On Formal Versus Commonsense Semantics 
David Israel 
AI Center and CSLI 
SRI International 
There is semantics and, on the other hand, there is seraan~ics. And then there is the 
theory of meaning or content. I shall speak of pure mathematical semantics and real se- 
mantics. I have very little idea what "formal" means in "formal semantics"--unless it 
simply means semantics done rigorously and systematically. 1 I have even less idea what is 
meant by "commonsense semantics". I shall not speak much of the theory of meaning. The 
distinction between these two modes of semantics, the mathematical and the real, is not 
meant to be a hard and fast distinction--nor, most assuredly, is it intended to be rigorous 
or systematic. As I see it, the distinction has primarily to do with purposes or goals, deriva- 
tively, with constraints on the tools or conceptual resources considered available to realize 
those goals. In particular, real semantics is simply pure mathematical semantics with cer- 
tain goals in mind and thus operating under certain additional constraints. Another way 
to put the point: some work in pure mathematical semantics is in fact a contribution to 
real semantics; however, it does not have to be such to make a genuine contribution to 
pure mathematical semantics. 2 Hence, since real semantics can be executed with the same 
degree of rigor and systematicity as must all of pure mathematical semantics, it should be. 
Have I made myself clear? Not entirely, perhaps. Let's try a more systematic approach. 
Pure mathematical semantics is either a part of or an application of mathematical logic. 
Real semantics, even though an application of mathematical logic, is a part of the theory of 
meaning or content. Contributions to real semantics had better cast some light on naturally 
occurring phenomena within the purview of a theory of meaning--on such properties and 
relations as truth analyticity, necessity, implication. 
Traditionally (indeed, until Montague, almost undeviatingly) the techniques of pure 
mathematical semantics were deployed for formal or artificial languages. But this by itself 
is of no importance. These languages were invented, and are of interest only because, 
or insofar as, they are plausible and illuminating models, in the intuitive sense, of real 
phenomena in thought and in informal language. Consequently, the nature of the languages 
studied need not make an essential difference. 3 What does make a difference is the purpose 
1This is mildly disingenuous; talk of "formal semantics" is usually grounded in one or another idea of 
"logical (or, more generally syntactic) form". But one should beware the overly eager application of such 
notions to the semantics of natural languages. 
20f course, problems pursued for purely technical, mathematical reasons often turn out to be related to 
important questions and issues in real semantics. 
3Indeed, the two examples I shall consider concern the semantics of formal languages. 
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or end of the study and the fact that the end imposes constraints on the choice of means. 
In doing work in semantics, the logician has a range of tools available and certain 
criteria for choosing among them. In pure mathematical semantics, the only criteria are, 
in a suitably broad sense, technical. There are no nontechnical constraints; anything 
goes. That is, even if the development of a class of logics was inspired by an attempt 
to model various pretheoretic notions, work on the pure mathematical semantics of the 
languages can still diverge quite far from that original motivation. The objective is to 
provide a systematic way of assigning mathematicaUydescribable entities to the nonlogical 
expressions and mathematically characterizable operations to \[or correlating them with\] 
the logical constants so that the best proofs of the strongest and most general results may 
be achieved. Not so for work in real semantics. There the choice of tools and conceptual 
resources should be grounded somehow in the nature of the phenomena to be analyzed or, 
to pilt it differently, problems in real semantics generate not-purely-technical criteria for 
choosing among technical means. 
This, I realize, is all rather vague and airy-so let's get down to cases. The first illustra- 
tion is from work on higher order logics, in particular Henkin's proof of completeness in the 
theory of finite types \[I\]. The intended interpretation of the relevant language is that the 
individual variables-those of type 0-range over some domain of elements To, and that for 
each n, Tn+1 is the power set of Tn, that is, the set of all subsets of Tn. Monadic.predicate 
variables of type n+l range over all elements of the power set of Tn, m-place predicate 
variables of that type range over the entire power set of the rn th Cartesian product of T~. 
The theory of finite types can therefore be regarded as a \[perhaps noncumulative\] version 
of a part of impure set theory, that is, it formulates a conception of an "initial segment"-up 
to rank w-of the set-theoretic universe over some domain of individuals. Now it is a fairly 
immediate corollary of GSdel's proof that second-order logic-let alone w-order logic, which 
is what we are now concerned with-is incomplete relative to this intended interpretation. 
