(appeared
in: J. Bernard-J. Kelemen: Zeichen,
Denken, Praxis, Institut fur Sozio-Semiotische Studien: Vienna, 1990, pp. 249-267.)
"Are
quantification and cross reference in English well represented by the
quantifiers for ‘every’ and ‘some’, the usual propositional connectives and the
equals sign? It’s my impression that many philosophers and logicians think that
- on the whole - they are. In fact, I suspect that the following view of the
relation between logic and quantificational and referential features of natural
language is fairly widely held: No one (the view begins) can think that the
propositional calculus contains all there is to logic. Because of the presence
in natural language of quantificational words like ‘all’ and ‘some’ and words
used extensively in cross reference, like ‘it’, ‘that’ and ‘who’, there is a
vast variety of forms of inference whose validity cannot be adequately treated
without the introduction of variables and quantifiers, or other devices to do
the same work. Thus everyone will concede that the predicate calculus is at
least a part of logic. Indispensable to cross reference, lacking distinctive
content, and pervading thought and discourse, identity is without question a
logical concept. Adding it to the predicate calculus significantly increases
the number and variety of inferences susceptible of adequate logical treatment.
And now (the view continues), once identity is added to the predicate calculus,
there would not appear to be all that many valid inferences whose validity has
to do with cross reference quantification and generalization which cannot be
treated in a satisfactory way by means of the resulting system. It may be
granted that there are certain valid inferences, involving so-called
"analytic" connections, which cannot be handled in predicate calculus
with identity. But the validity of these inferences has nothing to do with
quantification in natural language, and it may thus be doubted whether a logic
that does nothing to explain their validity is thereby deficient. In any event
(the view concludes), the variety of inferences that cannot be dealt with by
first-order logic (with identity) is by no means as great or as interesting as
the variety that can be handled by the predicate calculus, even without
identity, but not by the propositional calculus."[1]
I
think it is significant, and generally characteristic of the change of
attitudes taken by logicians and philosophers of language in the last two
decades towards the relationship between quantification theory and natural
languages, that George Boolos, from whom this lengthy quotation derives, gives
this detailed description of what may be called the paradigmatic view of this
relationship only to raise several objections to it.
This
change of attitudes was mainly prompted by the recognition of a steadily
growing body of anomalies in the application of quantification theory to
natural languages. These anomalies may be gathered, roughly, under the
following headings:
As
is well-known, natural language sentences of evidently the same syntactic
structure are represented by formulae of quantification theory of entirely
different structure, while the same formula may have different
"readings", expressible by natural language sentences of widely
different syntax.
Regarding
these discrepancies, of course, one might say that there is no justifiable need
of a strict correspondence between the syntactic structure of natural language
sentences and the formulae representing them. After all, a logical semantics,
which is to be a general semantics for all kinds of human languages, should
precisely disregard accidental grammatical features of particular natural
language expressions, and hence also the delusive grammatical structure of
natural language sentences in general. All that is required for correspondence
is that the formula should state correctly the truth conditions of the sentence
which it represents, since it is only these truth conditions that determine the
logical relations of sentences among each other.
Along
these lines, mismatch of syntax may be made to appear entirely harmless, by
making a distinction between logical form on the one hand, and grammatical form
on the other, placing much confidence in the capability of quantification
theory to express the former, and thereby justifiably ignoring the latter.[2]
There
is, however, a further set of anomalies, which comes as a fatal blow to this
interpretation of the relationship between quantification theory and natural
languages. For, as it turned out, some apparently simple quantified sentences
of natural languages are demonstrably unrepresentable in first order quantification
theory in the sense that no first order formula is able to give their correct
truth conditions.[3]
As
is well-known, examples of such sentences are those containing the determiners
‘most’ or ‘more than half of’, and so on. But if there are no formulae giving
the correct truth conditions of such sentences, then quantification theory is
simply unable to supply their logical form, and so the above-mentioned
rationale for drawing the distinction between logical and grammatical form
breaks down with these sentences.
But
there are also other types of natural language sentences that pose a serious
challenge to the claim that quantification theory has all types of
quantificational and cross-referential resources that natural languages may
possibly have. Recent discussions of the troubles caused by the so-called
"donkey-sentences" supply ample evidence against the claim that
variables of quantification theory can do everything that natural language
pronouns can do.[4]
To
be sure, the above-mentioned "anomalies" may be considered as such
only because they pose problems to quantification theory that everyone feels it
should handle but cannot. It was clear from the beginning that there are large
portions of natural language reasonings that simply fall outside the authority
of quantification theory, namely those involving intensional contexts.
Nevertheless, Frege’s relegation of modal notions to the sphere of psychology
notwithstanding, logicians have been working on expanding formal logic even to
these contexts. Possible worlds semantics produced interesting results
concerning modal notions and still seems to have some resources concerning
tensed modal contexts. However, in virtue of the coarse-grained character of
intensions available in possible worlds semantics, several intentional
contexts, namely those created by attitude verbs, seem to defy analysis in
terms of these intensions.[5]
The
growing recognition of these and similar difficulties in the application of
quantification theory to natural languages gave occasion to some
historically-minded logicians to make comparisons between quantification theory
and traditional logic showing traditional logic in a much more favorable light
than before. The well-known differences between traditional and
quantificational analyses of categorical propositions resulting in the
invalidation by quantification theory of the Square of Opposition and several
syllogistic forms were no longer regarded by these logicians as revelations of
mistakes of an antiquated theory in the light of a better, new theory, but
rather as adding to the growing evidence against the capability of
quantification theory to represent natural language reasonings.[6]
As
a matter of course, all the above-mentioned troubles occasioned several new
developments providing more or less conservative extensions of, or more or less
radical departures from the usual construction of quantification theory.
Richard
Montague’s grammar and intensional logic may be regarded as answers to the
challenge posed by mismatch of syntax and intensional contexts.[7]
Generalized quantification theory, taking a cue from Montague, is intended to
cope with the troubles caused by ‘pleonotetic’ determiners and common noun
phrases of natural languages in general.[8]
Donkey sentences provided the main motivation for discourse representation
semantics.[9]
Also several efforts have been made to construct systems in keeping with some
basic principles of traditional logic that would match in power the resources
of quantification theory.[10]
Game theoretical semantics tries to take an entirely fresh look at questions of
natural language semantics in general.[11] Situation
semantics, beyond trying to provide answers to all semantic troubles of
classical quantification theory and intensional logic, intends to account also
for pragmatic aspects of communication within the framework of a general theory
of meaning and information.[12]
There are even attempts at breaking with extensionalism and set theory in logic
in general, by constructing intensional logics with an avowedly Platonistic
ontology claimed to be the most fitting model for handling intensional and
intentional contexts of natural languages.[13]
So
what we experience nowadays in formal semantics may be described in Kuhnian
terms as the splitting up of an old paradigm in consequence of the accumulation
of its unsolved puzzles and a search for new unifying perspectives.
