The logical-philosophical basis of logical systems

Tamás Mihálydeák

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University of Debrecen Indiana University Bloomington Department of Philosophy Department of Philosophy 4010 Debrecen, Egytem tér 1. Sycamore Hall, 1033 E Third Street Bloomington, Indiana 47405-7005 www.unideb.hu www.iub.edu [email protected] Adviser: Prof. Karen Hanson ......

The most general questions are what modern logic regards the logical role of compositionality, how it works in two—component logical . After showing different versions of the compositionality of natural language we analyze the possible appearances of the principle of compositionality in two—component logical semantics. Finally, some of the most fundamental notions of intensional logical semantics are given in the way of maintaining the priority of compositionality concerning sense.

1. Background It is more than 125 years that a booklet language of pure thought modeled on that of by a hardly thirty-year old logician/ arithmetic41, which occupies a special place philosopher was published. So something in philosophical (and not only in philo- began. Something, which was not really sophical) culture. 20th (and 21st)-century a continuation, it revealed brand new logic cannot exist without Frege’s life- horizons for thinkers. This booklet is work. In the preface of Begriffsschrift, at Gottlob Frege’s Begriffsschrift, a formula the birth of modern logic, Gottlob Frege

41 See Preface and Part I translated by M. Beaney from Begriffsschrift, eine der arithmetischen nachgebildete Formelsprachen des reinen Denkens (L. Nebert, Halle, 1879) in Beaney 1997, pp. 47-78. 111 AY 2002-2003 The logical-philosophical basis of logical systems writes he disregards expressing anything grammatical structures by structures from the meaning of its constituents. creativity/understandability’ cita- that has no significance from the point relying on the functor-argument decom- (Szabo 2000, p. 488) tions.” (Pelletier 2001, p. 111) of view of consequence. He calls what position from the logical point of view • The Substitutivity Principle: If two Secondly, I have to mention there is a very he finds solely important conceptual the main question is whether there is a expressions have the same meaning, wide—ranging scientific discussion about content. Summing up my intentions, I theoretical limit of the functor-argument then substitution of one for the other the compositionality of natural language.46 can word my most general questions as decomposition. The limit can be found in a third expression does not change The place and role of the principle of what modern logic regards the logical in the semantic mirror of the functor- the meaning of the third expression. compositionality in linguistics is quite role of compositionality, how it works in argument decomposition, in semantic (Szabo 2000, p. 490) controversial. It is regarded as a method- two-component logical semantics, how it compositionality.43 The structures which First of all I have to note that the prin- ological principle, or a basic linguistic- structures conceptual content. are relevant from the logical point of view ciple of compositionality (and each of its philosophical law, or supervenience. In Frege’s view, one of the most important (i.e. logical structures) must have been versions) is referred to as ‘Frege’s principle’ (See, for example, Partee 1984 and Szabo inventions of Begriffsschrift is the created in a compositional way.44 (or ‘Fregean principle’). Philosophers and 2000.) replacement of the subject— logicians dealing with Frege have been decomposition by the functor-argument 2. Compositionality discussing for many decades whether Figure 1 one. He wrote the following: What is compositionality, what is meant by Frege accepted the principle of composi- The very invention of this compositionality? Each of the following tionality (or at least one of its versions), 45 Begriffsschrift, it seems to me, has four formulations appears as the principle and if he did, to what extent. I don’t advanced logic. … [L]ogic hitherto of compositionality. Different names want to analyze the whole story of the has always followed ordinary language (used in Szabo 2000) for different versions discussion, I would like to mention only and grammar too closely. In particular, are appropriate to show small differences two attitudes, the first from Hans Rott, I believe that the replacement of the between them. and the second from Jeffrey Pelletier: concept subject and predicate by • The Principle of Compositionality 1. “The principle is certainly very Fregean argument and function will prove (in a wide sense): The meaning of a in spirit … However, as Janssen (1997, itself in the long run. It is easy to see complex expression is determined by p. 421) points out, Frege does not how taking a content as a function the meanings of its constituents and seem to have stated compositionality of an argument gives rise to concept by its structure. (Szabo 2000, p. 475) as a principle in any of his writings.” formation. … The distinction between • The Function Principle: The meaning (Rott 2000, p. 625) subject and predicate finds no place in of a complex expression is a function 2. “Frege may have believed the prin- 42 my representation of a judgement. of the meaning of its constituents and ciple of semantic compositionality, (Frege 1997, pp. 51, 53) of its structure. (Szabo 2000, p. 484) although there is no straightforward Are we aware of the significance of the • The Building Principle: The meaning evidence for it and in any case it does step? How can we survey and evaluate of a complex expression is built up not play any central role in any writing In figure 1 the situation before the its consequences? Before Frege it was of his, not even in the ‘argument form mentioned distinction between gram- generally accepted that grammatical 43 Mihálydeák 2003 deals with the formal syntactical matical structure and logical structure structures were also logical ones, and what 45 Here I don’t deal with the principle of contextuality, but and semantic properties of the most general type- made by Frege can be surveyed. The is more all logical structures had to prove as Janssen (2001, p. 115) mentions many philosophers theoretical language relying on functor—argument principle of compositionality holds and to be grammatical ones. Having replaced understand the principle of compositionality as the decomposition. works on the level of natural language. principle of contextuality. That principle is mentioned 44 Here I am speaking about the compositionality of Logical structures originated from at the beginning of Frege’s Grundlagen der Arithmetik 42 I use the expression ‘functor’ instead of ‘function’ natural language. The modern version of compositionality in the following way: “never to ask for the meaning of a 46 See for example the thematic issue of Journal of in order to differentiate an incomplete expression of a appeared tacitly in Frege’s Über Sinn und Bedeutung word in isolation, but only in the of a ” Logic, Language and Information (Vol. 10, 2001) on language from its semantic value. (Frege 1892). (Frege 1980, p. x). compositionality.