Yet Henkin proved completeness for a system of w-order logic. How? 
By ingenious hook and ingenious crook, is how. He introduced a wider class of models 
(interpretations) according to which the sole requirement was that each Tn be nonempty; 
otherwise, the interpretation of the Tn's was arbitrary. In particular, it is not required 
that each Tn be the power set of the immediately preceding type. This approach made 
it possible for Henkin to reduce w-order logic to a many-sorted first-order logic, thereby 
allowing him to obtain soundness, completeness, compactness, and LSwel~heim-Skolem 
results. Henkin's work was an exercise in pure mathematical semantics. The task before 
him was to provide a class of models for an axiomatic system in such a way as to provide 
soundness, completeness, and other results-and to do so in whatever way worked, without 
any thought being given to the interpretation on which the real significance of the system 
was based. 4 
Now let's move on to the treatment of languages for propositional modal logics. 5 
40f course, quite independent of tIenkin's motivations, it could have worked out that the class of models 
he focused on was indeed of real semantical interest. It just didn't work out that way. 
SThere is an interesting twist as regards motivation in this case. C. I. Lewis, the founding father of 
modern modal logic, was interested in different conceptions of implication (or the conditional), not in differing 
conceptions of contingency and necessity. Of course, on the conventional view, implication simply is validity 
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Modal logics have been studied as axiomatizations of different conceptions of necessity 
or possibility-or to put it somewhat differently, as axiomatizations of different conceptions 
of modal facts. The current standard semantical account is in terms of Kripke structures. 
For our purposes, let us think of these as ordered pairs < K, R >, with K a nonempty set 
and R a binary relation on K. (Kripke structures are now usually thought of as triples, 
the third element of which is a distinguished element of K. I'll return to this briefly later.) 
Roughly speaking, what happens is that the elements of K are used to index classical 
propositional models or interpretations-that is, assignments of T or F to the sentence 
letters-and the relation It, which is correlated with the modal operator, is a relation 
among such indices. (There may be more than one modal operator in which case there 
will be more than one binary relation.) Now if one thinks of the models as representing 
ways the world might be or alternative possibilities (or some such), it is not really such 
a bizarre exercise to follow Kripke's heuristic; the set K of indices of models is a set of 
possible worlds and It is a relation of accessibility or relative possibility among worlds. 
This heuristic results in a version of an old idea due to Leibniz: Necessity is truth in all 
possible worlds. 
The work on model-theoretic semantics for modal languages and logics using Kripke 
structures is a bit of pure mathematical semantics that is arguably also a real contribution 
to real semantics. Moreover, the techniques involved have shown themselves to be widely 
applicable. Thus, besides work on temporal logics, in which the set K is understood to be 
a set of times or time slices and It the relation, say, of temporal precedence, we have work 
on provability interpretations in which, for example, K is the set of consistent recursively 
axiomatized extensions of Peano arithmetic and TII:tT2 if and only if the consistency of T2 is 
provable within T1. There is also, of course, a good deal of purely technical, mathematical 
work on the Kripke-style semantics for modal languages. As Kripke asks, "What is wrong 
with the purely technical pursuit of mathematically natural questions even if the original 
motivation is philosophical?" 
Still, the philosophical questions, questions from metaphysics and the theory of mean- 
ing, keep insinuating themselves, as they must. If the work on Kripke structures is to be 
taken seriously as a piece of real semantics, something must be said about these entities 
called possible worlds and about the relation between them and the classical models they 
index. 6 It will not do simply to say, as we can when doing pure mathematical semantics, 
that K is just some nonempty set and It is just some binary relation on K, that meets such 
and such conditions. You've got to put up about possible worlds or shut up. I would argue 
that when you do put up, the best net result is to postulate a family of structures like 
or necessity of the conditional. 
6For instance, the distinction between models and indices is crucial, but that very distinction leaves room 
for the following possibility; there can be distinct possible worlds which are exactly alike as ways the world 
might be. That is, one and the same model can be paired with more than one index. Is this just an artifact 
or is it supposed actually to represent something? If so, what? There are things to he said here, things about 
representing contingent relations to propositions. Never mind what they are though; the point is that when 
taking work in the model-theoretic tradition seriously, one has to keep in mind that what is being done is, 
precisely, modeling. One must think seriously about what aspects of the proposed model are merely artifacts 
and what not. 