Historically,
in such and similar situations scholars tend to seek for exemplars from earlier
paradigms: as is well-known, the Copernican revolution was almost as much
prompted by Copernicus’s sympathies with pre-Aristotelian, Platonic and
Pythagorean cosmological ideas as by his calculations.
To
be sure, such historical examples alone in our case would by no means justify
more than mere historical interest in traditional logic.[14] After all, even if
quantification theory has its own problems in its application to natural language
semantics, it has sufficiently proved its superiority over traditional logic in
its capacity to handle inferences involving relational expressions and multiply
quantified sentences like the following:
1.
A man sees every horse
2. A
horse of the king is a horse
———————————————
3.
Thus, a man sees a horse of the king
Now
I think that the mediaeval flavor of the example, familiar to many at least
from Umberto Eco’s best-seller, The Name of the Rose, already suggests my
intention to raise certain doubts concerning the usually unquestioned
superiority of quantification theory in these matters.
As
a matter of fact, the example derives from Jean Buridan’s Tract on
Suppositions,[15]
where the famous 14th-century master is not at all at a loss to account for the
validity of this inference in terms of the mediaeval theory of reference, the
theory of supposition. Indeed, supposition theory is only one, although
unquestionably the most important one, of those highly sophisticated,
peculiarly mediaeval semantic theories that place mediaeval logic high above
the relatively shallow standards of the so-called traditional logic of the last
century, recently eliciting an ever growing appreciation of the achievements of
mediaeval among contemporary scholars.[16] The increasing
contemporary interest in supposition theory is amply testified by the
proliferation of both historical and systematic studies on this theory, as well
as of its reconstructions in terms of, or comparisons with modern logic.[17]
As far as I know, however, thus far nobody has tried to use supposition theory
as what in my view it was really meant to be: namely (at least a starting point
of) a unified theory of reference in natural languages.[18] Now my intention
is to do precisely this in the rest of this paper.
Supposition
theory, as it appears in mediaeval logic textbooks from the 12th century up to
the 17th, usually begins with a series of definitions and divisions exhibiting
sometimes considerable variations from author to author, or even explicit
disagreements among the authors. So, properly speaking, there are several
theories of supposition held together by a common phraseology, a common stock
of background assumptions rooting mainly in Aristotelian metaphysics,
psychology and epistemology, and a common intention to give a unified account
of the referring function of terms in widely different contexts. For our
present purposes, however, these various teachings possess a sufficient unity,
so that I shall treat supposition theory rather indistinctly, even at the risk
of some slight historical incorrectness to be noted when necessary. I base my
treatment mainly on the accounts given by William Ockham and Jean Buridan, the
two most influential authors in late mediaeval logic. Nevertheless, most of
what I will say applies quite well to mediaeval authors of logic texts in
general.
Supposition
was commonly characterized by our authors as a property of terms in
propositions, namely, the taking of a term for something in a proposition, that
is, as we would put it, its referring function. Generally three main types of
supposition were distinguished: 1. material, when the term in a
proposition stands for itself (or for some other token-term of the same type),
like the term ‘man’ in the proposition ‘man is a noun’, 2. simple, when
the term stands for a universal, whatever a universal is, like ‘man’ in ‘man is
a species’ and 3. personal, when the term is taken for those things upon
which it is imposed, and of which, consequently, it is truly predicated.
Personal supposition was commonly divided further into discrete and common supposition. Discrete supposition is the referring function of singular terms, which, by reason of their meaning can be truly predicated only of one thing. Examples of this type are proper nouns, say ‘Socrates’, or common terms combined with demonstrative pronouns like ‘this man’ or ‘this horse’, pointing at a particular man or a particular horse. Common personal supposition is the referring function of common terms in propositions, which, by reason of their meaning, can be truly predicated of many particular things, like ‘man’ or ‘horse’.[19]
Now
common personal supposition was divided further according to the different
manners in which common terms may refer in different propositional contexts.
These different manners, and correspondingly the different subdivisions of
common personal supposition, were characterized by late mediaeval logicians by
so-called suppositional descents, descensus ad inferiora; that is to say, by
certain types of inferences in which the common term, of which the mode of
supposition is being characterized, is replaced by singular terms falling under
it, appearing in either nominal or propositional conjunctions or disjunctions.
These several types of conjunctions and disjunctions of singular terms, or of
propositions formed with these singular terms, served then both to characterize
the mode of supposition of the original common term under which the descent was
made and to give the truth conditions of quantified sentences in terms of the
truth or falsity of several singular ones. The main divisions of common
personal supposition may be given as follows:
Some
later schoolmen also added a fourth mode of supposition:[21]
and
also conversely,
That
is to say, a term has determinate supposition in a proposition if one can
descend under it to the singulars with a disjunctive proposition and
conversely. A term has confused and distributive supposition, if one can
descend under it by a conjunction of singular propositions, and conversely
(with the exception of the controversial case of 2.b., of which, however, see
n.26. below). A term has merely confused supposition, if one can descend under
it with a proposition with a disjunct term (and conversely) but one cannot do
the same by a disjunctive proposition. Finally, a term has copulative supposition
if one can descend under it by a proposition with a conjunct term and also,
conversely, one can ascend from this proposition to the original one, but the
same cannot be done with a conjunctive proposition.[22]
I think
there are two things that should strike the modern logician in these descents:
the first is their suggesting the idea of restricted quantification, and the
second is the problem whether in some sense they give a complete set of
truth-conditions for categorical sentences. Let me elaborate on these points.
I
think the idea of replacing a common term by a series of demonstratives in
these descents should remind a modern logician of the way variables of
quantification theory pick up their values from the domain of a model. Indeed,
we might even say that the several assignments of values of a variable may be
conceived as several acts of pointing at several individuals, thereby
associating a variable with these individuals. In this way, we may explain the
function of a variable in different assignments as that of a demonstrative
pronoun in different acts of pointing at a thing. So, for example, the formula
representing the sentence: ‘Every man is an animal’, namely the one which reads
‘For every x, if x is a man then x is an animal’ ((x)(Mx®Ax)) may be explained as
saying: this thing, if it is a man, then it is an animal and that thing, if it
is a man, then it is an animal ... and so on, pointing at each and every thing
in the world. And this explanation of the quantificational formula, in
comparison with the suppositional descents presented above, shows us
immediately the basic difference between the mediaeval and the modern approach:
while the variables of quantification theory range over all the objects of the
universe, the common terms of mediaeval logic range only over objects falling
within their extension: that is, they function as restricted variables.