112 113 AY 2002-2003 The logical-philosophical basis of logical systems natural language directly, based only on Figure 2 example, the origin of information content constituent expression.” (Salmon 1994, the compositionality of natural language. of identity statements. In his most famous p. 112)50 Carnap was the first to attribute Received logical laws get meaning through semantic paper, Über Sinn und Bedeutung both versions of the principle explicitly associated structures and compositionally (Frege 1952c) he introduces an extensive to Frege. He wrote the following in his joined natural language meanings. version of two-component semantics, he fundamental semantic book, Meaning and Therefore, there is no separated room differentiates sense and (or Sinn Necessity (Carnap 1947): for the (either informal/pretheoretic or and Bedeutung).49 “Frege Principles of Interchangeability: formal) principle of compositionality Taking the principle of compositionality • … First principle … the nominatum of concerning the logical features of natural seriously two questions may arise: the whole expression is a function of language expressions. Natural language 1. How should we modify the principle the nominata of the names occurring compositionality acts not only on the of compositionality (of the informal in it. level of natural language but on the level level)? • … Second principle … the sense of the of logical investigation as logical compo- 2. How does the functor—argument whole expression is a function of the sitionality. It produces the main patterns decomposition producing the main senses of the names occurring in it.” of logical structures. The logical system logical structures work with the system (Carnap 1947, p. 121)51 is closed, at least in some sense, in the of semantic values of two-component Now let us turn our attention to the prison of natural language. semantics? second question, i.e. the behavior of the If we try to answer the first question, we functor-argument decomposition in the will have to take into consideration that case of two principles of compositionality. 3. Compositionality many philosophers duplicate the principle Anybody may think there is no problem and two—component and attribute both principles (concerning at all, we have two different principles of semantics play a crucial role in the system of logi- the reference and the sense of compound compositionality, and we can use them in Frege broke up the identity between the cally relevant semantic values. It can be or complex expressions) to Frege. “Crucial determining logically relevant semantic grammatical structure of natural language said – regarding Begriffsschrift –, that to Frege’s theory are a pair of principles value. But which of them do we have to expression and its logical one. The gram- the system of semantic values which is in concerning the referent and sense of take into consideration, and how? In matical structure may only appear as one the informal background is dominated by complex expressions. These are the Prin- logical investigations we are interested of the possible logical structures, but an truth values in the sense that we receive ciple of Compositionality (Interchange) in the truth value of a sentence, i.e. its expression (or in a more sophisticated the semantic values of functors from the of Reference and the analogous Prin- reference. The reference can be produced 47 way, the expressed conceptual content) set of truth values and the set of objects. ciple of Compositionality (Interchange) by means of the first principle concerning 48 may have, and usually does have logical In his semantic writings Frege recog- of Sense. They hold that the referent or reference, and the function occurring in it structures different from the grammatical nized that a more flexible system of sense of a complex is a function only of is asking for the reference of arguments. structure of the expression. semantic values was needed to explain, for the referents or senses, respectively, of the Frege recognized that in some cases the reference of the whole expression cannot be determined by means of the In figure 2 there is a special box for 47 As is well—known Frege considers sentences as 49 There is no standard terminology for different of its parts. Sometimes we need to take into informal compositionality (of infor- a special type of names, and he puts the possible semantic values. In the literature many pairs appear: mation/conceptual content). What does semantic values of sentences, i. e. truth values, into the sense-reference, meaning-reference, sense-meaning, it mean? In order to understand it we set of possible semantic values of names, i. e. the set of sense-nominatum, sense-denotatum, meaning- 50 Quoted by Pelletier (2001, p. 88). have to define the notion of semantic objects. This unification proved problematic later, and denotatum, -, intension-factual value. 51 Quoted by Pelletier (2001, p. 89). The principles are value, and specify what kinds of semantic therefore the development of logical semantics hasn’t While on the informal level I use on similar to the function principle, which is the second values we have. Naturally, truth values followed Frege in that sense. the formal level I do intension and extension. version of the principle of compositionality. 48 For example:Frege-(1952a, b, )

