A Defense of the Significance of Intentional Logic - Research Paper Example

?Meaning, Truth, and Identity: A Defense of The Significance of Intensional Logic Submitted by: Logic concerns itself with truth. It is known that the manner in which the truth of an identity statement, interpreted within formal logic, is explicated purely in terms of its logical structure. This kind of explication leads us to the concept of extension, wherein it is assumed that the assignment of extensions to statements is purely compositional. Hence, it follows that given the principle of compositionality, two statements are bound to have the same meaning if and only if they have the same truth-value; and it is plausible to explain the meaning of a statement with reference to its truth-value. However, if we only consider extensions, a problem would arise. For extensional logic fails to provide an adequate analysis of meanings, and an adequate explication wherein intensional contexts are needed as in the case of its treatment of identity statements. Thus, a distinction must accordingly be drawn between intension and extension. The aim of this research paper is to show that an intensional logic can be done, for it seems to be necessary for an adequate analysis regarding the problem of identity statements between singular terms or variables. In doing so, identity statements will be used to serve as the test case for showing that if meaning is what is sought, then extensions will not suffice. The researcher contends that it is indeed possible to establish an intensional logic. And not only is it possible, but is also necessary for interpreting the truth of non-trivial identity statements between singular terms or variables of the form (a = b). Literature Review The review of related literature in this study focuses on the nature of identity statements flanked by the connective “=” between names, which can be represented by the use of variables. The review examines the scope and limitations of my main inquiry, as well as related issues within the field of the philosophy of logic dealing with identity in the context of intensional and extensional logics. This survey is intended to show related studies and works done in the past regarding the logic and nature of identity. A. Frege, Gottlob. “Sense and Reference.”In The Philosophical Review, Vol. 57, No. 3 (May, 1948) : 209-230. Frege begins by questioning the concept of identity. His main claim is that “meaning” is possible between identity statements for it contains two aspects: its sense and its reference. In a nutshell, sense is its mode of presentation and reference, its mode of designation. However significant Frege is in the philosophy of language and logic, my topic will not focus on Frege’s theory of meaning and his take on the logic of identity. My reasons for such is that Frege’s concern is not that of the nature of “=”, which is the symbol used for identity statements, but rather, identity statements itself, that is, the “sense” of what is on the left and right of “=”. However, his contributions are very much useful in dealing with identity as a relation, for he does not take on the path of identity statements as a relation between names or objects, but rather introduces another concept; its sense, that is, its mode of presentation or representation. Although Frege himself did not explicitly say whether or not his take on identity is intensional or extensional, I believe that this is one instance wherein an intensional reading of identity statements is called forth, to have a better understanding of the nature of identity. B. Geach, Peter Thomas. “Quine on Classes and Properties.” In The Philosophical Review, Vol. 62, No. 3 (July 1953): 409-412. Geach’s article emphasizes on Quine’s account on how classes stand in relation to properties, and does so with the attempt to point out Quine’s mistake in identifying properties with classes. Inasmuch as my topic is concerned with both the relation of classes and properties, Geach’s take differs from my main inquiry insofar that he stresses on Quine’s formulae to express number-statements, but is similar to my concern regarding the supposed proper reading of “F’s” in any given identity statement involving the predicate “F” such as (Fa = Fb). I agree with Geach in his article in two respects, that is, (1) the need for a clarification and distinction between classes and properties and (2) his emphasis on F’s as used predicatively to express a property, rather than Quine’s reading on F’s as pertaining to that of a class. C. Wittgenstein, Ludwig. 