THE NATURE OF UNDERSTANDING FROM THE LOGICAL POINT OF VIEW 
Pavel Materna 
I~DIS Praha, Czechoslovakia 
In this paper I want to sketch an analysis of the con- 
oept of understanding (in the semantic sense of word). No com- 
paration with other attempts at such an analysis wall be made. 
Since the most adequate logical tool for analyzing natural 
language is, to the author's opinion, Tioh~'s "transparent 
intensional logic" (TIL, see, e.E., P. Tich~: The Logic of 
Temporal Discourse, Linguistics and Philosophy 3 (1980), 343- 
369), I shall first adduce a brief exposition of some relevant 
concepts of TIL. 
1. T r_a_ns__~arent intensional___ logic. ~X~ is consequently 
intensional system that exploits partial theory of types and 
a modified version of ~ -calculus. Omitting technical details 
(however important they are) we shall summax~ze some princip- 
les of TILa 
Objects which are supposed to be denoted by the express- 
ions of a (natural) language are type-theoretical objects 
over an "epistemtc basis", where elementary types are the 
universe of discourse (L), the set of truthvvalues (o), the 
set of time moments or real numbers (~), and the logical 
space (~ ; members of ~ are "possible worlds")i compound 
types are sets of (partial and total) functions. The obJedts 
are the members of the particular types. The objects denoted 
(named) by "normal" expressions of a language are intensions, 
i.e., functions whose domain is ~. Thus definite descriptions 
name individual concepts (members of ~ + (~ I-' ) ), sentences 
name propositions (members of ~ ÷ (T * o) ), some nouns name 
properties (members of ~÷(T~(n÷o) ) , where n is a type),etc. 
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In TIL there is introduced a concept of key importance= 
the concept of construction. Intuitively, a construction is 
a way in which an object can be given. Atomic constructions 
are objects themselves (an object _A constructs A_) and variabl- 
es of the given type (a variable v-constructs an object de- 
pendently on the valuation v). Non-atomic constructions are 
applications of functions to their arguments, and ~-abstract- 
ions. Constructions are defined inductively, so that an in- 
finite hierarchy of constructions with embedded constructions 
82'iseB. 
Distinguishing between constructions and objects is one 
of main contributions of TIL. Every object is a construction 
(of atomic constructions!) but the variables and non-atomic 
constructions are not objects. 
The interrelations between lans~age expressions, objects 
and constructions are stipulated as follows: 
Let E be a language expression: E expresses a construct- 
ion, say, CE, and names (denotes) the object, say, OE, which 
is constructed by C E- 
There are, however, some expressions whose role differs 
from the role of "normal", semantically analyzable expressions. 
This concerns 
i) expressions whose role is solely a syntactic one, 
ii) interjections, 
ill) "egocentric expressions" such as "I", "you", "her.n, 
"thiS", etc. 
The category ii) is uninteresting in our context. As 
for i), transform~ E into C E is generally impossible without 
the expressions from this category. With ili), a pragmatic 
el~nent appears: the transformation into C E is possible only 
if we are acquainted with the situation in which E has been 
uttered. 
A special category of expressions is, from the semantic 
viewpoint, the category of "formal expressions", such as the 
mathematical ones, e.g., "two times three" or "four minus two 
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equals two". These expressions are supposed to denote direct- 
ly constructions. 
2, U_nder_st.and_i_n~, It may be stated that there is a ge- 
neral agreement among llngulsts, as well as among logicians, 
that understanding (in the semantic sense) is a relation 
between an individual and an expression. Any explication must, 
however, specify this relation, With respect to what has been 
said above, there are two principal possibilities of such a 
specification- 
a) (the individual) A understands E iff A asociates E 
with the object (0 E) denoted by E! 
b) A understands E iff A associates E with the construct- 
tion (~) expressed (or - in case of a mathematical 
expression - denoted) by E. 
Thus let A hear or read a sentence S. In the case a) 
we would say that A understands S iff he knows that S names 
a proposition P. In the case b) we should say that A under- 
stands S iff he knows the construction C S that constructs P 
(or any structure preserving the meanings of "atomic express- 
ions" and isomorph with CS). 
It is clear that understanding in the sense of b) 
implies understanding in the sense of a) wherever both these 
senses are thinksble.We can show, however, that the implicat- 
ion does not hold vice versa. Indeed, take the English sen- 
fence 
(S) John owns a cutlass or he does not own a cutlass. 
Even those who do not know what a cutlass is will know that 
(S) denotes the proposition "verum", i.e., the proposition 
which is true in every possible world at every time moment. 
Thus not knowing the construction C S (because of not knowing 
an atom being part of it) the above individuals know the 
object (i.eo, the proposition) denoted by (S). One argument 
against explloatlnE understanding in the sense a) is that we 
would probably hesitate to say that who does not know the 
meaning of "cutlass" does all the same understand (S). 
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Another argument against a) is strictly bound ~o our 
conception and can be formulated ms follows: in the case of 
mathematical expressions a) is not applicable, since such 
expressions generally name constructions rather than objects. 
The last argument in favour of b) again refers to our 
intuition: we feel that one can "more or less" understand an 
expression. This "more or less" is excluded if we connect 
understanding with the objects named by expressions (or~ at 
most, we must confine ourselves to the cases of "more or less" 
clear meanings of particular atomic expressions). When connect- 
ing understanding with constructions we can explicate tbls 
"more or less" rather intuitively. We shall sketch this ex- 
plication (technical details are omitted again): Let the 
given expression E contain n "atomic", i.e., unanalyzable 
meaningful (sub/expressions) including, as the case may be, 
the expressions from the category iii) ) el,...,e n. A necess- 
ary condition for A's understanding E (in the sense b) ) is 
that A associated the appropriate atomic constructions, i.eo, 
objects with el,...,e n. (The second necessary condition 
consists in A's correct transformation of E into a construct- 
ion schema according to the grammar of the given language). 
Now, the degree of A's understanding E can be, among others, 
measured by 1 - k/n, where k is the number of those subexpress- 
ions among el,...,en, which are associated >y A with no object 
at all! analogically, the degree of A's misunderstanding E 
can be measured by k'/n, where k ° is the number of those 
subexpressions among el,...,en, which are associated by A 
with an inappropriate object. (Clearly, k~" k'< n.) 
Thus it seems more appropriate to accept the position 
b) and to claim that understanding is a relation between an 
individ~.al and an expression which holds iff the individual 
correctly associates two structured entities: the grammatical 
(or: tectogrammatical) structure of the expression and the 
(logical) structure of the corresponding constructions 
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