114 115 AY 2002-2003 The logical-philosophical basis of logical systems consideration not only the reference of an also have to use the second principle in approach reference cannot be identified is that those functors are extensional ones argument but also its sense. However, at the determining the reference of the whole with sense, thus we have to speak about which affect the reference of their argu- first glance this contradicts the principle expression, we need the principle not the sense of an expression occurring in an ments in order to get the reference of of compositionality concerning reference. only in the case when we are interested indirect context and its sense can be the the output (of the whole expression), and How did Frege try to get rid of the prob- in the sense of the whole expression but sense of its ordinary sense (etc.). those functors are intensional ones which lematic situation? As is well-known he also when we determine the ordinary The next question is how the mentioned are not extensional and affect the senses differentiates between direct and indirect reference. Frege cannot avoid the use of problem connected with context-depen- of their arguments. It may seem that we occurrence, ordinary (direct or customary both principles but he didn’t mention how dence (or more precisely occurrence- are ready and the problem of context- in Frege) and indirect (oblique) reference to apply the second one, how to get the dependence) of the type of reference can dependence (or occurrence-dependence) respectively and he says if the occurrence of sense of the whole expression and what be avoided. I would like to emphasize that is solved. We may also realize that the an argument is indirect, then its reference the connection is between the principles. the problem is not the context-dependence notion of extensional functor corresponds is its ordinary sense. At this point we can differentiate two of reference. The real problem is that in a to the first principle of compositionality Let us pay all our attention to functors. main types of functors and introduce typical fixed situation (where the ordinary concerning reference; however, the notion The output of a function occurring in the the notions of Fregean intensional and reference and the ordinary sense are given) of the intensional one doesn’t correspond principle of compositionality concerning Fregean extensional functor, which will in some cases the ordinary reference is the to the second principle concerning sense references is the reference of the whole prove very useful further on. Let a functor reference and in other cases the indirect because an intensional functor produces expression. We saw that it might depend be extensional or intensional in the Fregean reference i.e. the ordinary sense is the not the intension but the reference of its on either the reference or the sense of its sense if the occurrence of its argument is reference. output. argument. Two types of functors appear: direct or indirect, respectively. Of course a Carnap recognized the problem in Is there any way to embed the second 1. If a given expression occurs directly functor is extensional in the Fregean sense Frege’s approach, and tried to choose principle in this picture? I think we have to in an expression, then the given if and only if it is not intensional in the another way. He characterized the differ- find the way, because the second principle expression can be taken as an argument Fregean sense.52 We have to note that all ences with Frege’s method as follows: is more fundamental than the first one. and the remaining part of the whole functors in Frege’s semantic theory (called A decisive difference between our Sense is the most fundamental semantic expression, i.e. the functor concerns Fregean functors) affect the reference of method and Frege’s consists in the value. In order that an expression may the ordinary reference of the given their argument and their results are usually fact that our concept, in distinction to belong to a natural language, in order that expression. a given type of reference. Frege’s, are independent of the context. it can be an expression of a given language 2. If a given expression occurs indirectly The subtle distinction between direct An expression in a well-construed it has to be meaningful. Accepting Kripke’s in an expression, then the functor and indirect occurrence has a very prob- language system always has the same approach, only proper names can be (the remaining part of the whole lematic consequence: The reference (and extension and the same intension: but exceptions in the manner that they may expression) concerns the indirect therefore the sense) of an expression [in Frege’s theory] in so me context be expressions of a given natural language reference (i.e. the ordinary sense) of depends on the context in which it occurs it has its ordinary nominatum and its without having sense. (Usually nobody the given expression. and, of course, we have to determine not ordinary sense, in other contexts its wants to ‘understand’ a proper name, We can say that the latter type of functors only the indirect reference of an argument, oblique nominatum and its oblique everybody wants to use it for referring to affect the (ordinary) sense of their argu- but also its indirect sense. In the Fregean sense. (Carnap 1947, p. 125) something.) It can be said that except for ments. Since we attach reference to The definitions of Fregean intensional proper names there is no expression in reference, everything seems to happen and extensional functors apparently any natural language which has reference 52 That type of definition of the Fregean intensional and according to the first principle. However, need to be modified just a little bit to get but no sense because meaningfulness is extensional functors is not usual. Generally the notion indirect reference is ordinary sense, and ones which are applicable to Carnap’s the crucial characteristic of an expression of extensional functor is defined at first, and non— in order to get ordinary sense the second approach. We only have to transfer the that belongs to a given language. In two- extensional functors are intensional ones; the defined principle should be applied. Thus we sensitivity of the type of semantic value component semantics an expression may functors are not always Fregean. from occurrences to functors. The result have sense without having reference,

