LOGIC AS A STUDY OF
CONCEPTS
Pavel Materna
Institute of Philosophy
Academy of Sciences of Czech Republic
Prague
Kyiv 23.5. – 25.5.
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LOGIC AS A STUDY OF CONCEPTS
• The only tools that Philosophy can use are concepts.
• What are concepts? (Better: How to aptly explicate the notion
of concept?)
• Two groups of explications:
– According to objectivist or logical conceptions, concepts exist
independently of individual human minds, e.g. As self-subsistent
abstract entities or as abstractions from logical practices. According to
subjectivist or psychological conceptions, concepts are mental
phenomena, entities or goings-on in the mind or in the head of
individuals.22
– (Glock 2009, 5-6)
Clearly, the latter explications are used in cognitivist theories.
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LOGIC AS A STUDY OF CONCEPTS
• Only the former kind of explications can be accepted by logic.
• One option (not only) Frege: concepts are simply universals,
i.e., what is denoted by predicates (prime number(s), colour(s),
• but also cat(s), tree(s) …).
• BUT a pitfall: What would be the distinction between concepts and classes
/ relations?
• This has been a problem for Gödel (1944, see Feferman… 1990, p. 140 and
others; see Materna 2007). The notions of meaning, property, class,
concept are not well explicated, at least in mathematics. Gödel himself is
not happy with this situation, he would welcome a theory that would
consist „in trying to make the meaning of the terms „class“ and „concept“
clearer , and to set up a consistent theory of classes and concepts as
objectively existing entities“.
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LOGIC AS A STUDY OF CONCEPTS
• One of the most important notion in logic (as well as
in mathematics) is the notion of function. Could we
perhaps say that concepts are functions?
• Here is why not:
• There are indefinitely many concepts of one and the same object. Take for
example the concept that is given by the definition
• D1: Natural numbers greater than 1 and divisible just by 1 and itself
• and the concept given by the definition
• D2: Natural numbers that have just two factors
• Clearly, bot D1 and D2 define the class of prime numbers. Let us suppose
that this class is given by two distinct concepts (a highly intuitive
assumption).
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LOGIC AS A STUDY OF CONCEPTS
• Now we can define the function F1 from D1, i.e.,
define a function such that returns True just on such
numbers that satisfy the definition D1.
• Similarly, the function F2 can be derived from D2.
• If concepts were functions then our intuitive
assumption would be falsified. In contemporary logic
it holds (f, g are variables for functions):
• fg (x1…xn (f(x1…xn) = g(x1…xn))  f = g)
• So that F1 is the same function as F2.
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LOGIC AS A STUDY OF CONCEPTS
• Thus while we wanted to get two distinct (but equivalent)
concepts we have got one function.
• Such situation is frequent. One forgets that from the viewpoint
of contemporary logic functions are pure extensions and if
they are used to analyze an expression E then the resulting
semantics throws the meanings of the subexpressions of E into
a ´black hole´. What counts is always the product of some
procedure, never the procedure itself (which is handled at most
verbally in some meta-language).
• In this way many semantic puzzles and misunderstandings
arise.
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LOGIC AS A STUDY OF CONCEPTS
• Our intuition in the last example has it that there are two
distinct concepts here: one which is derivable from D1, the
other one derivable from D2. In other words:
• Concepts should be ways to an object (if any). Thus we can
admit that in our example there is really one function (as a
mapping) in the sense of distribution of truth-values of the
characteristic function of the class of prime numbers whereas
there are two ways how to get this function.
• A possible generalization:
•Concepts are ways how to get objects.
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LOGIC AS A STUDY OF CONCEPTS
• But then
• the set-theoretical paradigm that dominates so much of
contemporary logic should be abandoned, i.e., the view
• that logic studies exclusively results of some operations and
that these results are satisfactorily captured by set-theoretical
expressions (preferentially by 1st order expressions).
• To abandon this view does not mean that
• a) the results of modern logic accepting the paradigm are not important or
even wrong
• b) we have to accept Dummett-like intuitionist anti-realism.