1974. Tractatus Logico-Philosophicus. London: Routledge & Kegan Paul. In Wittgenstein’s Tractatus, he explicitly states that “Russell’s definition of ‘=’ is inadequate...” In what follows is what Wittgenstein thought was an adequate version of “=” or of the ‘identity-sign’, since this matter is of crucial significance to what my main inquiry aims to answer as well. What was not clarified in Wittgenstein’s account is that when we use the term “identical,” its sense is unambiguous. That is, when we say that A and B are the same, do we mean that A and B are numerically or qualitatively the same? For it seems that Wittgenstein reserves the use of “=” for his metalinguistic notion of identity as seen in propositions 4.241 and 4.242, which points out to say that any expression of the form x = y is just another way of saying that x and y are two names for one and the same entity. He disallows the use of identity as used in language, and that through language, we can only show it but not speak of it. Finally, Wittgenstein’s contention in proposition 5.533 seems to me to dissolute the said problem, rather than to solve it, due to ambiguity of the identity sign. Hence, the problem still remains open as to the nature of “=”, which will be my main inquiry. D. Geach, Peter Thomas. 1980. Reference and Generality, 3rd ed. Ithaca, New York: Cornell University Press. Geach, in sections 30, 34, and 110 of one of his earliest work advocates his view of relative identity as opposed to classical identity. Classical identity is most commonly understood as a relation, which obeys Leibniz’s Law. It also takes identity as an “absolute relation”, that is, to assert the meaningfulness of identity relations of the form x=y, it should not be interpreted in the form “x is the same F as y”. Several philosophers have attempted to reject the classical view of identity. Geach, one of the earliest and well-known advocate of the theory of relative identity explains the theory as follows: first, that statements of the form x=y are incomplete and therefore ill-formed. A proper identity statement has the form “x is the same F as y”. Second, states of affairs of the following sort are possible, “x is an F, y is an F, x is a G, y is a G, x is the same F as y, but x not the same G as y”. These two propositions, when taken as a conjunction, imply that first, “x’s being the same F as y does not guarantee that x is indiscernible from y”, and second, “x is the same F as y” is not analyzable as “x is an F, y is an F, and x=y”. This is Geach’s theory of relative identity. Geach’s take on identity statements serves as a foundation and as well as a starting point in finding as answer to my main inquiry regarding identity, for it seems that the classical view of identity cannot hold as true, or at least has its own flaws, given the theory of relative identity. E. Fetzer, James and Paul Humphreys. 1998. The New Theory of Reference: Kripke, Marcus, and Its Origins. Netherlands: Kluwer Academic Publishers. The traditional theory of reference implies that names express descriptive senses. This new theory introduces or rather emphasizes notions such as direct reference, rigid designators, identity across possible worlds, the necessity of identity, a posteriori identities, essentialism, arguments dealing with the failure of substitutivity in modal logic, which are of much relevance to my main inquiry. The main point of this book is to present the origins of this new theory, which is more attributed to Marcus than to Kripke; however, they share the same positions. Here are some main concepts discussed here: proper names, definite description, referential theory of meaning, the necessity of identity, rigid designators, a-posteriori necessity and synthetic a posteriori. All these concepts which both Marcus and Kripke advocates will be of much relevance in defending my stand of an intensional reading of identity statements between names, flanked by the connective “=”. However, I shall not touch on essentialism, which is discussed by both philosophers in their respective works on identity. What perhaps will differ with regards to the study presented here and with my own is that my research on identity will not only cover the intensionalist stand but will also go through the extensionalist position, which Quine elegantly established, with the attempt to find a mistake in his position. However difficult it will be to disprove the theories established by Quine in defense of extensionality, it is in my contention that given a purely extensionalist view, we tend to fail to account for certain fundamental attributes that makes a thing identical to some other thing or to itself. F. Anderson, C. Anthony. “Alonzo Church’s Contributions to Philosophy and Intensional Logic”. In The Bulletin of Symbolic Logic, Vol. 4, No. 2 (June 1998): 129-171. In section 3, Anderson discusses Church’s logic of intensional entities. Here, Church asks: what is the criterion of identity for propositions? From this question alone, one can see the relevance of Church’s work to my present research. For when Church attempts to inquire into the criterion of identity, he is simply asking: under what circumstances do identity statements express the same expressions? Synonymy is outright rejected as an answer, for it is a mere vague notion. Another proposed answer to the question is that two sentences express the