Now
since common terms as restricted variables pick up their values from their
extension, the question naturally arises: what is their value when their
extension is empty? Well, the answer is quite simple: nothing. For a value,
that is, a suppositum of a term in a proposition, according to the mediaevals,
is a thing of which, when pointed at, the term is truly predicable by means of
the copula of the proposition.[23]
For example, the term ‘centaur’ in the proposition: ‘Every centaur is running’
refers to nothing, for whatever is pointed at we cannot truly say: ‘This is a centaur’.
But then even the singular terms: ‘This centaur’ or ‘That centaur’ refer to
nothing, and thus, all the singular propositions formed with them, like ‘This
centaur is running’ and ‘That centaur is running’ are false, in the same way as
Russell’s ‘The present King of France is bald’ is false. But in this way even
the universal proposition: ‘Every centaur is running’ must be false, since all
the singulars to which we can descend from it and from which we can ascend to
it are false. So we can easily understand why the mediaevals attributed
existential import to universal affirmatives, and why they held the relations
among categorical propositions to be those determined by the Square of
Opposition.
Now
as I have shown in some of my earlier papers, we can give formal expression to
these informal ideas by a rather conservative extension of standard
quantification theory. All we have to do is the following:
1.
we have to add restricted variables to the language of the theory, that is,
terms formed from open sentences by the following rule: if v is a variable and
A is a formula in which v occurs free, then ‘v.A’ is a term,
2.
we have to extend the definition of assignment to these terms so that they pick
up individuals as their values of which their matrix is true, and nothing, that
is, a zero-entity, if their matrix is true of nothing, by the following clause:
f(v.A)=f(v) if f(A)=1, otherwise f(v.A)=0, where 0 falls outside the domain of
the model, and
3.
we have to adjust the clause determining the value of a quantified formula in
an assignment as follows: f((Qv.A)(B))=1 iff for Q’u (u being an element of
RGf(v.A)), f[v.A:u](B)=1, where Q’ is the natural language equivalent of Q, and
RGf(v.A), the range of v.A with respect to f, is either identical with the extension
of A with respect to v and f, if it is not empty, or is a set containing the
zero-entity alone, if this extension is empty.
With
these clauses added to a standard construction of quantification theory we get
a powerful system, which, beyond restituting the Square of Opposition and all
the syllogistic forms previously invalidated by quantification theory, is able
to handle problems caused by complex noun phrases with relative clauses using
any types of determiners and the problems caused by anaphoric pronouns in
‘donkey-sentences’ in perfect accordance with what the mediaevals said
concerning the supposition of relative pronouns.[24]
If
we also add terms representing common terms combined with demonstratives and
interpret them relative to an index function (modeling the acts of pointing at
different objects) we can provide faithful representations of the above
descents. Indeed, it can be shown that these descents along with the
corresponding ascents give the correct truth conditions of the corresponding quantified
formulae (except for the much debated case of 2.b., but this is why we need the
addition of 4.)[25]
And this remark leads us to the other point I mentioned above, the problem of
the completeness of suppositional descents.
If
we examine carefully the above descents, then we can see that they are
basically of four kinds. Two of them lead to conjunctive and disjunctive
propositions, while two of them lead to propositions with conjunctive and
disjunctive terms. The conjunctive forms result from what we would call
universally quantified terms, while the disjunctive forms from existentially
quantified ones. The difference between the propositional and the
term-descents, as can be seen, is that of scope: if the quantifier binding the
term under which the descent is made has wider scope than the quantifier
binding the other term, then the descent is propositional, if, however this
quantifier has narrower scope than the other, then the descent is to be made to
a proposition with a disjunctive or conjunctive term.[26] Schematically, if
x and y are variables ranging over some countable domain, that is, from the
point of view of supposition theory, variables representing terms of universal
extension,[27]
related by a relation R and Arabic numerals are names of individuals of the
domain, we have the following four cases:
(1)
($x)(y)(R(x)(y)) Û (y)(R(1)(y))Ú(y)(R(2)(y))Ú...
(2)
(y)($x)(R(y)(x))
Û ($x)(R(1)(x))&($x)(R(2)(x))&...
(3)
(y)($x)(R(y)(x))
Û (y)(R(y)(1Ú2Ú...))
(4)
($x)(y)(R(x)(y))
Û ($x)(R(x)(1&2&...))
To
be sure, for a correct incorporation of these equivalences into a formal theory
we should interpret the formula schemata standing on the right side of these
equivalences as standing for formulae with an appropriate number of conjuncts
and disjuncts that are materially equivalent to the left hand side formulae in
particular models. (Of course, using restricted variables in the proper sense,
this appropriate number will be the cardinality of their range in the given
model.)[28]
But if we do give this interpretation, then these equivalences provably hold.
Now given the mediaeval logical-grammatical analysis of categoricals as
consisting of two terms prefixed by an explicit or implicit universal or
particular determiner joined by the copula (interpreted by late mediaeval as
expressing identity), these equivalences provide complete truth conditions for
any conceivable categorical sentence. And note here that apparent
counterexamples with verbal predicates were explained away by analyzing verbs
into copula and participle, and that the two terms were conceived to be of any
complexity possibly involving relative clauses of any sentential complexity, so
this conception involves a large class of natural language sentences indeed.
So
far, so good, one might say, but, despite my sweeping claim about the possible
fundamental role of supposition theory, all I have done thus far was not so
much using supposition theory as a foundation of a unified theory of reference,
as using it as an informal motivation for a particular sort of restricted
quantification theory and using this formal theory for a (rather sketchy)
reconstruction of suppositional descents. So instead of using the horse to pull
the car, I fixed the car to pull the horse (admittedly, taking tips from the
horse).
Well,
I accept this criticism regarding what I have said thus far, so to substantiate
my claim let me show now where I think suppositional descents may, indeed
should, have priority over restricted quantification, for the reason that they
can serve as explanations for the behavior of certain common noun phrases much
better than the idea of restricted quantification. Indeed, I wish to show how
common noun phrases as restricted quantifiers can be interpreted as special
cases of such descended forms, and why this interpretation is preferable
particularly with regard to two special contexts: namely the context of
intentional verbs and the context of numerically quantified ambiguous
sentences.