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Figure 3 but we cannot understand an expression 4. Conclusion which has reference but no sense. From the logical philosophical point of Relying on the two principles of compo- view, introducing the functor-argument sitionality we can say that the reference Bibliography decomposition and accepting its domi- (nominatum) of a functor is the function Beaney, M. (ed.). 1997. The Frege Reader. nance enforce the informal composi- which gives us the reference (nomi- Blackwell, Oxford. tionality of conceptual content which natum) of the whole expression with the Benthem, J. van and Ter Meulen, differs from the compositionality of A. (eds.). 1997. Handbook of Logic references (nominata) of different argu- natural language. In two-component and Language. Elsevier, MIT Press, ments, and that the sense of a functor logical semantics two different principles Amsterdam, The Netherlands, is the function which gives us the sense have to be represented. We showed that Cambridge, Ma. of the whole expression with the senses Carnap, R. 1947. Meaning and Necessity: the first principle of compositionality of different arguments. We can say that A study in semantics and modal logic. concerning references may work in the every expression has sense, but what about University of Chicago Press, Chicago, case of extensional functors. The second IL its reference? The reference of an exten- one concerning senses is more general Frege, G. 1892. Über Sinn und sional functor is given by the first prin- than the first. It holds for all functors. Bedeutung, Zeitschrift für Philosophy und ciple of compositionality directly, but we philosophische Kritik, no. 100, pp. 25-50. In intensional logical semantics we need have to suppose that the reference of the Dunn, J. M. and Hardegree, G. M. both principles. It was clear that for main- argument is defined. However, for inten- 2001. Algebraic methods in philosophical taining the priority of the second principle logic. Oxford University Press, New sional functors the notion of reference we have to permit semantic partiality to York. cannot be defined because there is no introduce the extensional—intensional Frege, G. 1952a. Function and function which gives the reference of the differentiation. (Figure 3) Concept. In Geach and Black (1952), output from the reference of the input. pp. 21-41. Translated by P. T. Geach What happens when an argument of an form ‘Funktion und Begriff’, H.Pohle, In figure 3 a new box appears (at least extensional functor is an intensional one? Jena, 1891. in comparison with figure 2), the box of Frege, G. 1952b. On Concept and To solve this problem we have to permit logical semantics. In order to embed the Object. In Geach and Black (1952), semantic value gaps, which means that two principles of compositionality, to Acknowledgements pp. 42-55. Translated by P. T. Geach from ‘Über Begriff the sense of a functor is not a total but a und Gegenstand’, Vierteljahrsschrift für wissenschaftliche represent them formally, to differentiate I am very grateful to the Fulbright partial function. Therefore, the sense of Philosophie 16, 1892, pp. 192-206. extensionality and intensionality we have Foundation for the opportunity to work an extensional functor would be a partial Frege, G. 1952c. On Sense and Reference. In Geach and to create the whole system of logically on this paper at Indiana University, Black (1952), pp. 56-78. Translated by M. Black from ‘Über function on the possible sense of its argu- relevant semantic values. As is well-known, Bloomington. I thank my colleagues for Sinn und Bedeutung’, Zeitschrift für Philosophy und phil- ments that is not defined for the sense of osophische Kritik 100, 1892, pp. 25-50. 53 possible word semantics offers a very their valuable comments and remarks on intensional functor . Frege, G. 1980. The Foundation of Arithmetic. A logic-math- useful possibility to treat intensionality the previous versions of this paper and for ematical enquiry into the concept of number. Basil Blackwell, (and extensionality) in logical semantics. their other help. Oxford, second revised edn. Translated by J. L. Austin, from Nevertheless, there are some theoretical ‘Grundlagen der Arithmetik. Eine logisch-matematisch differences in the role of the two prin- Untersuchung Über den Begriff der Zahl’, W. Koebner, ciples of compositionality, which appear Breslau. not only in the logical-philosophical Frege, G. 1997. Begriffsschrift, a formula language of pure thought modelled on that of arithmetic. In Beaney (1997), background but in the formal system as pp. 47-78. Selections (Preface and part I). Translated by M. 53 Compare this case with the functors ‘Peter believes well. …’ and ‘Peter does not believe …’.

