Language, Logic, and the Brain
R.E. Jennings
ABSTRACT
Language is primarily a physical, and more particularly a biological phenomenon. To say
that it is primarily so is to say that that is how, in the first instance, it presents itself to
observation. It is curious then that theoreticians of language treat it as though it were
primarily semantic or syntactic or some fusion of the two, and as though our implicit
understanding of semantics and the syntax regulates both our language production and
our language comprehension. On this view the brain is both a repository of semantic and
syntactic constraints, and is the instrument by which we draw upon these accounts for the
hard currency of linguistic exchange. With this view comes a division of the vocables of
language into those that carry semantic content (lexical vocabulary) and those that mark
syntactic form (functional and logical vocabulary). Logical theory of the past 150 years
has been understood by many as a purified abstraction of linguistic forms. So it is not
surprising that the “logical” vocabulary of natural language has been understood in the
reflected light of that formal science. Those internal transactions in which “logical”
vocables essentially figure, the transactions that we think of as reasonings, are seen by
many as constrained by those laws of thought that logic was supposed to codify. Of
course no vocabulary can be entirely independent of semantic understanding, but whereas
the meaning of lexical vocabulary varies from context to context (run on the treadmill,
run on the market, run-on sentence, run in her stocking, run down, run the tap etc.)
logical vocabulary has fixed minimal semantic content independently of context.
A biological view of language presents a sharply contrasting picture. On an evolutionary
time-scale the human brain and human language have co-evolved. So we have prelinguistic ancestors, some of whose cunning we have inherited, as we have quasilinguistic ancestors and early linguistic ancestors whose inherited skills were enhanced
and made more effective by the slow acquisition of linguistic instruments of control and
coordination. Where in this long development does logic enter? On the shorter time-scale
of linguistic evolution, we know that all connective vocabulary descends from lexical
vocabulary, much of it from the language of spatial relationship. We can now say, more
or less, how that happens. We can even find many cases of mutations in logicalized
vocabulary, semantic changes that come about in much the way that biological mutations
occur in molecular biological processes. These changes proliferate to yield a wide
diversity in the evolved uses of natural language connectives. Just as surprisingly, we
discover, we don’t in general understand connective vocabulary, nor do we need to for
the purpose of using it correctly in speech. And by no means do our automatic uses of it
coincide with those that would be predicted by the syntax/semantics view. Far from
having fixed minimal semantic content, logical vocabulary is semantically rich, contextdependent, and, partly because we do not in general understand it, semantically extremely
fragile.
1. INTRODUCTION AND BACKGROUND
George Boole (1815-1864) called his seminal study The Laws of Thought. Boole took the
title seriously as a description of the relationship between logic and thought. Cognitive
scientists should not. Why not? is the subject of this address.
Boole’s ambitions for the treatise are clear enough, and are repeated several times in the
course of his introduction. This is the opening sentence:
The design of the following treatise is to investigate the fundamental laws
of those operations of the mind by which reasoning is performed1; to give
expression to them in the symbolic language of a Calculus, and upon this
foundation to establish the science of Logic, and construct its method.
(Boole, 1)
The difficulty is in getting at those fundamental laws. Boole presents his solution to the
problem, which is to get at them through an examination of the language of reason, as a
tactic of convenience. A fastidious reader might wish that Boole had spent fewer words
on a more perspicuous justification of the method.
…It will not be necessary to enter into the discussion of … whether
language is … an essential instrument of reasoning … It is the business of
Science to investigate laws; … whether we regard signs as the
representatives of things and of their relations, or as the representatives of
the conceptions and operations of the human intellect, in studying the laws
of signs, we are in effect studying the manifested laws of reasoning. …
For though in investigating the laws of signs a posteriori, the immediate
subject of examination is Language, with the laws which govern its use;
while in making the internal processes of thought the direct object of
inquiry, we appeal in a more immediate way to our personal
consciousness, it will be found that in both cases the results obtained are
formally equivalent. Nor could we easily conceive, that the unnumbered
tongues and dialects of the earth should have preserved through a long
succession of ages so much that is common and universal, were we not
assured of the existence of some deep foundation of their agreement in the
laws of the mind itself. (24-5)
Boole’s contemporary, Augustus De Morgan (1806-1871), had he understood what we
now understand of such things, would have been forced to judge Boole in this matter as
he judged the philosophers of his day, who, he said, ‘speak with the majesty of
ignorance.’ It is merely an article of faith that we could have access to something called
thought independently of our use of language, though we could empirically investigate
some human capacities by experiments that required no explanation as to protocol.