Consider
the following sentence: ‘I owe you a horse’. According to one of its possible
interpretations, this sentence is true even if no horse is such that I owe it
to you - namely, when my obligation does not concern some particular horse
(possibly specified by name or description in a contract), but only a horse in
general, that is, any horse whatsoever.[29] However, if we try
to formalize this sentence in quantification theory, whether we use restricted
or unrestricted quantification, we cannot give the correct truth conditions for
this interpretation. (For ($x)(Hx&O(a)(x)(b), would
read like this: ‘Something is a horse and I owe it to you’, while ($x.Hx)(O(a)(x.Hx) (b)) like
this: ‘Some horse is such that I owe it to you’, which are clearly not
equivalent to the intended interpretation. ‘x.Hx’ is a restricted variable
picking up its values from the extension of ‘Hx’ in a model. Cf. my papers
referred to in nn. 24. and 25.) Notice that with this example it would be
highly unintuitive to try something similar to Montague’s trick with ‘John
seeks a unicorn’, analyzing it essentially in terms of ‘John tries to find a
unicorn’, since for this sentence there seems to be no obvious paraphrase of
this kind, and, in any case, even if there were such a paraphrase, the formal
analysis would apply only to the exponent sentence, leaving the semantic
function of the problematic noun phrase in the original unexplained.[30]
Mediaeval
logicians, instead of trying to avoid accounting for the semantics of this
sentence by paraphrasing it away in terms of "easier" ones, faced
directly the problem in terms of supposition theory. In fact, the Latin
equivalent of the above sentence receives extensive treatment by Buridan in his
Sophismata, while Ockham in his Summa Logicae discusses at some length the
supposition of ‘horse’ in a similar sentence: ‘I promise you a horse’.[31]
In his discussion Ockham writes as follows:
"...
we have to say that propositions like this: ‘a horse is promised to you’,
‘twenty pounds are owed to you’, according to their proper meaning are false,
because any of the singulars is false, as is clear inductively. However, if
their terms like these are placed on the part of their predicate, they can be
conceded in a sense. And then we have to say that the terms following these
verbs, in virtue of these verbs have merely confused supposition, and so we
cannot descend to the singulars by a disjunctive proposition, but only by a
disjunct predicate, enumerating not only present things, but also future ones.
So this is not a valid inference: ‘I promise you a horse, therefore I promise
you this horse or I promise you this horse and so on ... So we have to know
that in such a proposition ... the common term in question does not supposit
determinately, taking ‘suppositing’ in the sense in which also a part of an
extreme can supposit, that is, you cannot descend under that term to the
singulars by a disjunctive proposition, but only by a proposition with a
disjunct extreme, or a disjunct part of an extreme."[32]
Now
comparing Ockham’s analysis with the descent schemata above we can clearly see
why we have troubles with these propositions in a quantificational approach: a
term having merely confused supposition, in (restricted) quantification theory
is like a quantified variable bound by a narrow scope existential quantifier;
but in this case the quantifier binding (the restricted variable representing)
the term ‘horse’ should have narrower scope even than the verb, indeed, the
quantifier should not get out of the argument place of the verb, which is
impossible already for mere syntactic reasons in any sort of quantification
theory.
Indeed,
the same is shown further if we consider the sentence ‘I owe you two horses’,
which, in the vein of Ockham’s above analysis, is clearly not equivalent to
‘Two horses are such that I owe them to you’, which, however, is the only
possible reading of the corresponding quantified formula.
(‘(2x.Hx)(O(a)(x.Hx)(b))’)
On
the other hand, ‘I owe you a horse’ seems to be intuitively clearly equivalent
to ‘I owe you this horse or that horse and so on’ without being equivalent to
‘I owe you this horse or I owe you that horse and so on’.
Again,
‘I owe you two horses’ seems to be equivalent in the same way to ‘I owe you
this horse and that horse or that one and that one and so on’ without being
equivalent to ‘I owe you this horse and that horse or I owe you that one and
that one and so on’.
So
in this case (a generalized form of) Ockham’s account seems to be clearly preferable
to a quantificational account, provided that we are able to explain why and how
these verbs cause merely confused supposition in contradistinction to other,
extensional verbs, and that we can supply a working semantics for the nominal
disjunctions and conjunctions involved. So let me turn to these topics.
In
his treatment of intentional verbs, Buridan explains the peculiarities of these
verbs in the framework of his theory of appellation, which may be characterized
roughly as a general theory of connotation.[33] However, without
going into the details of this otherwise highly interesting doctrine, let me
deal here only with that part of it which concerns the context of intentional
verbs. According to Buridan, the peculiarity of these verbs is that they make
the terms following them connote their rationes, i.e., the concepts according
to which they signify external things.[34]
In
some of my earlier papers I made a proposal concerning how an exact
reconstruction of Buridan’s concepts or rationes can be given within the
framework of a general formal semantics, so that we shall have no troubles in
the identification of concepts in a semantic model. But lack of space does not
allow me to elaborate this proposal here.[35] Nevertheless,
whatever we take Buridan’s rationes really to be, it is quite clear that
insofar as we are able to identify them and correctly distinguish them from one
another, they may present a good explanation for the peculiar behavior of noun
phrases in the context of intentional verbs.
For
if we suppose that we give an account of these rationes according to which the
term ‘horse’ and the disjunct term ‘this horse or that horse or ...’ (giving a
complete enumeration of horses including even future ones, as Ockham said) have
the same ratio, while all the singular terms of these disjunctions have
different rationes, and we determine the truth conditions of sentences with
intentional verbs so that they should depend also on these rationes, then
clearly, substituting the complete disjunction for the term following such a
verb will not affect the truth value of the proposition, while substituting any
of the singulars will. But it is precisely substitutions of these kinds that we
make in the different descents: when we descend from ‘I owe you a horse’ to ‘I
owe you this horse or that horse and so on’ (giving complete enumeration), we
substitute for ‘horse’ a term with the same ratio, so this substitution
preserves truth value, consequently the inference is valid; however, when we
descend to ‘I owe you this horse or I owe you that horse and so on’ in each
member of this disjunction ‘horse’ is replaced by a term with a different
ratio, so each member of the disjunction may be false while the premise is
true, whence the consequence is not valid, just as Ockham said.[36]
However,
to complicate matters, at one place Buridan does not allow descent even to a
proposition with a disjunct term, because he probably does not take the ratio
of this term to be identical with that of the original one.[37] On the other hand,
contrary to Ockham, he allows the inference from ‘I owe you a horse’ to ‘A
horse is such that I owe it to you’, indeed, to ‘Every horse is such that I owe
it to you’ for the reason that through the
general concept of ‘horse’ my obligation is related to every particular horse,
which, however, does not imply that I have to give you every particular horse.[38]
But
this difference between their particular intuitions and decisions on this
matter notwithstanding, Buridan’s theory, as we could see, can be used to
explain even Ockham’s rules. And even further, if we took sides with Ockham, we
could explain even the apparent validity of the passage from ‘I owe you two
horses’ to ‘I owe you this horse and that horse or that one and that one and so
on’ (giving a complete enumeration of all pairs of horses) without committing
ourselves to the truth of ‘Two horses are such that I owe them to you’, or ‘I
owe you this horse and that horse or I owe you that horse and that horse, and
so on’, provided we would work out an account of the rationes of the noun
phrases involved parallel to the above case. But without going into the
technical problems of assigning the appropriate rationes to these noun phrases,
one thing may be interesting in these descents even regarding other contexts,
namely the analysis of a numerical quantifier in terms of a disjunction of
conjunctions. So let us turn now to this topic.