• We will show that there is such an explication of the notion of concept that
does not call for anti-realism and does not accept the set-theoretical
paradigm.
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LOGIC AS A STUDY OF CONCEPTS
• Objectivity of concepts
• Bolzano 1837: concepts are a kind of Vorstellungen an sich.
• (No mental entities!)
• Church: concepts are possible senses (Frege´s Sinn) of
expressions (…anything which is capable of being the sense of
some name in some language, actual or possible, is a
concept“(1985, 41).
• The sense of an expression E is a concept of the
denotation of E (See Church (1956, 6)
• Cf. the example with two definitions of primes!
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LOGIC AS A STUDY OF CONCEPTS
• We will use the term meaning as synonymous with sense.
• We have seen that Church (when correcting Frege´s definition of sense) has
construed concepts as objective, language-independent entities which can
become senses (meanings) of an expression.
• If such entities were some sets we could not explain how come that distinct
concepts (possible meanings) can be linked to one and the same object. No
set can be a ´way´ to an object.
• Such problems, frequently connected with attitudinal texts (see already
Frege 1892), made Carnap (1947) uncertain as for his method of intensions
and extensions and led Church to his criticisms of Carnap and to inspiring
„Alternatives“ – see Anderson 1998) – similar in a way to the solution
proposed by Tichý and TIL.
• It seemed that the solution would come from Cresswell (1975), (1985) with
his slogans „hyperintensionality“ and „structured meanings“.
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LOGIC AS A STUDY OF CONCEPTS
• The idea that meaning is structured is attractive. As soon as
concepts are related to meanings in the sense of Church (1956)
the idea of structured meanings implies that concepts are
structured as well.
• Cresswell´s attempt at the realization of his idea has not led to
really structured entities, as Tichý has shown in (2004, 839842), in more details can be this critique found in Jespersen
(2003) and DJM (2010).
• The idea itself is in its generality sound. We have stated that no
set-theoretical object can be a way to anything. Sets are simple
objects. Being structured means being not a simple object.
• Some historical remarks:
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LOGIC AS A STUDY OF CONCEPTS
• Aristotle invented concepts by building up his theory of
definitions. Two principles are essential:
• a) Definiens (i.e. what we would today call concept) is never
simple.
• b) No object can have more than one definition.
• The point a) is clear. Some comments will be useful though:
• Aristotle´s definition is not Russell´s definition-abbreviation. Aristotle´s
definiendum is not a mere abbreviation of definiens where defiiniendum
would get its meaning just by definiens. Aristotle´s definiendum is
something like a vague term: we understand it in some way but we do not
know its essence, and definiens tries to find this essence. This finding the
essence is naturally not simple.
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LOGIC AS A STUDY OF CONCEPTS
• Still more interesting is b). Unlike Russellian
definitions Aristotle´s definitions may be right or
wrong (Man as a not feathered biped vs. Man as a
rational being): the essence can be aptly captured or
not.
• Interestingly, more than 2000 years after Aristotle a
modern counterpart of this principle can be found in
the writings of some logicians who begin to discover
the possibility of a hyperintensional view.
• The following quotations are from Bealer (1982):
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LOGIC AS A STUDY OF CONCEPTS
• „…there have been two fundamentally different
conceptions of properties, relations, and propositions.
On the first conception intensional entities are
considered to be identical if and only if they are
necessarily equivalent…On the second
conception…each definable entity is such that, when
it is defined completely, it has a unique, non-circular
definition.“
• And Bealer´s example is radical enough:
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LOGIC AS A STUDY OF CONCEPTS
• „(c)
x is a trilateral iff x is a closed plane
figure having three sides.
• (d)
x is a trilateral iff x is a closed plane
figure having three angles.