1
The emphasis is mine.
2. FOLK SEMANTICS
The notion that universal semantic features of language reveal themselves at the level of
connective vocabulary is more likely an artefact of our practices of translation. One still
reads in introductory logic tests, that English has one word for or, whereas Latin had two.
Latin has at least half a dozen words (including vel, aut, sive, seu, and enclitic ve) that we
can translate as or, and so does English. (They include unless, alternatively, but, and
otherwise.) One might ask whether each of these uses is usefully thought of as a
representative of a conception or operation of the human intellect. Boole, like many more
recent authors2, in the course of expounding over-simple theories about natural language
connectives, himself quite unselfconsciously uses the very connectives they are speaking
of in ways that their theories do not comprehend. In the Boole passage cited, notice the
whether …or construction. In every case that I know of, the judgement about the
connectives of natural languages arises, not from a study of natural languages, but from a
fascination with or a devotion to a formal technology. Psycholinguistic studies are not
free of this taint. In one study3 the dogma that the satisfaction-conditions of nonexclusive disjunction and those of exclusive disjunction exhaust the uses of English or
premisses a test administered to young children; nevertheless the explanation of the
experimental protocol offered to the young subjects is framed with a use of or that is not
admitted by the experiment. Unremarkably, the four-year-olds had sufficient command of
that use of the word, that they were able to understand the test. Unremarkably, none of
the adult academics noticed.
Here is an example: suppose that a waiter says to you ‘You can have tea or you can have
coffee.’ Is his pronouncement an exclusive or4 an inclusive disjunction?’ Audiences
generally divide on this subject into two camps: a majority who think the former, and a
minority who think the latter. But although when they are explicitly asked there is a
division on the question, when asked immediately thereafter whether in such a situation
they would infer that they could have tea, there is unanimity that anyone to whom a
waiter said this would be in doubt that he or she could have tea. But this implies that in
practice we do not understand the sentence as either sort of disjunction, because from a
disjunction (inclusive or exclusive) we cannot infer either disjunct. Our answer to the
abstract question is an artefact of our academic training; our inference in the event is an
outcome of our natural understanding of the language.
In general our naturally acquired linguistic understanding does not require and does not
confer a semantic understanding. This evident in the case of lexical vocabulary: as
children we acquire both the ability to be the first to use a vocable in conversation, and
the ability to make responses to its introduction in suitable ways that nevertheless do not
involve our using the vocabulary. Gradually, our uses and responses occasion neither
puzzlement nor correction. But we have almost all made it adulthood with some
eccentricities of vocabulary selection that our interlocutors have not corrected or queried.
There is no semantic audit in adolescence by which our uses of the language are held up
2
Paul Grice (1913-1988) is another.
Braine and Rumain, 1981.
4
Notice this interrogative use of or.
3
to independent expert scrutineers. There can be no such scrutineers; for all of us are, in
this respect, in the same semantic pickle. Unless we specifically make a study of it, we do
not understand the connective vocabulary of our own natural language. We merely know
how to use it and to how respond to its uses.
Experts with sufficient academic training in logic and the formal representation of natural
language sentences can write textbooks of logic. But few have themselves much
understanding beyond what they have acquired from teachers similarly placed who
themselves wrote textbooks. The result has been an academic tradition of glib recitals of
uninvestigated myths: myths about English or, myths about Latin aut and vel. They are
to be found in virtually every introductory logic text. Though venial in themselves, they
also both skew and miscolour academic perceptions of natural language, and its role in
our cognitive life. In the first place, they falsely teach us that language is primarily
semantic, though they give us no usable theory of meaning, and in the second, they
elevate the language of thought, belief, intention, inference, and reasoning to a theoretical
role that it cannot serve.