Recently
several papers appeared that were addressed to the problems involved in the
analysis of numerically quantified ambiguous sentences like ‘Two examiners
marked six scripts’.[39]
In this section I only try to indicate very briefly how I think a generalized
theory of suppositional descents could provide a unified framework for handling
sentences of this kind.
The
basic idea can be put in one sentence as follows: we can treat all common noun
phrases with numerical determiners as nominal disjunctions of nominal
conjunctions having as many members as the cardinality of the numerical
determiner, while we can determine scope relations by allowing further descents
to disjunctive and conjunctive propositions. Semantically, we can determine the
import of such a complex nominal phrase by saying that a complex predicable is
true of a nominal disjunction if and only if it is true of at least one of its
members, while it is true of a nominal conjunction, if and only if it is true
of each of its members. But this latter holds only of the distributive reading
of nominal conjunctions: further ambiguities can be accounted for by
distinguishing between distributive, collective and divisive readings of
nominal conjunctions, or rather of argument places of predicates in which these
conjunctions occur, just as the mediaevals did.[40] In this way from
the general nominal descent scheme of an ambiguous numerically quantified
sentence we can get specifications of its possible readings by the further
possible propositional descents and these distinctions.
So
e.g. the general nominal descent scheme of ‘Two examiners marked six scripts’
may be given as follows:
(e1&e2
Ú
...)M(s1&s2&s3&s4&s5&s6
Ú ...)
or,
in general, for any terms S and P, and any relation R,
(si&sj... Ú sk&sl...)R(pm&pn... Ú po&pq...)
where
the number of conjuncts is that of the numerical determiner, the range of the
numerical subscripts relative to a model is identical with the cardinality of
the extensions of the original terms (in our example the terms: ‘examiner’ and
‘script’) in that model, while the number of disjuncts is to be such that the
set of referents of the singular terms should be identical with the extension
of the original common term in this model, if the set of singular terms
occurring in the conjunctions varies from disjunct to disjunct, and arbitrary,
if the same set of terms makes up the conjunctions in each disjunct. Indeed, we
may take this as a degenerate case, and take here a one-member disjunction
instead of one with several members, that is, one conjunction alone. As a
matter of fact, this treatment of degenerate cases shows us that noun phrases
with the ordinary quantifiers can be regarded as degenerate cases of the above
general scheme. A universally quantified noun phrase may be regarded as a
one-member disjunction of conjunctions in which all members are different and
their referents together exhaust the extension of the quantified term. An
existentially quantified noun phrase may be regarded as a disjunction of
one-member conjunctions (that is conjunctions with the same members), but such
that the referents of the disjuncts are different, and together exhaustive of
the extension of the quantified term. I think it is easy to see how several
other determined noun phrases of natural languages could be defined along these
lines, but I do not want to linger on this point here. Instead, I would like to
indicate how we can get from the above general nominal descent scheme the
possible different readings of the same ambiguous sentence.
As
we could see from the four types of descent schemata (1)-(4) above, descent to
propositional disjunctions and conjunctions expresses the larger scope of a
noun phrase in comparison with descent with nominal conjunctions and
disjunctions. So descending to disjunctive propositions once under the left and
once under the right side argument of M gives us two scope-differentiated
readings of the restricted quantifier analysis of ‘Two examiners marked six
scripts’, satisfiable either by a situation possibly involving two examiners
and twelve scripts, each of them being marked by one of the examiners, or by a
situation involving six scripts and twelve examiners each of the scripts being
marked by two examiners and each examiner marking exactly one script.
However,
we can descend by disjunctive propositions also on both sides, so that neither
of the noun phrases of the original sentence gets wider scope than the other
like this: ‘this and that examiner marked this and that and so on, that is,
these six scripts, or that and that examiner marked those six scripts, and so
on’, in which case our sentence says that we have some set of two examiners and
some set of six scripts each of which was marked by each of the examiners,
which is the branching quantifier reading of this sentence.
But
we can get even further possible readings if we consider the collective and
divisive interpretations of nominal conjunctions, or rather of argument places
of predicables in which these conjunctions occur, as I have said. For, as is
well-known, certain predicables can apply only to groups of individuals without
applying to the members of these groups. For example, even if we can truly say
that six wolves surrounded two deer, it is not true of any of these wolves that
it surrounded two deer, or for that matter, any number of deer. So in this case
we cannot think of the predicable ‘surrounded two deer’ as applying to the
conjunction enumerating six wolves if and only if it applies to all of its
members, but as applying to what the conjunction as a whole applies to, namely
the six wolves enumerated by it together. In general, we can say that a
predicable is true of a nominal conjunction taken collectively if and only if
it is true of what the conjunction as a whole applies to, namely of the
collection of the individuals enumerated in the conjunction. Note here that
while the first argument place of ‘surrounded’ is necessarily collective, the
other may be taken either as collective or as distributive. In the latter case
the sentence ‘Six wolves surrounded two deer’ may be true in a situation in
which one deer is surrounded by six wolves and another by other six wolves.
But
it may also be the case that six wolves so surround two deer that three of them
surrounds one deer and the other three the other one. In this case neither six
wolves taken one by one, nor six wolves taken together can be said to have
surrounded two deer, rather we can say that some subgroups of a sum total of
six wolves surrounded some one-member ‘subgroups’ of a sum total of two deer.
That we should think here of one-member subgroups of deer instead of individual
deer is easily seen if we take examples with higher numbers. But, of course, a
predicable is true of such a one-member group if and only if it is true of its
one member. Note here, that divisive reading of one argument place of a
relation forces divisive reading of the other too, while distributive and
collective readings could freely combine with one another. In general, we can
say that if we attribute divisive readings to the two argument places of the relation
R, then the truth condition of (the particular formula instantiating in a
particular model) the general descent scheme of a sentence (NS)R(MP) is that
there be some together exhaustive subconjunctions of the N-member and M-member
conjunctions of (the formula instantiating in that model) the general scheme
such that R holds of all of these subconjunctions either collectively or
distributively. For example, on a divisive reading of ‘Twelve wolves surrounded
three deer’ this sentence may be true in a situation in which, say, six wolves
surrounded one deer and six others surrounded two other deer.