• On the first conception both (c) and (d) count
as correct definitions since they both express
necessary truths. On the second
conception…(d) does not count as a correct
definition; only (c) does.“
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LOGIC AS A STUDY OF CONCEPTS
• This ´hyperintensionalist manifesto´ is in harmony with
Aristotle. It could seem however that it is incompatible with
our intuition that various concepts can determine one and the
same object. But it is not: We can reformulate the last
quotation as follows:
• We have got two concepts: a closed plane figure having three
sides, and a closed plane figure having three angles. If what
we are interested in is the number of sides we have to use the
former concept, if we are interested in the number of angles
we have to use the latter.
• (Compare: if the physical appearance is what we are interested
in we use „not feathered biped“, if the distinction man – beast
is important we use „rational being“.)
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LOGIC AS A STUDY OF CONCEPTS
• Logic and Logical analysis of natural language that abandons
the set-theoretical paradigm becomes a procedural logic. A
most systematic procedural approach to logic that is however
not connected with anti-realism is represented by Tichý´s TIL.
• A detailed information on TIL can be found in Tichý´s
writings, especially in his (1988), and in many articles and
books of his followers, in particular in Duží, Jespersen,
Materna (DJM) (2010) („Procedural Semantics for
Hyperintensional Logic“), Springer.
• In what follows we will a) bring a brief information about TIL,
• b) finish the explication of concept, c) show that Logic really
is a theory of concepts.
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LOGIC AS A STUDY OF CONCEPTS
• So we have learned from Aristotle and Bealer that concepts
should be not simple and that necessary (logical, analytical)
equivalence is not a sufficient criterion of identity.
• Let us return to our great Bolzano. In his (1837) he elaborated
a remarkable theory of concepts as a kind of Vorstellungen an
sich, where he argued that they are structured. Traditionally,
what concepts are was not clear or even was psychologically
explained, but every student knew that concepts possess
content and extent (Inhalt, Umfang). Talking about content
Bolzano suggests that while the content of a concept consists
of some components[1]it does not determine the way in which
these components combine.
• Thus the concept is just this way. That this interpretation is right can be
justified, e.g.,by a remarkable place in §148 of (1837) where Bolzano
distinguishes between the concept, say, TRIANGLE1, as defined in terms
of having three sides, and the concept, say, TRIANGLE2, as defined in
terms of having the sum of its angles equal to 2R.
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LOGIC AS A STUDY OF CONCEPTS
• Giving together some desiderata for explicating concepts we
can appreciate Pavel Tichý, who already in 1968 recognized
that one of the thinkable ways to satisfy them (and probably
the best one) consists in handling not only results of
procedures but also procedures themselves.
• In 1968 and 1969 he realized this view by modelling
procedures as Turing machines (in the case of analyzing
empirical expressions O-machines), later – already in
University of Otago (Dunedin, New Zealand) – he created a
highly expressive system (TIL)
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LOGIC AS A STUDY OF CONCEPTS
• Standard approach to analyzing expressions:
• (Cf. Montague after 1970) expressions of a natural language are
translated into a logical system (intensional logic, IL) and semantics is
defined due to standard rules of interpretation in IL. The other approach is
represented by Montague´s 1970 and, mainly, by Tichý´s Transparent
intensional logic (TIL), which defines semantics directly.
• Tichý´s approach consists in associating expressions of a
natural language with abstract procedures (constructions),
which make up their meanings. What is missing is the
translation to IL.
• This ´direct´ approach has been argued for in Tichý (1988):
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LOGIC AS A STUDY OF CONCEPTS
• There is no intrinsic relation between a formula and the construction it
represents.
• Hence if anything said about the formula is to have a bearing on things
mathematical, the relation of the formula as a whole, or of its constituents,
to mathematical objects must be explicitly stipulated. In order to put a
stipulation into words, one has to name entities of both kinds: the
mathematical objects and the linguistic expressions corresponding to them.
Hence the need for a metalanguage, distinct and separate from the original
notation in question. But the metalinguistic expressions themselves signify
constructions. One thus faces a choice: one can either acquiesce in these
higher-order constructions, or one can
• ignore them too and look instead at the meta-meta-expressions
corresponding to them. If the first option is chosen the question arises why
the same treatment cannot be applied at the bottom level, thus avoiding the
original linguistic detour as well. And if the second option is taken one is
obviously caught in an infinite regress of ever higher metalanguages.