In general, we have at best an imperfect semantic understanding of natural language
connective vocabulary. As difficult a lesson as that is to learn, it is also true of mental
vocabulary. Like natural language connectives vocabulary, mental vocabulary is learned
without the benefit of explicit semantic theory. Part of the reason for this is that we
require no explicit semantic understanding in order to use the vocabulary with standard
proficiency. It is perhaps the failure to recognize that fact that sends us haring after
semantic theories. It is that fact among others that makes it a certainty that we will not
find one. For if its existence is not dictated as a requirement of the transmission of
language, nothing remains that is capable of enforcing or preserving it. One might more
plausibly argue that the overburden of such a requirement would effectively prevent the
transmission of language. We can, however, explain why a language has the particular
connectives and the particular mental terms that it has; moreover, as we better understand
how we come by the terms, we also better understand the futility of attempting to gain an
understanding beyond that required for its conversational use. We will consider
connectives in this light in due course. As for mental terms, I merely illustrate the point
with the verb intend. English owes the word to Norman French and ultimately to Latin,
and we may suppose without grief that there were uses in these ancestral languages that
roughly correspond to our own. But the earliest uses of its Latin forebear, as any Latin
pupil can confirm, and its principal non-figurative understanding had to do with the
drawing of a bowstring, an understanding that was certainly still present in sixteenthcentury English. A secondary but still physical understanding, namely to point towards a
target exploited a usual incidental feature of the primary action of drawing the string. Its
initial mental uses were also crudely spatial: one aimed one’s soul, and in another use,
one aimed in one’s soul. Thus anyone who supposes that in its descent from those earliest
figurative uses, it has become transformed into a theoretically useful explanatory
instrument is bound to say how that transformation came about. Its history records only
the loss of its original lexical understanding, and the ellipsis of explicit mention of anima
in its use. No doubt educated sixteenth-century English speakers still distinguished
figurative uses from non-figurative ones. But the language seems to have lost the non-
figurative use since the advent of firearms. Paraphrase of the residual use requires
similarly problematic mentalistic vocabulary in such constructions as minded to, and
figurations, as in have it in mind to.
3. THE FRAGILITY OF SYNTAX
All of the so-called “logical” connectives of English (and, or, if, but, unless and so on)
are typically morphologically reduced residua of lexical vocabulary, typically
prepositional or relative adverbial, but in any case, typically representing a spatial,
temporal or other natural relationship. Or, for example is a reduced form of oþer (other),
which we would translate as second, as in “every second (every other, every alternate)
day”. The original survives in such words as otherwise and either. But is a reduced form
of by outan, then butan (outside), unless, a reduction of the longer prepositional phrase
on [a condition] less (than that…).
Consider but as a representative example. Its early ancestral uses were closely
constrained to structures such as dwellings. The first stage of its functionalization is the
slow generalization of its use in application to geographical, and by assimilation, to
institutional boundaries (but (i.e., outside) the country), then to categorical and unit
boundaries (nothing but rubble, no one but his mother, could not but smile), thence to
circumstantial boundaries (but for the rain, but that the rain ended the search). At this
point there stands only the ellipsis of that between prepositional uses of but and its first,
disjunctive, connective use It doesn’t rain but it pours. Compare the dialectic It doesn’t
rain without (≈ outside) it pours. Of course this is a crude sketch of one path, and there
are more than one by which language of physical relationship yields up in its maturity
what some would call a propositional connective. Each of these stages represents an
exploitation or abaptation5 of a previous use.
Our familiarity with the word but tells us that the story does not end there, since the uses
mentioned so far do not exhaust the familiar list. In fact our explanation can be extended
to include even a short list of linguistically problematic uses of but; these are, however
discussed more fully elsewhere. For present illustrative purposes we take but also as a
clear illustration of a second phenomenon of linguistic development: mutation. Although
mutations in language transmission are more unlike DNA mutations than like, the use of
the term mutation is anything but metaphorical and very particularly applied to string
replication errors, so more like frameshift mutations than like other biological sorts. Like
biological mutations, they are sometimes insignificant, sometimes unimportant, and
sometimes capable of introducing major and widespread syntactic novelty.