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those to be found in the situation-theoretic universe. But that's an argument for another 
occasion. 
I want to make one more point about propositional modal logics. Oddly enough, 
structures that yielded models for propositional modal logics had been made available 
as a result of research in Boolean algebra by Jdnsson and Tarski \[2\]. This work had 
nothing to do with the issues of necessity and possibility; the research was not in the 
least concerned with modal facts, nor, in fact, with modal languages. As a result of this 
work (and thanks to the perspicacity of hindsight) structures for modal languages can be 
seen to be/proper/relation algebras. Proper relation algebras are a special case of Boolean 
algebras with operators; work on them is directly related to results in universal algebra, the 
metamathematics of algebra, and category theory. For my purposes, though, the crucial 
aspect of this work is precisely its austere abstractness and generality. This is work in 
mathematical semantics at its purest. In this framework, even the set K rather disappears 
into the background-to be replaced by binary relations on K, those being the elements of 
the algebra. 7 Once again, the Kripke heuristic is available; it's just farther removed from 
the mathematical action, s 
The point to stress is a simple, but an important, one: the "reading" of the set K as a 
set of possible worlds, and of R as a relation of accessibility among possible worlds plays 
no part in the technical development. That heuristic enters precisely when claims of a 
real semantical nature are to be made for, or on the basis of, the technical results in pure 
mathematical semantics. And those claims cannot be extricated from more general issues 
in the theory of meaning. 9 
Earlier I suggested that I had just about no idea what is meant by commonsense 
semantics. Alas, this too was disingenuous of me. Sad to say, my guess is that most 
adherents of commonsense semantics are convinced that rigorous, systematic accounts of 
the semantics of natural languages are unattainable. In this regard, Schank and Chomsky 
are bedfellows, however strange. I know of no good arguments for the Schank-Chomsky 
view. l° Rather than canvass the various bad arguments that have been trotted out, let me 
conclude by citing four crucial sources of confusion that may have led many astray. They 
all have to do with the scope and limits of semantics. 
The first is to think that a semantic account of a natural language has to say everything 
there is to say about the meanings or interpretations of expressions of the language, with 
meaning' and interpretation understood very broadly and informally. A theory of the 
7Very nice work exists on relating Kripke-structures for modal logics to relational algbras-or, more specif- 
ically, to modal algebras. 
SAs Professor Scott reminded us all at TINLAP, the prehistory of the model-theoretic semantics of modal 
logic is quite rich and complex. It starts \[more or less/ with Tarski's work on topological interpretations 
of intuitionist logic, continued by Tarski and McKinsey in a more general algebraic setting in which they 
could draw illuminating connections to one of Lewis's systems ($4). But a more complete telling of this tale 
deserves both a more proper occasion and a better story-teller. 
9All these questions arise much more sharply in the case of quantified modal logics. Many of these have 
been canvassed in an important series of papers by Kit Fine, \[3\]. In any event, work in quantified modal 
logic simply has not developed in the robust way as has work in propositional modal logic. 
1°That's not to say, though, that the naysayers might not, in the end, be right. There are no guarantees 
of success in this business." 
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\ 
semantics of a natural language, e.g., English , is not (nor is it intended to be) the whole 
story of that language, minus its syntax \[phonology, morphology, etc.\] A semantic account 
may be the whole story about a formal language, minus the account of its syntax. But 
that is because formal languages are studied, not used. 11 A semantic account of \[the 
declarative fragment of\] English should be one that, in a systematic and rigorous manner, 
relates various facts about \[or aspects of\] the circumstances in which sentences of English 
are used to various facts about \[or aspects of\] the truth conditions of those uses-that is, 
to various facts about the states of the world those utterances are intended to describe. 
This is a central aspect of meaning or interpretation-where, again, these are pretheoretical 
notions-but it does not exhaust the subject. 