So,
in this way from the general nominal descent scheme of a numerically quantified
ambiguous sentence by means of the further possible propositional descents and
by distinguishing the three possible readings of nominal conjunctions we can
generate apparently all possible readings of these sentences. We could also see
how sentences with the ordinary quantifiers and possibly also with others can
be regarded as special cases of these general descent schemata. We could even
see how these descents might work in the thorniest intentional contexts. I
think it is also quite easy to imagine how, with reference to the divisive
readings of nominal conjunctions, these descent schemata could account for
plurals. So I hope by now it seems not so exaggerated to claim that the theory
of suppositional descents may indeed serve at least as a starting point of a
unified theory of reference in natural languages. But further elaboration of
this claim would exceed the limits of this paper.
In
the first two sections I presented the state of our art as characterizable in
terms of the crumbling of an old paradigm in view of the accumulation of
anomalies and, at the same time, by a quest for new unifying perspectives. I
took justification from this description for seeking different new
orientations, occasionally with a view to old exemplars. I repelled an
objection to seeking our historical exemplars in traditional logic by pointing
to the enormous difference between the traditional logic of last century logic
textbooks and that of the mediaeval masters of logic. Making reference to the
growing contemporary interest in the mediaeval theory of supposition, I set out
to show how in my view this theory could be used also in modern
logico-linguistic research as what it was originally meant to be, as a
foundation of a unified theory of reference in natural languages. After a brief
presentation of the basic definitions and divisions, I pointed out the
fundamental agreement of the doctrine of suppositional descents with the idea
of a particular sort of restricted quantification. I even sketched how the
theory of descents can be reconstructed, and how the completeness of descents
in giving the truth conditions of categorical sentences with complex noun
phrases can be shown within the framework of such a restricted quantification
theory. Then I tried to show that rather than using restricted quantification
to explain suppositional descents, we should use descents to explain the
behavior of common noun phrases both in cases in which the quantifier analysis
works and in those in which it fails. I selected as test cases the contexts of
intentional verbs and those of numerically quantified ambiguous sentences. In
the former case I argued that the quantificational analysis should fail of
necessity, already for mere syntactical reasons. Then I indicated that Ockham’s
analysis of these contexts supplemented with Buridan’s theory of appellation
may give satisfactory results, and may even explain the opposing intuitions of
the two authors. However, instead of trying to elaborate here the technical
details of appellation theory, I turned to the analysis of numerically
quantified sentences in terms of suppositional descents. I tried to show how
common noun phrases may be regarded as nominal disjunctions of nominal
conjunctions of singular terms in general, and so how the common noun phrases
with the usual quantifiers may be regarded as special (degenerate) cases of
these nominal disjunctions of conjunctions. I have also made proposals as to
the semantic import of these nominal disjunctions and conjunctions in
determining the truth conditions of sentences in which they occur, indicating a
threefold distinction of the possible readings of nominal conjunctions. Then I
tried to show how the several possible readings of a numerically quantified
ambiguous sentence may be generated from such a general nominal descent scheme
by further possible propositional descents and by making use of the
distinctions between the possible readings of nominal conjunctions. I have also
remarked that the divisive readings of nominal conjunctions may be useful in
the analysis of plurals.
Of
course, several claims I made could not receive appropriate treatment within
the confines of this paper. Most importantly, I think the following points need
further elaboration:
1.
an account of Buridan’s appellatio rationis, which would allow us to explain
the peculiarities of the possible descents under common terms in the context of
intentional verbs
2.
a systematic account of the semantics of nominal conjunctions and disjunctions
in general
3.
an account of plurals in terms of the divisive reading of nominal conjunctions
The
general semantic framework for the elaboration of these points may be, I think,
also classical model theoretical semantics. But it is also a tempting idea to
regard the several possible descents under common terms in several contexts as
describing particular semantic games for the evaluation of a sentence in which
these terms occur in a particular model. So from this point of view it seems
that a combination of the theory of suppositional descents with game
theoretical semantics may provide even more interesting results.[41]
[1] G. Boolos: "To Be Is
To Be a Value of a Variable (or To Be Some Value of Some Variable)", The
Journal of Philosophy, 8(1984), pp.430-431.
[2] Indeed, this distinction
renders mismatch of syntax the rationale of an interesting research program for
linguists: if quantification theory expresses logical form, i.e., deep
structure of natural language sentences, then the task of linguistic research
may appear to be to reveal the intricate connections between surface and deep
structures in particular languages. Cf. e.g. G. Englebretsen: "Logical
Form and Natural Syntax", Indian Philosophical Quarterly, 11(1984),
pp.229-254. What is more, logical form may provide means to distinguish among
several senses of crucial concepts hidden by grammatical form, such as the
several senses of ‘to be’. For discussions of Frege’s "ambiguity
thesis" in a historical context cf. J. Hintikka-S. Knuuttila: The Logic of
Being, Helsinki, 1986.
[3] For proof see J. Barwise
& R. Cooper: "Generalized Quantifiers and Natural Language",
Linguistics and Philosophy, 4(1981), pp.159-219.
[4] See e.g. G. Evans:
"Pronouns, Quantifiers and Relative Clauses" and
"Appendix", Canadian Journal of Philosophy 7(1977) pp.467-536. and
pp.777-796., B. Richards: "On Interpreting Pronouns", Linguistics and
Philosophy, 7(1984), pp.287-324.
[5] Cf. D. Lewis: "General
Semantics", in: B. Partee: Montague Grammar, New York-San
Francisco-London, 1976.
[6] Cf. D.P. Henry: Mediaeval
Logic and Metaphysics, London, 1972; D.P. Henry: That Most Subtle Question - (Quaestio
Subtilissima): The Metaphysical Bearing of Medieval and Contemporary Linguistic
Disciplines, Manchester, 1984; F. Sommers: The Logic of Natural Languages,
Oxford, 1982; G. Englebretsen: "The Square of Opposition", Notre Dame
Journal of Formal Logic, 17(1976), pp.531-541.
[7] R. Montague: "The
Proper Treatment of Quantification in Ordinary English", in: J.
Hintikka-J. Moravcsik-P. Suppes: Approaches to Natural Language, Dordrecht,
1973.
[8] Cf. J. van Benthem-A. ter
Meulen: Generalized Quantifiers in Natural Language, Foris Publications, 1985.