Frameshift mutations occur as a result of the deletion or addition of some small number
of nucleotides in a DNA string. The alteration causes a shift in the reading frame of the
resulting mRNA transcription. Unless compensated for by some other mutation later in
the string, all codons below the site of alteration are affected. As an example consider the
5
Abaptation in language use is akin to exaptation in evolutionary biology except that in the former case, an
earlier function is exploited in a later analogous one, whereas in exaptation a morphological feature adapted
to one function adapts to later functions in a disanalogous environment.
syntactic disturbance resulting from the deletion of the underscored pyrimidine, uracil in
the following RNA string:
... UUU UCU UAU UGU ...
... UUU UCU AUU GU ... .
Such a mutation is significant if it results in the production of some non-functional
protein, an outcome that may or may not be deleterious.
Linguistic use also involves replication of syntax. The brain of the recipient must (or
rather does in general) transcribe the product of the brain of the speaker in such a way
that the syntax of the transcription matches the syntax of the produced speech. The
speaker reinforces syntactic construal by prosodic presentation, guiding the hearer to the
correct construal by appropriate variations of pitch contour, syllable lengthenings, and
stress. Sometimes, however that syntax is in some respects both indeterminate and
immaterial, as that of
No trees have fallen over here,
so that no disruption is occasioned if the speaker would restate the claim as
Over here no trees have fallen
while the hearer would restate it as
Here no trees have fallen over.6
Sometimes the consequences become apparent in the later word-formation patterns of the
language. Under this heading we may note the transitions form a numpire to an umpire,
from an once word to a nonce word, from an ick name to a nickname, and from a
naperon to an apron. Though we might be misled in our understanding of the origins of
the newly-formed word, it makes little practical difference that one word has replaced
another. But these trivial mutations do share a feature with more interesting mutations
that is worth noticing: namely that a mutation requires an incubation period in which it
remains undetected and therefore uncorrected, and during which it gradually establishes
itself within a critical proportion of a linguistic community. It is only after such a period
that the capacity for correction is overwhelmed, and the new form the new transmitted
orthodoxy. That it should take hold at all requires a degree of ignorance as to linguistic
history that is readily attainable.
A theoretically more interesting form of mutation is one which also requires for its
survival and propagation an achievably general degree of ignorance. That level of
innocence is attained in large part because of semantic ignorance and syntactic
indeterminacy. In the case of connectives those two conditions are nurtured by the
6
The example is due to Mary Shaw.
morphological reduction of the vocabulary from its original, recognizably lexical form,
and by the scope indeterminacies created by the presence of other sentence elements,
notably negations and modal elements. Since our example is to be but, we might note the
prevalence of negations (no one but, nothing but, etc.) in the later prepositional uses of
the particle and in the earliest sentential uses where modal auxiliaries are also found in
negativizing roles. The following illustrate the patterns (with that’s) restored
parenthetically:
I would have come earlier but (that) I was delayed
He does not visit but (that) he begs money of me
In the former the scope of the modal would have extends to the end of the sentence,
likewise the negative does not in the second. Both are conditional in character (I would
have come earlier had I not been delayed / If he visits, he begs money of me.) Now
suppose that (the that being absent) the audience takes the scope of the modal or the
scope of the negation to extend only to the main clause. How does this syntactic
misconstrual distinguish this hearer from the hearer whose construal retains long scope of
the modal and the negation? There is sufficiently little difference between the construals
that it could remain undetected. On the correct construal of the one hearer, a subordinate
clause suggests that there was a delay in the one case and suggests that the visitor did
actually beg money. On the incorrect construal of the other hearer, a second main clause
asserts that that there was a delay in the one case and a second main clause asserts that
the visitor begs money. There may be few occasions on which the hearers actually agree
in their syntactic construal of similar constructions; nevertheless they might agree on the
occasions in which the constructions are appropriately produced. It is this agreement
among speakers on occasions of use, and disagreement on syntax that characterizes the
first stage of a mutation. The novel syntactic construal need not require some clearly
enunciated reconstrual of the use of but. There is in general no presumption that our
interlocutor has a clear understanding of his vocabulary; only that the use is not wildly
eccentric. So we need ourselves satisfy only that presumptive standard. This first stage of
mutation, because the new use is lies hidden beneath the old, could be called the
succubinal stage. The concealment of the novel construal permits an increase in the subpopulation of language users for which the novel construal is the correct one.