The phenomenon to be studied is the use of language or, if you like, the phenomena to 
be studied are kinds of uses of English sentences. Each such use is a complex event with 
many aspects. Those aspects enter into many different kinds of regularities or systematic 
connections. There are syntactic facts, morphological facts, phonological facts, facts about 
the surrounding discourse, facts about the nonlinguistic circumstances of the utterance, 
facts about the connections between states of mind of the participants, and facts relating 
such states to various objects, properties, and relations in the environment. These facts 
are related to one another in a wide variety of ways, some of which are the province of 
semantics and some not. For instance, any theory of language use had better make room 
for a distinction between the semantically determined information content of an utterance 
and the total information imparted by that utterance. The former is not the latter; the 
latter includes information imparted by the utterance in virtue of the interplay of \[broadly 
speaking\] pragmatic factors. In short, acknowledging the possibility of real mathematical 
semantics for natural languages does not imply acceptance of semantic imperialisrn. 
Second, semantics, even construed as part of a theory of language use, is not directly a 
theory of processing. Any real semantics for natural language should comport with good 
theories about the cognitive capacities and activities of users of such languages. But no 
theory of semantics can constitute or be directly a part of such a psychological theory. 
That a semantic theory does not attempt to be a processing theory, or more generally, a 
part of a psychological theory, is thus no cause for complaint. 
The third point is largely about the scope and limits of lexical semantics. The point 
is that there are limits. Lexical semantics does not yield an encyclopedia. Any semantic 
account worth its salt will yield a set of \[analogues of\] analytic truths, sentences such that 
the truth of utterances of them is guaranteed by the meanings of the words occurring in the 
sentence (plus, of course, their modes of combination), together with what might be called 
"the demonstrative conventions of the language" .12 Any such semantic account, then, 
will have to distinguish between analytic truths and world-knowledge. Consequently, no 
such semantic account will say everything there is to say about the objects which are the 
denotations of lexical items. A brief point about the connection between the current and 
the previous points is well worth making. A good theory of natural language processing will 
11Still, we should remember what was said earlier about the purposes for which these languages are devised. 
12If we have picked out a small set of lexical items as logical constanfs, then those analytic truths will be 
logical truths. Of course, we can make that set as large and as heterogeneous as we want. 
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have to explain how relevant world-knowledge is accessed, and used in processing. GOOD 
LUCK to such theorists/In any event, their job is not the semanticist's job. 
Fourth, and last: any real semantics for natural language should be a part of or be 
accommodable within a general theory of meaning--indeed a general theory of mind and 
meaning. Nevertheless no theory of the semantics of natural language can itself constitute 
such a general theory. Return with me now to the example of the mathematical, in 
particular, model-theoretic semantics of modal languages. As remarked earlier, the classical 
Lewis systems of modal logic might be said to express different conceptions of modality, but 
they don't express them in the sense that they constitute theories of modal facts. Nor do 
Kripke-style, model-theoretic treatments of those logics constitute theories of modality. The 
latter constitute ways of thinking about modal facts expressible in the former--that is, they 
provide models, in the intuitive sense, of the phenomena of modality. Kripke, for example, 
has presented bits and pieces of a theory of modal facts in Naming and Necessity, a piece 
which contains no mathematical semantics. David Lewis presents another conception of 
modal facts in his recent A Plurality of Worlds; that book too is devoid of the machinery 
of model-theoretic semantics. Those different theories may lead to the adoption of different 
mathematical treatments of modal languages. They will do this precisely by motivating 
choices among alternative pure mathematical semantic treatments-that is, by providing 
criteria of choice of a real- semantics for modal constructions. 13 
References 
\[1\] 
\[2\] 
\[3\] 
L. Henkin, "Completeness In The Theory Of Types," Journal of Symbolic Logic, 
Vol. 15, No. 1, pp. 81-91 (March 1950). 
B. Jdnsonn and A. Tarski, "Boolean Algebras With Operators, Part I," American 
Journal of Mathematics, Vol. 73, pp. 891-939 (1951) and "Boolean Algebras With 
Operators, Part II," American Journal of Mathematics, Vol. 74, pp. 127-162 
(1952). 
K. Fine, "Model Theory for Modal Logic: Parts I, II, and III," Journal of Philo- 
sophical Logic, Vol. 7, pp. 125-156 (1978), Vol 7, pp. 277-306 (1978), and Vol. 10, 
pp. 293-307 (1981). 
13The research reported in this paper has been made possible by a gift from the System Development 
Foundation. 
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