sortal status of the

imbedded statement. The sortal status is due to the function of ex

ternal negation as a "sortally opaque" (or semantic) connective; the

statement formed does not receive the sortal status of the statement

operated upon. The connective "D" is a simple limited connective.

The truth-value of a statement governed by "D" is determined

classically (10). But the statement is sortally incorrect if either of its immediate components is sortally incorrect (13). Thus, "If today is less than ten, then today is Friday" is both true and sortally incorrect on Friday.

Internal negation is defined with the aid of the following con

nectives.

(17) 1A =?tAD -A

As consequence, we have

(18) v(lA) = Ti??v?A) = F

(19) v(y 14) = v(yA),

"~l" is a simple limited negation.

(20) A&B =df HAD IB)f

where "&" is simple limited conjunction. Then internal negation is

(21) -A =d{ lA&yA.

Internal negation is a composite limited connective, for

(22) v(-A) = T iff v(A) = F and v(yA) = T

(23) v(y-A) =

v(yA).

Finally, "y" is a composite unlimited connective. The truth-value

of a statement governed by this connective depends only upon the sortal status of the immediate substatement ((12)-(14)), and the statement formed is always sortally correct (15). "y" is also "sortally

opaque," or a semantic operator.

The way in which models have been defined allows for the intro duction of two modes of semantic entailment and of validity.21 The second correspond to logical, or universal, relations; the first are more

21 This distinction is drawn in Thomason, pp. 233 ff.




restricted. A set of statements T sortally entails a statement A

relative to R, T |hR A, if: for every model M relative to R, if the bivalent valuation v generated by M is such that v(B) = T for all

PET, then v(A) = T also. A statement A is sortally valid relative to R if $ | h R A. A set of statements T logically entails a statement A, r | \-A, if: for every logical space S and sortal specification R thereon, r |hR A. A statement A is logically valid if 4> \\- A.

Thus, the conflict of intuitions behind Mw and Ms is resolved by separating truth-value and sortal status. The present system may

also be related, in various ways, to classical logic. Note first that if

every atomic statement is sortally correct for some model M, then the

simple limited and composite limited versions of any connectives will have identical truth-conditions on that model. (This is readily veri

fiable.) Moreover, the truth-conditions for the simple limited con

nectives are identical with the classical truth-conditions. So on the

hypothesis, the resultant system would differ from classical logic only in the addition of a primitive "verum" operation y :v(y A) = T for all A.

Of course, not all models will make every statement sortally correct. So define the simple unlimited connective 'T" (read: "is

true") as

(24) TA =??--A.

We may replace every atomic statement A by TA, and define the connectives as in Lc (note that the clause (12) is then inoperative).

The resultant system is then classical with the addition of the verum

operation. That is, the simple limited and composite limited connec tives again collapse. Note that TA and A are not in general inter

changeable salva veritate; for it is possible that for some v and A,

v(yTA) = T and v(yA)

= F.

V

A full quantificational system Lq may also be defined along the same lines. The vocabulary of Lq includes, in addition to the vocabu

lary of Lc, the following:

Individual variables xu x2, . . .

Possible individual variables wu w2, . . .

Dyadic predicate "="

Logical symbol "V."



78 MERRIE BERGMANN

The terms of Lq include the individual constants and both varieties of variables. The formulas are defined recursively:

If for some n, tu . . . ,tn are terms and P is an n-adic predicate, then Pti . . . tn is a formula.

If ii, t2 are terms then tx =

t2 is a formula.

If A andP are formulas, then -A, y A, and A D B are formulas.

If A is a formula and z is a variable (of either variety), then

(Vz)A is a formula.

The statements of Lq are exactly those formulas in which no variable

occurs free.

Let S (a non-empty set) be the logical space for Lq. Let R be a

sortal specification on S, as defined for Lc. Where D is some non

empty subset of S, let D be the domain for L q. We will assume that D

normally represents the set of actual individuals. Let ref be a func

tion defined over the individual constants of Lq: for arbitrary constant

a, ref (a) E D. Let M = (R,D, ref) be a model relative to the sortal

specification R on S. The definition of the bivalent valuations generated by the model

is twofold, since we have introduced quantification. First, let an

assignment function d on M be any function defined over the terms of

Lq with range S, such that for any individual constant a, d(a) = ref (a),

and for any individual variable x, d(x) E D. Relative to each assign ment function d, a satisfaction function sd is defined for the model M: for every formula A, sd(A) E {T,F}, and (where t j is a term, and y is a

variable of either variety)

(26) sd(P^ . . . ?i) = T iff (diU), . . . , d(t?) E ?HPj1)

(27) sd(A D B) = T iff sd(A) = F or sd(B) = T

(28)sd(-A) =

Tiff*d(A) = F

(29) sait, = t2) = T iff d(ix) = d(t2)

(30) sd((Vi/)A) = T iff sd, (A) = T for all assignment functions

d' on M which are the same as d except possibly in the value of

d(y)

(31) s?(y(Pfli . td) = T iff (?(ij, . . . , dtfi)) E ?W)

(32) sd(y(A Di)) = T iff sd(yA) =

sd(yB) = T

(33)sd(y-A) = T




(34) s^yt^t,) = T

(35) sd(y(Vy)A) = T iff s* (yA)

= T for all d' defined as in (30)

(36) sd(yyA) = T.

Finally, define the bivalent valuation v generated by M on S as: for

any statement A, v(A) = T iff sd(A) = T for all satisfaction functions

sd on M; otherwise, v(A) = F. The two modes of entailment and

validity are defined as earlier.

The reason for the introduction of two varieties of variables is that sortal specifications are then reflected in the language. Since individual variables range only over the domain in the logical space, the relations on the space determined by sortal specifications may not be adequately reflected in a language containing only individual varia

bles. In general, the cardinality of D may be smaller than that of S.

Suppose, then, that for some monadic predicates P and Q, R2(P) C R2(Q). Suppose also that for some D c M, [R2(Q)

- R2(P)]nD

= (?>. Then for the individual variable x, the statement (\/x)(-yPx

D -yQx), as well as (\fx)(yPx D yQx), will be true in that model. That

is, as far as the members of the domain are concerned, the sortal

ranges of P and Q seem to be identical. But they are not. The pos sible individual variables come into play here: if w is a possible indi vidual variable, then (Vw)(-yPw D -yQw) is not true in the model. Other relations in the sortal specification are also thus expressible. Let P and Q be monadic predicates, w a possible individual variable,

R a sortal specification on a space S,Ia model relative to R, and y

the valuation generated by M. Then

(37) R2(P) n R2(Q) =

<f> iftv((4w) -

(yPw&yQw)) = T.

(38) R2(P) = R2(Q) i??v((4w)(yPw

= yQw))

= T.

(39) R2{P) U R2(Q) = S ffiv(Qtw)(yPw v yQw))

= T, and so on.22

Dartmouth College.

22 The ideas contained in this essay were developed as part of my Ph.D. dissertation "A Presuppositional Theory of Semantic Categories," Uni

versity of Toronto, 1976. I should like to acknowledge my debt to my dissertation supervisor, Dr. B. C. van Fraassen, and advisor, Dr. H. G.

Herzberger.



Bergmann Logic and Sortal Incorrectness

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