In the second, migratory stage, the new construal emerges from concealment into
environments in which no scope ambiguity hides its new construal. In the case of but, it
now finds uses in negation-free constructions:
He visited his brother regularly, but he always begged money.
No doubt mavens of the old school object to this newly emerging use, and decry it as
nonsense; however, the use survives because for a sufficient number of the linguistic
community it is the natural generalization from the use in negative environments. On the
other hand, the generation for which there can be no such generalization are partly right.
The construction is clearly conjunctive, by which is meant that either clause can be
detached; however the new generation would be hard-pressed to explain why but should
have been selected rather than and. Nevertheless but inherits a niche from its earlier use,
and although that has perhaps changed, it has distinctive uses in present English speech.7
For conservative speakers, but has competing readings in negative environments, and illread young or non-native English speakers can be difficult to persuade that the proverb
It doesn’t rain but it pours
is to be read as
If it rains it pours.
That new ambiguity in negative environments typically represents a third stage of
mutation in which the mutated reading gets marked in some such way as by the affixing
of an extremal adverb. Thus we have, for example, just any, even if, for all, and others, to
distinguish mutated uses from ancestral ones. By far the most common mutations yield
conjunctive readings where there were only conditional ones, or conditional ones where
there were only conjunctive ones. But and without (both, incidentally descended from
vocabulary of outsideness) afford examples of the former. (Contrast (She’ll die without
medical help and She’ll die without betraying her comrades)8 Unless affords an example
of the latter. Some mutated connectives, such as but are minimally or only prosodically
marked. A comma suffices to mark the conjunctive reading in written English. Without is
marked variously, but marked. In some cases, such as that of unless, the ancestral reading
does not survive. Canadian political English affords an instance that shares with but the
feature that it is semantically difficult, and with unless the feature that the original is lost.
The prime minister initiates a general election by drawing up a writ. The verb phrase
draw up has been heard as drop, with the result that one now speaks of the PM ‘dropping
the writ’ rather than ‘drawing it up’, as has having finally ‘dropped the writ’ rather than
as having ‘drawn it up’. As to what the dropping consists in, no one is quite clear.
I have spoken of ‘stages of mutation’, but these stages, like the generations that produce
them, are convenient fictions. Populations change by proportionally minute additions and
deletions, and stages emerge as populations of language users do. The result is that
understood population-wise, the syntax of a language is, at many points, indeterminate.
Moreover a mere thirty-five independent points of difference would suffice to provide
every human on the planet with a unique syntactic understanding of the English language,
and many more than thirty-five such points of difference are present among the Englishspeaking minority of the Earth’s population. If the natural language of syntax is the
language of thought, then the notion of a universally shared language of thought is simply
a non-starter. Even in a small linguistic community, there is no reason to suppose that all
7
I say no more on the subject of its present use, as it is more fully discussed elsewhere, notably in Jennings
2004. The reader may, of course, have views on the subject. I remark only that any explanation of the role
of but must apply to such constructions as “He got here, but he got here late”, in which the second clause
implies the first, and to “He got here late, but he got here” in the first clause implies the second. The
explanation must also tell us why the construction is non-commutative, that is, why the two quoted
sentences are not conversationally equivalent.
8
For a fuller account of mutations in the history of without, as well as examples of mutations in Breton, see
Jennings and Schapansky 2000.
of its members work within the same set of syntactic generalizations, nor even that they
operate with a syntax, as distinct from a set of familiar strings.9
4. THEORY AND PRACTICE AGAIN
The familiarity of strings depends upon instruction and memory, and both memory and
instruction let us down. We can all offer examples from our own missteps, so the
example that I quote will not be taken to disoblige its author. In fact he himself offers it
in the course of making a point closely related to the general one that I am trying to
convey. The author is M.C. Corballis (2002) distinguishing human skills at abstract and
applied reasoning. His main point is well taken. The point makes reference to and adapts
‘a simple test of reasoning’: the Wason10 test.
