1 logic as a study of concepts pavel materna institute of philosophy academy of sciences of czech...

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LOGIC AS A STUDY OF CONCEPTS

Pavel MaternaInstitute of Philosophy

Academy of Sciences of Czech Republic

Prague

Kyiv 23.5. – 25.5.

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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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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• 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, 839-842), 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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• 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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• 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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• „…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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• „(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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• 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 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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• 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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• 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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• 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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• 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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• 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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• Constructions• Variables x, y, z,… names of Variables• (Double) Executions (2)X• Trivialization 0X

• Composition [XX1…X]

• Closure x1…x X

• Examples

• 3 + (5 – 7) [0+ 03 [0- 05 07]]• Some pupils do not believe that the Moon is smaller than the Sun

wt [[0Some 0Pupilwt]x [0 [0Belwtx wt [0Smwt0Moonwt

0Sunwt]]]]

• Constructions are procedures, they do not contain brackets or letters (unlike their records).

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Higher-order typesRamified 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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• 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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• 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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• 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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• Concepts (as any meaning for that matter) are context-independent. 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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• 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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• 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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• 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.

phil75

(which are they? This is not a trivial problem, see Bolzano 1837).

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• Quod

–Erat

Demonstrandum

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• References

• 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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• 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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1 logic as a study of concepts pavel materna institute of philosophy academy of sciences of czech...

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