[9] See H. Kamp: "A Theory
of Truth and Semantic Representation", in: J. Groenendijk et. al. eds.:
Formal Methods in the Study of Language, Amsterdam, 1981.
[10] As two main attempts in
this direction, Lesniewski’s Ontology and Fred Sommers’ term logic should be
mentioned here.
[11] See e.g. J. Hintikka:
Logic, Language-Games and Information, Oxford, 1973.
[12] J.Barwise-J.Perry:
Situations and Attitudes, The MIT Press, 1983.
[13] G. Bealer: Quality and
Concept, Oxford, 1982.
[14] On the other hand, I think
quite good general motivation could be taken from the "natural logic"
character of mediaeval logic, in the sense of G. Lakoff: "Linguistics and
Natural Logic", in: D. Davidson-G. Harmann: Semantics of Natural Language,
Dordrecht, 1972. For an excellent discussion of mediaeval logic from this point
of view see: A. de Libera: "La Logique du Moyen Age comme Logique
Naturelle (Sprachlogik)", in: B. Mojsisch: Sprachphilosophie in Antike und
Mittelalter, Amsterdam, 1986. Nevertheless, I do not wish to pursue this line
here. Instead, I am going simply to try to show this character of the mediaeval
approach.
[15] Buridanus: Tractatus de
Suppositionibus (ed. Maria Elena Reina), Rivista Critica di Storia della
Filosofia, 1957, pp.177-208. and pp.323-352; see pp.191-195.
[16] Characteristic of this
attitude is the following short evaluation of Buridan’s philosophy from Peter
King’s Introduction to his translation of two logical tracts by Buridan:
"Buridan’s mediaeval voice speaks directly to modern concerns: the attempt
to create a genuinely nominalistic semantics; paradoxes of self-reference; the
nature of inferential connections; canonical language; meaning and reference;
the theory of valid argument. It is to be hoped that Buridan can reclaim his lost
reputation among contemporary philosophers for his penetrating and incisive
views on these and other matters."
P. King: Jean Buridan’s Logic, D. Reidel Publishing Company, 1985. p.4.
[17] For good bibliographies on
the topic see N. Kretzmann-J. Pinborg-A. Kenny: The Cambridge History of Later
Mediaeval Philosophy, Cambridge, 1982; E.J. Ashworth: The Tradition of
Mediaeval Logic and Speculative Grammar, Toronto, 1978.
[18] This characterization of
supposition theory, which among several other attempted characterizations I
find the most fitting one, comes from J. Ashworth. Cf. her "Promitto tibi
equum", Vivarium, 16(1976), pp.62-78. For a similar account with good
theoretical and textual support see also G. Priest-S. Read: "Merely
Confused Supposition: A Theoretical Advance or a Mere Confusion?",
Franciscan Studies, 40(1980), pp.265-297.
[19] The addition ‘by reason of
their meaning’ in the characterization of singular vs. common terms is needed
for the reason that it may well be the case that a proper noun is predicated of
many, but only when used equivocally, being imposed upon different persons. It
may also be the case that a common term can be predicated only of one thing
because there can be only one thing of its kind. Nevertheless, this occurs not
due to the meaning of the term, but because of the nature of the thing, like in
the case of the term ‘God’. So this determination is added for the exclusion of
such apparent exceptions. Cf. Albert of Saxony: Perutilis Logica,
Hildesheim-New York, 1974. c.X.4d.
[20] Concerning the development
of the role of the converse ascents in supposition theory and, in general, the
problems involved in the requirement to descend to an equivalent proposition
see the excellent discussion in G. Priest-S. Read: "Merely Confused Supposition:
A Theoretical Advance or a Mere Confusion?", Franciscan Studies, 40(1980),
pp.265-297. For a good summary of the arguments against presenting supposition
theory as a sort of quantification theory, giving the truth conditions of
quantified sentences in terms of descents, precisely on account of the failure
of ascents see M. McCord Adams: William Ockham, Notre Dame, 1987, pp.367-377.
Adams’ alternative proposal is that "the divisions of common personal
supposition are not the means to the end of giving a contextual definition of
quantifiers nor for stating the truth conditions for propositions containing
quantified general terms; rather the divisions of supposition generally were
marshalled into service for the task of identifying fallacies". op. cit.
p.382. To be sure, the development of supposition theory from its very origins
was motivated by the need to detect fallacies, as it was convincingly shown by
L.M. de Rijk: Logica Modernorum, I-II. Assen, 1967. Nevertheless, it may be
argued that the need for fallacy detection developed also a relatively
independent interest in the referring function of terms in general, which,
during the development of supposition theory, evolved, among other things, the
explicit requirement of analyzing quantified sentences in terms of equivalent
descended forms, as we can clearly see this in such later authors as e.g. Paul
of Venice. (Cf. his Logica Magna, tr.2, ed. A.R. Perreiah, St. Bonaventure,
1971.) However, without going into the debate concerning its real historical role
and purpose, let me propose as a "conciliatory characterization" of
supposition theory the following: supposition theory is (aimed to be) a unified
theory of reference with the original intent of fallacy detection, in its most
mature form having the capability of giving (as I shall argue) a complete set
of truth conditions for quantified sentences in terms of equivalent descended
propositions. It is this inherent capability that I wish to develop in this
paper.
[21] See G. Priest-S. Read, op.
cit., esp. pp.289-295.
[22] To be sure, this
characterization of this mode as such cannot be found in the authors. (Cf.
however Paul of Venice op. cit. pp. 90-92.) Nevertheless, considerations of
completeness to be discussed below seem to require it. Cf. n.25. As for the implicational
order of these modes, see again Priest-Read, op. cit.
[23] Cf. e.g. Buridan:
Sophismata, ed. T.K. Scott, Stuttgard-Bad Cannstatt, 1977, p.50.; Marsilius de
Inghen: Treatises on the Properties of Terms, Dordrecht-London-Lancaster, 1983,
p.52.
[24] See "General Terms in
their Referring Function" in my Ars Artium: Essays in Philosophical
Semantics, Mediaeval and Modern, Budapest, 1988.
[25] See "The Square of
Opposition, Common Personal Supposition and the Identity Theory of Predication
within Quantification Theory" in the collection of my papers mentioned in
the preceding footnote.