Suppose that you are shown four cards, bearing the symbols A, C, 22, and
17, and are told that that there are also symbols on the other side of each
card. You are then asked which two cards need to be turned over to check
the truth of the following claim: “If a card has a symbol on one side, then
it has an even number on the other side.” If you’re like most people, you’ll
choose the cards displaying the A (a vowel) and the 22 (an even number).
It is indeed rational to turn over the A, but turning over the 22 is not really
very revealing, since whatever is on the other side cannot disconfirm the
statement. The better strategy is to turn over the 17, because the presence
of an A on the other side would falsify the statement. (Corballis, 97).
Corballis goes on to compare our ineptness at solving the abstract problem with our
superior talents when the case is described more concretely.
But now suppose it is explained that the symbol A stands for ale and C for
coke, and it is explained that there are people’s ages on the other. If asked
which two cards to turn over to check the truth of the statement: “If a
person is drinking ale, he or she must be over 20 years old”, most people
easily understand that the critical cards are those bearing A and 17.11 The
task is formally the same as that involving the letters and digits as abstract
symbols, but now the policeman in you readily understands that you
should examine what the seventeen-year-old is drinking if you want to
stamp out underage drinking.
Now it is by no means to the discredit of Corballis to offer this comparison as an
excellent example of his own point. The concrete case does not parallel the more abstract
one, and while his solution to the practical problem is indeed correct, his solution to the
more abstract one is by no means so. In the abstract case we are told only that there is a
symbol on each side of each card; in the practical one we are told that are beverages on
one side and people’s ages on the other. With a little charity we can accept, in the
9
Examples of large-scale syntactic schism in Korean and Japanese are studied by ….. in ……
Wason 1966
11
Example adapted from Cox and Griggs 1982
10
concrete case, that no beverages are coded numerically, and that no ages are given in
Roman numerals. Failure of that condition would indeed change the case. But in the
abstract case, we must turn over three cards, not just two, for we are given no assurance
that there is a numeral on the other side of the C card. Therefore it too must be flipped to
check whether the other side bears only a vowel. In fact, if we take from his statement of
the case only what can be strictly inferred, flipping the A card might falsify the claim in
either of two ways. It would be falsified if there were no even number on the other side;
but assuming that the side initially displayed contains only an A, the statement would
also be falsified if on the reverse there were an even number and a vowel.
Again, to point this out is to say nothing to the discredit of the author. He readily
acquiesces in the view of Tooby and Cosmides12 that ‘most of us are poor logicians’. We
are poor logicians because we are not trained in logic. We fare ill in Wason-like tests,
because we are not practiced in Wason-like tests. Even if, as De Morgan maintained, the
study of logic tends to reduce the difference in difficulty between the familiar and the
novel, and even if logic has found its own technological sphere of supremacy, logic is
itself unnatural. An ability to extend sequences of sentences to proofs by the application
of available rules is not what makes clever people clever. To be sure clever people learn
logic with ease, but that is because the extensions of sequences of sentences to proofs
provide an uncluttered arena for their cleverness. The steps in a proof do not themselves
require extensions of sequences of sentences behind the scenes. Sequent-introduction, the
formal practice that most closely approximates this model, does not depend upon a
hierarchy of meta-sequent-introductions.
5. WHENCE REASONING?
Where then do our practical reasoning skills come from? Were we looking for a
promising clue, we could do worse than to resurrect from the dust the writings of
Augustus De Morgan, and browse through his lectures on the syllogism.
In his Cambridge lectures, Augustus De Morgan complained about traditional syllogistic
that the only relation it was prepared to accept was that associated with the copula verb,
that all other relational terms must be relegated to the predicate. So, for example, the
syllogistic demands that we represent the sentence
[Three] is greater than [two]
as
All [three] is [a thing greater than two].