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LOGIC AS A STUDY OF CONCEPTS
• Constructions in TIL are well-defined abstract procedures
´working in´ a type-theoretical setting. The apparatus of TIL
reminds us of applying lambda calculus by R. Montague.
• Simple hierarchy of types uses the base {, , , } and
functional types:
• ( 1,…, n): Classes of partial functions with  … type of the
value
• i … type of the ith argument.
• Every object has got a function (maybe nullary) as its type.
• Examples: (((  ) ) ), abbrev. ()  : a property of
individuals
– in general intensions:  for any type .
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LOGIC AS A STUDY OF CONCEPTS
•
•
•
•
•
•
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Constructions
Variables
(Double) Executions
Trivialization
Composition
Closure
Examples
• 3 + (5 – 7)
•
•
•
x, y, z,… names of Variables
(2)X
0X
[XX1…X]
x1…x X
[0+ 03 [0- 05 07]]
Some pupils do not believe that the Moon is smaller than the Sun
wt [[0Some 0Pupilwt]x [0 [0Belwtx wt [0Smwt0Moonwt 0Sunwt]]]]
Constructions are procedures, they do not contain brackets or letters (unlike their
records).
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LOGIC AS A STUDY OF CONCEPTS
Higher-order types
Ramified hierarchy of types: by defining which types are to be assigned to
constructions it makes it possible
to construe constructions as objects sui generis.
Idea:
Constructions of order n are defined (they construct objects of a type of order
n).
n is the set of constructions of order n.
n and types of order n are types of order n + 1.
Example: x let be a numerical variable. It is a construction of order 1. , a
member of 1, so it is an object of a type of order 2. 0x constructs x and its
order is 2, so it is an object of a type of order 3.
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LOGIC AS A STUDY OF CONCEPTS
• Concepts approximately: closed constructions.
• They can become meanings of expressions that do not contain
indexicals.
• So they are structured meanings (Cresswell).
• The principle of identifying functions is (slide 5)
• fg (x1…xn (f(x1…xn) = g(x1…xn))  f = g)
•
For constructions in general and for concepts the following
principle holds (c, d: concepts):
• cd (c = A & d = A & 0c  0d)
• So there are indefinitely many concepts of one object.
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LOGIC AS A STUDY OF CONCEPTS
•
•
•
•
•
•
•
Empirical and non-empirical concepts.
A Fregean corrected semantic ´triangle´ :
An expression E
expresses a construction
this construction is the Fregean Sinn (sense, meaning) and
constructs an object (if any), which is the denotation of E.
Empirical concepts construct non-trivial intensions, they have
always denotations.
• Example: the man that is higher than Eiffel tower
• This expression denotes an individual role (type    ).
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LOGIC AS A STUDY OF CONCEPTS
• So empirical concepts always construct a function.
• (What may be missing is the value of the denoted
intension in the actual world – „reference“.)
• Mathematical concepts construct extensions
(numbers, classes, other concepts,…), they may
construct nothing (the greatest prime…)
• Non-empirical concepts containing empirical
concepts construct trivial intensions (the case of
analytic sentences).
• (Example: All bachelors are men.)
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LOGIC AS A STUDY OF CONCEPTS
•
Concepts (as any meaning for that matter) are contextindependent. Any expression expresses a concept
independently of the context. What is dependent on the context
S is the way the concept is handled in S.
• There are three kinds of S in this respect (let C be the given
concept):
•
hyperintensional: C is in S mentioned, i.e. in S we have
0C.
intensional: C is used in S to construct a function.
extensional: C is used in S to construct the value of the
function on its argument.
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LOGIC AS A STUDY OF CONCEPTS
• Consider now any empirical science S.
• The purpose of S is to inform us about the real („actual“)
world. Thus S uses empirical concepts.
• Empirical concepts construct non-trivial intensions, i. e.
criteria. No empirical concept can construct some real object.
