PROOF AND CONVERSATION
R.E. JENNINGS
Abstract. Proof-theoretically, natural deductive rules are not rules of inference, but rules for extending a proof. They are in certain respects like certain
rules of conversation. Rules of conversation are rules for extending a physical
event.
1. Introduction
Many authors of textbooks of natural deduction present the rules of their chosen
formal systems as rules of inference. No doubt they wish them to be thought
useful or intuitive, and perhaps even think them so themselves. I confess that I
no longer cherish the kind of intuitiveness that depends upon little more than the
familiarity of language, and would postpone questions of usefulness to a time of
better understanding. So my own students find themselves presented with a more
austere account of the matter. The rules of a natural deductive system are, I tell
them, rules for extending a proof, no more. The justification is as follows:
We (I at any rate) don’t know what inference is. To be sure we know how to use the
word and its cognates. But even that much must tell us that we do not infer A ∨ B
from A. A and B are upper-case letters of the Roman alphabet and ∨ is heaven
knows what in a mysteriously varied font. Certainly A and B have been introduced
by this time as metalogical variables ranging over sentences of a language. But little
or nothing in that discussion has called upon my ordinary understanding of what a
language is. And the language of the formal system is just a recursively defined set
of strings of marks, call it what you may. Nothing in our ordinary understanding
of what a language is helps us in the matter. And our ordinary understanding
of what a language is is probably pretty feeble anyway. By contrast, given the
definition, our understanding of the formal language is anything but feeble. We
are better off without the well-intended references to anything beyond it. And
again, we might give or as a reading for “∨”, but that must be just a reading,
and not an interpretation. Whatever we might think inference is, we don’t infer
from a male guest’s saying ‘I must go’, that he must go or he’ll be late. Nor do
we infer from a waiter’s (improbably) saying ‘You can have tea’ that we can have
tea or we can have coffee, not at least as we would understand it if the waiter
were to say ‘You can have tea or you can have coffee’. In neither case do we even
suppose it to be correctly inferrable, certainly not implicit in what was said. So if
a student should take the reading as an interpretation, it will likely do more harm
to his understanding of natural language than it will good to his understanding of
Key words and phrases. proof theory, rules, introduction, elimination, semantics.
Such as it is, the understanding that this paper communicates owes a good deal to the many
discussions of these topics with Nathalie Prévost and with members of the Logic and Language
Group at Simon Fraser University.
1
2
R.E. JENNINGS
the formal one. It is difficult enough for the tiro to understand what he is about
when constructing a proof, without having first to free himself from an unnecessary
illusion of understanding. But there is another reason why or should be no more
than a reading of “∨”: this is that anything more is not proof-theoretic. And this is
in general also a reason for not calling the rules of such a system rules of inference.
If inference involves interpretation and semantic understanding, then it has no place
in proof theory.
So much, one might suppose, for the role of conversational language in an introduction to proof theory. Yet conversation itself yields an intuitive account of the
roles of introduction and elimination rules. There is a kind of understanding of
elements of natural language, which consists in being able to use those elements
in conversation. We can imagine stipulating two criteria of such conversational
understanding: The first is this: the subject should have command of those conversational situations in which the subject be able appropriately to introduce the
element into the conversation before any other party to the conversation has done
so. Call this the first-in requirement. The second is that, the element’s having
introduced into the conversation by someone else, the subject should know what
conversational responses lacking that element are appropriate ones. Call this the
first-out requirement. For elements of a natural language, first-in and first-out conversational skills would correspond to a proof-theoretic mastery of introduction and
elimination rules for connectives of a formal language. There would of course be
differences: we might also expect the logicianer to be able to explain the formal
rules, whereas we should not expect the conversationer to explain the natural ones.
But there may also be other parallels. It may be that command of conversational
I- and E- rules confer all other conversational command of the linguistic elements
under consideration. So for example, mastery of those rules for that element may
enable appropriate conversational exchanges in which all parties’ contributions have
occurrences of the element, as mastery of ∧-introduction and ∧-elimination (on one
formulation anyway) justify commutation.
As conversational understanding prompts helpful intuitive remarks about the nature of proof theory, so proof theory offers up intriguing suggestions about conversational understanding of the elements of natural language. So as not to trivialize
the metatheory in the logicianer’s understanding of it, one might stress in a lecture
that proof-theory can be studied and proofs constructed independently of any semantic representation of the underlying language, and we might studiously avoid
any intrusion of semantic language (truth, validity and so on) into the explanation
of the rules. ‘These are rules for the extension of a proof’, we might insist. Proofs
can be extended without semantic understanding. We are prompted to ask the
corresponding question about conversation: how much semantic understanding of
vocabulary is required for conversational mastery? The answer is in general surprisingly easy to formulate, and, though non-specific, revealing. We can reformulate
the question as: supposing we had an effective method, how much semantic understanding would we be guaranteed of finding were we to investigate a competent
conversationalist? The formal answer is, I think, along the lines of the following:
We must find as much semantic understanding as can be inferred
from the fact of the transmission of the language from one generation of language users to the next at the current rate of change for
the language.