But then the evidently valid argument
4 > 3; 3 > 2/∴ 4 > 2
12
Tooby and Cosmides 1989.
must be accounted syllogistically invalid, since when, as the tradition requires, we give
the copula its accustomed pre-eminence, the resulting representation commits the
syllogistic fallacy of introducing four terms: four, things greater than three, three, things
greater than two, and two. De Morgan did not quite have the means to make his point as
we would now most naturally put it, but his intention was clear enough. What makes
syllogistic work is not the particular relation that the copula represents, but rather some
assumed formal properties of that relation. From our vantage point, we see that there are
really two relations: that of inclusion, and that of set membership. So, to consider an
example, in general the first figure syllogism in Barbara is valid in virtue of the
transitivity of set inclusion, but in the special case in which the subject term is singular,
the validating property can also be understood as the monotonicity of membership along
inclusion. To say that a relation R is monotonic along a relation S is to say that for any
individuals x, y, and z if Rxy and Syz, then Rxz. (Thus to say that a binary relation R is
transitive is to say that it is monotonic along itself.) Membership is monotonic along
inclusion, because for any individual x, and any sets a, and b, if x ∈ a, and a ⊆ b, then x
∈ b. Contains is monotonic along identity because for any individuals x, and y, and any
set a, if x ∈ a, and x = y, then y ∈ a. De Morgan preferred the language of composition.
The composition, R S of two binary relations R and S is the set of pairs { x,y ∃z : Rxz
& Szy}. To say that R is monotonic along S is to say that R S ⊆ R.
Thus, properly understood, a first-figure syllogism in Barbara in which both the minor
and the middle term are singular owes its validity to that monotonicity. The language of
syllogistic made no special use of these distinctions, the undifferentiated copula serving
adequately for its rather limited purposes. However, De Morgan observed, seen from the
vantage point of sufficient mathematical abstraction, the first figure syllogism in Barbara
shares a valid form with the greater than argument cited, and with some arguments in
which distinct premisses introduce distinct relations, provided that the relations bear to
one another the right second-order relationships (inclusion, monotonicity, and so on) in
the right combinations. First-figure Barbara in which the major term and middle terms
are singular shares a valid form with
y < z; x = y /∴ x < z.
De Morgan’s exposure of the traditionalists was intended in part as a demonstration of
the uselessness of syllogistic as an instrument of scientific investigation and of human
reasoning more generally. He likened the syllogism to an ornamental cannon: ‘out of
mathematics, nearly all the writing is spent in loading the syllogism, and very little in
firing it’.13 It is easy to see why De Morgan would have regarded his own proposal as an
improvement on this score. Insofar as explanation involves decomposing causal
relationships into spatio-temporal component relationships, a general theory of relations
and their compositions would provide a general mathematical theory of scientific
explanation. But although he saw the possibility of composing relations in arguments, as
Lucas is a son of Laurie;
Laurie is a sister of Alison;
13
In ‘On the Syllogism: IV and on the logic of relations’ in Heath 1966, 239.
Martyn is a son of Alison;
therefore, Lucas is a cousin of Martyn
he made no special plea on behalf of decomposable relations, such as those of cousin,
nephew, niece, for purposes of detailed understanding. His point is the negative one that
inclusion relations between classes of items, whether objects or moments or states,
relations that depend upon identities between elements, provide inadequate resources.
Part of the reason is that identity, as deployed in the definition of inclusion, does not
allow for predictable transitions of states or for regularity of change.
On De Morgan’s account, ‘it is to Algebra that we must look for the most habitual use of
logic forms. … Here the general idea of relation emerges, and for the first time in the
history of knowledge, the notion of relation and relation of relation are symbolized. And
here again is seen the scale of gradations of form, the manner in which what is difference
of form at one step of the ascent, is difference of matter at the next. It will hereafter be
acknowledged that … the algebraist was living in the higher atmosphere of syllogism, the
unceasing composition of relation, before it was admitted that such an atmosphere
existed.’ (241)
De Morgan’s insight deserves to be better known and its importance more widely
acknowledged. The reason is this: the algebraic character of human reasoning is the
product of the algebraic character of its evolution. The distinction between vocabulary of
matter and vocabulary of form, if it exists, must itself be an evolved distinction, since all
linguistic organisms are descended from non-linguistic ones. Moreover the process of
functionalization of which I have earlier sketched an account must itself be descended
from an ancestral process through some higher-order development that produced the one
kind of change from the other. Again, this follows from our having had non-linguistic
ancestors. Whatever process it is alters the significance and morphology of vocabulary
has descended from a process that altered the (ancestor of) significance and (the ancestor
of) morphology of non-linguistic physical interventions. Each accession to a higher order
requires a longer time frame for illustration. So the number of degree-raising iterations
(changes in changes in changes etc.) that we can usefully make is strictly limited and in
any case beyond my expository skills. But if we are looking for the origins of these
peculiarly human skills of reasoning, then we must certainly look to pre-linguistic
conditions, perhaps to biological conditions that ground the earliest developments toward
language itself.