[26] As a matter of fact, this
understanding of the difference between nominal and propositional descents, as
expressing scope relations of quantified terms, gives also a clue to resolving
the notorious problem of attributing confused and distributive supposition to
predicates of O-propositions, like ‘Some man is not an animal’. (Cf. 2.b.
above.) For the real problem with the corresponding descent is that by
descending propositionally we attribute wider scope to ‘an animal’ over ‘some
man’ in this proposition, in which, however, clearly the converse scope
relation holds. So to set things right either we have to descend by a nominal
conjunction under ‘an animal’, or we have to take ‘some man’ to refer to the
same man in each propositional conjunct, i.e. read ‘some man’ referentially, as
if we had already descended under it propositionally to some determinate man,
following a ‘priority rule of analysis’. Indeed, both of these remedies were
considered in the literature. See Priest-Read, op.cit., Adams, loc. cit., King,
op. cit. p.51., A. Broadie: Introduction to Mediaeval Logic, Oxford, 1987,
p.24.
[27] You may even think of them as
x.x=x and y.y=y, respectively, I am using simple variables only to reduce the
complexity of the schemata below.
[28] In this case special care
needs to be taken of cases when these ranges either are infinite or contain
only 0. For the technical details see "The Square of Opposition, Common
personal Supposition and the Identity Theory of Predication within
Quantification Theory" in my Ars Artium: Essays in Philosophical
Semantics, Mediaeval and Modern, Budapest, 1988.
[29] Cf. P. Geach: "The
Perils of Pauline", in: Logic Matters, Berkeley and Los Angeles, 1972.
[30] Cf. R. Montague: "The
Proper Treatment of Quantification in Ordinary English", in: J.
Hintikka-J. Moravcsik-P. Suppes: Approaches to Natural Language, Dordrecht,
1973; J.D. McCawley: Everything that Linguists have Always Wanted to Know about
Logic - but were ashamed to ask, Oxford, 1981. pp.411-421.
[31] See Buridan: Sophismata,
ed. T.K. Scott, Stuttgard-Bad Cannstatt, 1977, pp.83-90; Ockham: Summa Logicae,
ed. Ph. Boehner, St. Bonaventure, N.Y., 1974,
P.I.c.72., pp.219-221.
[32] Ockham: Summa Logicae, ed.
cit., P.I.c.72., pp.219-221. cf. P.II.c.7. Cf. also Guillelmi de Ockham
Scriptum in librum primum Sententiarum Ordinatio, St. Bonaventure N.Y.,
1967-1979; d.2.q.4., pp.145-148. Cf. also the similar treatment of Albert of
Saxony: Perutilis Logica, Hildesheim-New York, 1974, 14ra.
[33] See King, op. cit. pp.
17-25. A. Maierú: "Signification et Connotation chez Buridan", in: J.
Pinborg (ed.): The Logic of John Buridan, Copenhagen, 1976.
[34] See Buridan: Sophismata, ed.
cit., c.4.pp.59-90; Tractatus de Suppositionibus, ed. cit., pp.184-185,
333-335, 343-347.
[35] See "Understanding
Matters from a Logical Angle" and "Socrates est species" in my
Ars Artium. Presently I am working on a detailed and, at least according to my intentions,
faithful formal reconstruction of Buridan’s theory of signification and
appellation as they are set to work in his analysis of ‘Debeo tibi equum’, in a
paper under this title to be presented at the 9th European Symposium of
Mediaeval Logic and Semantics, in St. Andrews, Scotland.
[36] For a similar analysis of
Ockham’s treatment see S.L. Read: "‘I promise a penny that I do not
promise’: The Realist/Nominalist Debate over Intensional Propositions in Fourteenth-Century
British Logic and its Contemporary Relevance", in: The Rise of British
Logic, ed. P. Osmund Lewry, O.P., Papers in Mediaeval Studies 7 (Toronto:
Pontifical Institute of Mediaeval Studies, 1985), pp.335-359.
[37] "... but in this kind
of confusion it is not permissible to descend to the supposits either by a
disjunctive sentence or by a categorical with a disjunct extreme, since such
verbs make the terms following them appellate their rationes, namely those
according to which they were imposed to signify." King’s translation (ed.
cit. p.145.) of Buridan’s Tractatus de Suppositionibus, ed. cit. pp.333-334.
[38] See Buridanus: Sophismata,
ed. cit. pp.83-90.
[39] See e.g. M. Davies: ‘Two
examiners marked six scripts: Interpretations of Numerically Quantified
Sentences’, Linguistics and Philosophy, 12(1989) pp.293-323. Davies supplies
also a number of further references.
[40] Cf. e.g.: "... this
sign ‘omnis’ <meaning: ‘all’ in the plural, as opposed to its distributive
meaning, translatable as ‘every’, which it has in the singular>, when it is
taken in the plural, may be interpreted either collectively or divisively. If
it is taken divisively, it denotes that the predicate truly applies to all
those things of which the subject is truly predicated, like by this: ‘All
apostles of God are twelve’ is meant that this predicate: ‘twelve’ is truly
predicated of all those of which the subject ‘apostles’ is truly predicated;
and so, since Peter and Paul are apostles, it follows that Peter and Paul are
twelve. <And in this sense the proposition is false, of course.> But if
it is taken collectively, it does not denote that the predicate applies to all
those to which the subject applies, but that it applies to all those things
taken together of which the subject is verified; so it means that these
apostles, pointing at all the apostles, are twelve." Ockham: Summa
Logicae, ed. cit., p.266. Cf. further: "Introductiones Montane
Minores", in: L.M. de Rijk: Logica Modernorum, ed. cit., II-2., pp.29-30.;
Buridan: Tractatus de Suppositionibus, ed. cit., pp.199-200.; William of
Sherwood: Syncategoremata, Mediaeval Studies, 3(1941), pp.46-93 esp. pp.84-89;
Walter Burleigh: De Puritate Artis Logicae Tratctatus Longior with a revised
edition of the Tractatus Brevior, St. Bonaventure, N.Y., 1955, pp.241-243,
252-253; Paul of Pergula: Logica and Tractatus de Sensu Composito et Diviso,
St. Bonaventure, N.Y.-Louvain-Paderborn, 1961. pp.152-153.
[41] As a matter of fact, this
paper is only a preparatory work for a joint project with Gabriel Sandu of the
University of Helsinki in which we try to work out the technical details of
this intuitive idea. (Appeared: Klima,
G.—Sandu, G. “Numerical Quantifiers in Game-Theoretical Semantics”, Theoria, 56(1990), pp. 173-192.) Earlier
drafts of this paper were presented at the Department of Linguistics and at the
Department of Philosophy of the University of Helsinki during my stay in
Helsinki as a member of the project ‘Ockham and the via moderna’ under the
chairmanship of Simo Knuuttila, in the Fall Semester of 1989. I wish to express
my gratitude to all the Finnish friends and colleagues for the inspiring
discussions and all kinds of help they provided.