PROOF AND CONVERSATION
3
The material answer is:
Not much.
The facts of proof theory prompt an even more challenging question: How much
semantic understanding can we hope ever to find by such an examination of an
accomplished conversationalist? And the answer is again that any such hopes must
be grounded in nothing more than the facts of the transmission of language at
its current rate of change.1 How much semantic understanding of the elements of
language is permitted by linguistic transmission? It may be, indeed it seems likely,
that the transmission of language from one generation to another actually depends
upon there being very little semantic understanding as a result of the transmission.
What is being transmitted, after all, is conversational ability.
How to go about answering such specific-seeming questions is a difficult problem.
We can confirm or disconfirm the particular estimation of “not much” by checking
how much semantic understanding we actually come across in users of the language,
though even that task is not a simple one. But we could apply monotonicity
hypotheses that would permit us to examine people that we would expect to be
paragons of semantic understanding and argue from their ignorance to that of
lesser folk. Call this the modus tollens approach. Alternatively, we could argue
from the semantic clarity of the dullest of us to the understanding of the rest: the
modus ponens approach. Now it is sometimes assumed that the so-called “logical”
vocabulary is the semantically easy part of natural language. And if anyone can be
expected to have semantic understanding of that vocabulary, the authors of logic
texts can. As to the more abstract vocabulary of natural language, that of mind and
ethics and metaphysics, if anyone can be expected to have semantic understanding
of this vocabulary, philosophers can. Not only have these two categories of thinkers
at least the conversational command of this vocabulary that everyone else has, they
have actually made a study of it, the latter for several thousand years. It is true
that the latter group have been at their task for much less time, but after all the
logical vocabulary is simpler.
By now the reader will have discerned the direction in which this discussion is tending. If we could gain a semantic understanding of such words as mind, intention,
thought, right, wrong, ought and so on, we would surely have it by now. That we
have not managed it is evidence that semantic understanding is no more necessary
for conversation than it is for proof theory. The transmission of the use of this
vocabulary from one generation to another at a normal rate of linguistic change
does not require that there be any way of acquiring semantic understanding. If no
such way is required, then there is no basis for the supposition that such a way
exists.
Children learn a use of the vocabulary from other children, from adolescents, from
adults, not even any very precise use, probably not quite the use of the oldest
adults with whom they share it. Older adults’ uses are guided by neural structures
including much earlier uses inaccessible to conscious memory perhaps, but accessible
nevertheless to the neural control of speech production. Their uses of vocabulary
still bear the imprint of the uses of their own grandparents and the uses of that
generation. This fund is unavailable to the child learner, whose uses of the same
vocabulary will be underwritten by a greatly attenuated set of connections with
1It should be added that the facts of language change, though they cannot be completely
ignored, do not figure very importantly in these estimations of our chances.
4
R.E. JENNINGS
the speech of earlier generations of speakers of the language. Each new generation
grows to adulthood without the benefit of what I have called elsewhere a semantic
audit, by which their uses are rendered consistent with other people’s uses, or
with one another, or tested against some acknowledged standard explicitly handed
down or even remembered from language users of the past. To give an example
from ethical vocabulary, it may not be a separation of many generations between
English speakers whose every use of the word blame reflected an understanding
or neural association, theirs or their parents’ or their grandparents’ or their great
grandparents’, of its connection with the word blaspheme of which blame is a worndown remnant. For such speakers, supposing they were besought not to blame some
fellow for some deed, and wished to comply, they would refrain from speaking ill of
the fellow on that account. Nor are we many generations removed from speakers of
the language whose usage of “answerable” reflected an accessible connection with
its Latin form “responsible” and vice versa. Even at such a remove, we can still
individually recover some of the connections and re-read earlier literature in the
light of the recovery; we can even, to some limited extent, reform our own speech
in acknowledgment of it. But we cannot return the language to the state in which
usage was supported by the neural set-up of the period before the connections
were lost. We cannot, for example, recover to general use the earlier neural map
of -ble and bility word endings more generally that affected the uses of and the
responses to such words as responsible and answerable. We cannot reissue the
neuro-linguistic map in which that conversational place of answer was identified.
In simple terms,we cannot collectively recover the collective linguistic understanding
of earlier generations of speakers of the language.
As connections are lost, so are distinctions. To give but one example, in the ordinary
way, the vocal majority do not carefully distinguish between deserts and rights.
“The people deserve to know” and “The people have a right to know”, even, God
help us, “The people deserve the right to know”2 and perhaps even “The people
have the right to deserve to know” are virtually interchangeable in the language
of political address. Politicians do not rally to separate theoretical banners who
say ‘deserve’ and who say ‘have a right’. Again, the difficulty is not that experts
cannot disentangle the confused strands; it is that the same experts cannot give
positive accounts of each in language of which we can have more than conversational
understanding, that is, in language not subject to the same pressures of attenuation.