Certainly we can find purely biological monotonicities that would have offered
exploitable traits in the ancestry of referentiality and they can serve as sufficient
illustration of the more general point. One such trait is the evolved response of fixing in
foveal attention and tracking singular motion detected within the visual field. The story is
a complicated one. In the nature of things, we cannot have a detailed understanding, but
we can make intelligent inferences that accord with the outcome and with the
anthropological evidence. It begins with the development of bipaedalism, the consequent
narrowing of the pelvis with changes in shape and size of the birth canal that resulted in
soft-cranial births and post-partum encephalization, and with the increase in the degree of
cranial flexion that grew out of the requirements of balancing a head on an upright body.
With these developments the forelimbs became available for non-locomotive uses and
among them, the articulated ballistic motions of tool-making and particularly of
throwing. Again, with these developments came a brain free to develop in size and
functional constitution to organize and initiate such motions.
An early legacy of that cluster of developments together with the more primitive tracking
response was the capacity to track thrown objects, and, one may safely presume, the
capacity to occasion tracking by throwing. Now a developing capacity for tracking
intermittently visible motion establishes a monotonicity between two relations: the graph
of successive vectors representing the direction of gaze, and the graph of the trajectory of
the tracked motion. The monotonicity is represented by the accurately anticipated
reappearances of intermittently visible objects. At its most highly developed, it would
permit the accurate anticipation of a trajectory solely from the perceived character of the
articulated ballistic motion that produced it. With this anticipation comes the increased
capacity of articulated motion, even when disarmed, to direct attention accurately to a
distant location, even one beyond dead ground. Finally, that ability to anticipate is
exploited through ballistic motion, now disarmed of any ballistic outcome, permitting the
motion to acquire a purely deictic significance.
On such an account, one can usefully see the development as an (ancestral)
functionalization of throwing. The disambiguation of the functionalized form evolves as a
natural consequence of the disarming: like the morphologically reduced profile of
functionalized vocabulary, it would have evolved into an evidently low-energy version of
a comparatively high-energy ancestral ballistic intervention.
6. CONCLUSIONS
We have all judged of this or that charlatan that he does not know what he’s talking
about. And we have all marvelled at that plausible fluency to which ignorance is no
impediment. Such remarks are reserved for the mischievous and the insidious. But at
quite an ordinary level, it has to be true of everyone who speaks a language. We need no
understanding beyond conversational fluency to engage in conversation in a natural
language. Since conversational fluency is what is transmitted from one generation of
language users to the next, the most that we are guaranteed is whatever is necessary for
that transmission. The difference between the speaker and the charlatan is that the
charlatan lays claim to a level of understanding that he, unlike others, has not attained.
The speaker’s naïve confidence though demonstrably unwarranted, is not mischievous.
He is merely caught up in a compelling illusion that almost all speakers tacitly share. The
mischief arises when the illusion is given charge of theory: linguistic theory, logical
theory, cognitive theory or any other.
One certain theoretical way out of the illusion is to ground cognitive theory in the
physicality and more particularly in the biological character of language. What does this
amount to? No one to whom the fact is pointed out will disagree that language is
primarily a biological phenomenon. The difficulty lies, not in acceding to the fact, but in
troubling to work to the theoretical standards that the fact imposes. This requires more
than verbal acceptance; it requires a commitment to understand language in the light of
linguistic data: not just data pertaining to how we find ourselves speaking, but the data
that reveals how it comes about that what we say has the physical significance that it has.
One consequence of this imposition is particularly unwelcome in some academic
quarters: it is likely that in some tolerated intellectual pursuits, the worse one’s biological
understanding of language, the sillier one’s theories.
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