Therefore we need experience:
• The way from the intensions-criteria to objects of the real
world at the given time is not a logical step; so we can
construct empirical propositions (i.e. criteria of truth-value)
but without an empirical stage we cannot verify our claims.
• Thus S uses empirical concepts not only intensionally but also
extensionally.
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LOGIC AS A STUDY OF CONCEPTS
• A simple example:
• An empirical sentence „The Mayor of Brno is corrupt“ uses empirical
concepts Mayor of, Corrupt. Let Brno be an individual (simplification).
The concept expressed by the sentence is
• w t [0Corrwt [0Mayorwt0Brno]]
• Assume that we know the („simple“) concepts 0Corr and 0Mayor, i.e. that
we know the procedure that leads to the property being corrupt and and the
procedure leading to the function of being the Mayor of…, assume also that
we know the town Brno. So we know the criteria (intensions) given by
these concepts. To verify / falsify our sentence we have to use our concepts
extensionally, i.e. to know the value of the respective intensions in the
actual world (+ time).
• This can be reached only empirically because no logical step can lead to
identification of the actual world (+ time).
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LOGIC AS A STUDY OF CONCEPTS
•
•
•
•
•
While empirical sciences use empirical concepts to learn
something about reality, i.e.,use empirical concepts
extensionally,
Logic does not use empirical concepts at all.
The only concepts that are used in logic (not only
intensionally but also) extensionally are logical concepts, i.e.,
concepts expressed by „logical words“.
Thus the (empirical) sciences use concepts to find out some
facts about the real world whereas logic (mentions and) uses
concepts to find out some properties and relations of concepts:
The results of extensional use of logical concepts never
concern the reality.
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LOGIC AS A STUDY OF CONCEPTS
• Quod
–Erat
Demonstrandum
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LOGIC AS A STUDY OF CONCEPTS
• References
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Anderson, C. A. (1998): ‘Alonzo Church's contributions to Philosophy and
Intensional Logic’. The Bulletin of Symbolic Logic 4 (2),
129-171.
Bealer, G. (1982): Quality and Concept, Oxford: Calendon Press.
Bolzano, B. (1837): Wissenschaftslehre, vols. I, II. Sulzbach.
Carnap, R. (1947): Meaning and Necessity, Chicago: Chicago University Press.
Church, A. (1956): Introduction to Mathematical Logic, Princeton: Princeton,
Church, A. (1985): Intensional Semantics. In: A. P. Martinich (ed.) The Philosophy of Language, Oxford
UP, 40-47
Cresswell, M.J. (1975): ‘Hyperintensional logic’, Studia Logica, vol. 34, pp. 25-38.
Cresswell, M.J. (1985): Structured meanings, Cambridge: MIT Press.
DJM: Duží, M, Jespersen, B., Materna, P. 2010. Procedural Semantics
for Hyperintensional Logic. Springer 2010
Frege, G. (1892): ‘Über Sinn und Bedeutung’, Zeitschrift für Philosophie und
philosophische Kritik, vol. 100, pp. 25-50.
Glock, H. J. (2009): Concepts: Where Subjectivism Goes Wrong. Philosophy 84, 5-29
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LOGIC AS A STUDY OF CONCEPTS
•
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Gödel, K. (1944) See
Feferman, S., J.W.Dawson, Jr., S.,C., Kleene, G.H.Moore,R.M.Solovay, J.van Heijenoort, eds.:
Kurt Gödel Collected Works Vol. II., Oxford UP 1990
Materna, P. (2007): Properties of mathematical objects. (Gödel on classes, properties and concepts)
In: Journal of Physics: Conference Series 82 (2007) 012007, 1-15
Tichý, P. (1988): The Foundations of Frege’s Logic, Berlin, New York: De Gruyter.
Tichý, P. (2004): Collected Papers in Logic and Philosophy, V. Svoboda, B. Jespersen, C.
Cheyne (eds.), Prague: Filosofia, Czech Academy of Sciences, and Dunedin:
University of Otago Press.
•
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