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Beaney from ‘Begriffsschrift, eine der arithmetischen nach- Partee, B. 1984. Compositionality. In F. Landman and F. gebildete Formelsprachen des reinen Denkens’, L. Nebert, Veltman (eds.) Varieties of Formal Semantics, pp. 281-312. The Most Frequent Value Halle, 1879. Foris, Dordrecht. Geach, P. T. and Black, M. (eds.). 1952. Translations from Pelletier, F. J. 2001. Did Frege Believe Frege’s Principle? the philosophical writing of Gottlob Frege. Basil Blackwell, Journal of Logic, Language and Information, vol. 10, pp. Method in Groundwater Oxford. 87-1114. Janssen, T. M. V. 1997. Compositionality. In J. van Benthem Rott, H. 2000. Words in Contexts: Fregean Elucidations. and A. Ter Meulen (eds.), Handbook of Logic and Language, Linguistics and Philosophy vol. 23, pp. 621-641. Modeling pp. 417-473. Elsevier, MIT Press, Amsterdam, The Neth- Salmon, N. 1994. Sense and Reference: Introduction. In M. erlands, Cambridge, MA. Harnish, (ed.) Basic Topics in the , pp. Janssen, T. M. V. 2001. Frege, Contextuality and Compo- 99-129. Prentice-Hall, Englewood Cliffs, NJ. sitionality. Journal of Logic, Language and Information, vol. Szabo, Z. G. 2000. Compositionality as Supervenience. 10, pp. 115-136. Linguistics and Philosophy, vol. 23, pp. 475-505. Péter Szûcs Harnish, M. (ed.). 1994. Basic Topics in the Philosophy of Thomason, R. H. 1999. Type Theoretic Foundations of Language. Prentice-Hall, Englewood Cliffs, NJ. Context, Part 1: Contexts as Complex Type-Theoretic ...... Hodges, W. 2001.Formal features of compositionality. Objects. In P. Bouquet, L. Serafini, P. Brézillon, M. Benere- Journal of Logic, Language and Information, vol. 10, pp. 7-28. cetti, and F. Castellani (eds.) Modeling and Using Contexts: Landman, F. and Veltman, F. (eds.). 1984. Varieties of Formal Proceedings of the Second International and Interdisciplinary University of Miskolc The University of Oklahoma Semantics. Foris, Dordrecht. Conference, CONTEXT’99, pp. 352—374. Springer-Verlag, Department of Hydrogeology School of Petroleum and Geological Engineering Mihálydeák, T. 2003. Tarskian and Husserlian models in Berlin. 3515 Miskolc-Egyetemváros Norman, OK 73019 general type-theoretical semantics. Manuscript, forthcoming. www.hidrotanszek.hu http://mpge.ou.edu/ [email protected] Adviser: Professor Faruk Civan ......

Abstract The Most Frequent Value Method (MFV) is applied to groundwater modeling as a robust and effective geostatistical method. The Most Frequent Value method is theoretically derived from the minimization of the information loss called the I-divergence. The MFV algorithm is then coupled with global optimization (Very Fast Simulated Annealing) to provide a powerful method for solving the inverse problems in groundwater modeling. The advantages and applicability of this new approach are illustrated by means of theoretical investigations and case studies. It is demonstrated that the MFV method has certain advantages over the conventional statistical methods derived from the maximum likelihood principle.

1. Introduction One of the main objectives of ground- by optimization (Lee, 1999). Frequently, water modeling is to determine the the inverse methods are used to determine properly working earth models in order the optimal parameter values of the to adequately explain the hydrogeological groundwater models. In Earth science observations. From the mathematical applications, the objective functions may point of view, such solutions can be found have multiple hills and valleys in the

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