And even if their disentanglement should provide an academic distinction, it does
not follow that the politicians or the population of language-users more generally
would take it up. A small band of academics cannot withstand the forces of language
change. In popular usage the rights of the people and the deserts of the people
will merge, as ineluctably as responsibility and blameworthiness have merged in
the restricted domain of disapproved activity, and as ineluctably as the cognates,
2This was a mere fancy until The Vancouver Sun reported ‘Gary Leier, president of the
Kelowna riding association, said, “I may be in the minority but I won’t be voting for a review.
I believe the prime minister deserves the right to decide when he wants to go.”’ (2002-07-31).
Again, ‘They [a group of loyal House members] argued the prime minister, having led the Liberal
party to three consecutive majority victories, deserves the right to retire at a time of his own
choosing.’ (2002-08-19). Stay tuned.
PROOF AND CONVERSATION
5
responsibility and response, have drifted apart.3 The only demands to which this
usage responds are those that the uses of the conversation dictate. If those usages
do not dictate coherence, no coherence can be predicted. If they do not dictate
semantic understanding, no semantic understanding is predictable.
Now there is nothing special in this regard about ethical language. Much the
same point can be made about mental, metaphysical, and religious vocabulary.
The language of intention and meaning, mind, thought and belief, faith and soul
must come under the same strictures. Nor is the connective vocabulary of natural
language exempt from corresponding observations. I do not argue here the case
that we don’t in general have a semantic theory of the connectives, or that in
particular we don’t have as our semantic theory the one that tabulations of graphs
give us for the truth-functors. These points are discussed at length elsewhere. (See,
for example, [1, 2, 3].) The evidence is that, if we are to give truth-conditional
semantics for connectives, we must give more than one per connective, and then
when we try to give examples of uses that match the semantic theories we favour,
in many cases we give examples that do not. Nature has not endowed us with
a semantic understanding of the connective vocabulary over which she has given
us conversational control. We do not understand it unless we make a study of it.
And when we come to understand it through study, our understanding is largely
historical and not specifically semantic.
‘But,’ one feels, ‘conversation cannot, like proof theory, be completely free of semantic involvement. Language is about the world. This aboutness is specifically
semantic, and whatever can said in explanation of the character of particular contributions to conversations, what is being explained is somehow connected with the
way things are.’
What can be said in response? This at least: conversation is physical activity. It is
as much a part of the way things are as anything that we take the conversation to be
about. Every contribution to a conversation, including the contributions to which
one is responding, is a contribution to the way things are, as well as a response
to it. A conversational contribution is a physical intervention that adds one more
set of sensory data to an already rich manifold of sensory data. If there are rules
of conversation, then were they to be formulable at all, they would have to be
formulable as rules for the extension of a physical situation or occasion. And so far
this does not distinguish the construction of a conversation from the construction
of a proof, which is also a physical activity. The difference is this: In a proof, all of
the previous lines are elements of a language. Were we to describe a conversation
by analogy with proof, we would have to say that a conversation is like a proof to
which linguistic activity has supplied only some of the lines; it is like a proof to
which other features of a perceptible physical situation have provided some of the
lines, and to which neural states of participants have supplied others still.
Now a philosopher might wish to insist that, on such an analogy, conversation must
be regarded as enthymematic. In conversation, as in general mathematics, good
style insists that not every move be made explicit, that something be left to the intelligence of the interlocutor, as in a demonstration something be left to the reader.
3On Tuesday, B.C. conservation officers continued to comb the area looking for the bear responsible. They have killed one bear already, but [RCMP Staff Sergeant Bryan] Reid said they
weren’t certain it was the right bear.
“The bear responsible for this attack was clearly aggressive and predatory in nature,” Reid said.
“The fact it returned twice indicates it’s clearly a problem. (The Vancouver Sun 2002-09-04.)
6
R.E. JENNINGS
But this, even if true, (for the sake of argument, let its truth be admitted) won’t do.
It supposes that every element of a physical setting that, as it were, contributes
lines to a conversation is accessible to observation and linguistically formulable,
and for neither of these suppositions is there any basis whatever. Some are both,
but most may be neither. We need only consider pheromonal and other subliminal influences, not to mention synaptic connections, over which, in the short term,
we have no control. And the admission goes nowhere toward requiring semantic
understanding.
References
[1] Jennings, R.E. The Genealogy of Disjunction. Oxford University Press. New York, 1994.
[2] Jennings, R.E. The Meanings of Connectives. to appear in S.Davis and B. Gillon. Semantics.
Oxford University Press. forthcoming.
[3] Jennings, R.E. The Semantic Illusion. to appear in A. Irvine and K. Peacock. Mistakes of
Reason. forthcoming.
—————————————————————Laboratory for Logic and Experimental Philosophy, Simon Fraser University, Burnaby BC V5A 1S6, Canada
E-mail address: jennings@sfu.ca