same proposition when they are both true or when they are both false. Moreover, the propositions expressed by sentences A and B are “=” tell us that the material equivalence between A and B is not merely true, but true necessarily. One important concept put forth in this article is the notion of “intensional isomorphism” as a criterion of identity. This was first proposed by Carnap saying that two entities are intensionally isomorphic if they are created in the same way from designators with the same intensions. Church criticizes Carnap using the principle of tolerance – this tells us that we can introduce two primitive symbols A, B as constants that are intensionally isomorphic, yet one can easily believe a sentence involving A without believing a sentence involving B. Thus, Church proposes an alternative method – synonymous isomorphism. Research Questions What is the nature of identity? In order to analyze this question is to understand the nature of an identity statement from a logical point of view. What is the relation between intensional and extensional logic when dealing with identity statements? Is it possible for us to reduce an extensional reading of identity to an intensional reading of identity and vise-versa? Why is there a need to clarify the concept of identity within the parameters of intensional and extensional logic? Finally, assuming that there is a need for such clarification, which of these two interpretations can really account for the nature of the logic of identity? Methodological Approach The main concern of this research is the tenability of maintaining a purely extensional logic. In arguing that a pure extensional program will not suffice, I shall first provide a background by which the readers or audience would be able to understand the problem at hand. I intend to begin my research paper with a story, in order to remove any personal biases towards the subject matter. Also, a popular logic puzzle that captures the problem of this research topic may be included here. Then I provide my task as researcher, as well as my goals for this research. Having said my main objective, I shall provide a comparative analysis of two main ideas: intension and extension. In relation to that, I shall then define my use of intension and extension with regards to intensional logic and extensional logic. Next, I shall enumerate possible objections to my position, as well as a logical analysis of those objections. Here, I shall employ formal techniques in symbolic logic when assessing various examples of arguments. Having laid down a foundation for this research topic, I shall briefly present an historical analysis of intensional logic, starting with Gottlob Frege. Here, an intersection between closely related fields in philosophy will be seen between philosophy of logic, linguistic philosophy, contemporary metaphysics, philosophy of mind and epistemology. From there, I will emphasize why my concern is only that of the first, that is, the philosophy of logic. A main technique that will be used throughout this research paper is argumentation based on logical and linguistic analysis. The reason why this method is highlighted is due primarily because of the nature of the said research topic, that is to say, logical. In order for me to defend the status of intensional logic, I will have to show where the arguments of a purely extensional logic breaks down. This will entail dealing with logical fallacies, forms and structure. Hence, what will count as “data” herein are the arguments presented by the philosophers. Also, logical proof’s of validity, rules of inference and replacement, symbolic representation of arguments, basic syntax and semantics of first order logic, and modal logic will be used. Materials, references and other necessary sources that will be used for this research are primarily, books, maximizing the present collection of the researcher. Also, several journal articles will needed, all of which, access is provided. Academic sources from the Internet are carefully selected. Online correspondence with contemporary experts within the field may also be necessary for the researcher to have a better grasp on the scope of the selected research topic. The intended design of this research paper is as follows: 1. Introduction 2. Intensional Logic a. Development of intensional logic Gottlob Frege’s Theory of Sense and Reference Rudolf Carnap’s Theory of Intension and Extension Hybrid between Intensional and Extensional Logic: Saul Kripke 3. Extensional Logic a. W.V. Quine’s Pure Extensional Logic Arguments of Quine 4. Failure of a Purely extensional logic a. Objections and Replies Against Quine’s arguments b. Defense of the necessity of intensional logic with regards to identity statements in logic 5. Final Assessment a. Conclusion Perhaps, a difficulty I will encounter may be the interpretation of different symbolic notations, as used by different logicians and philosophers. Thus, I shall specify which among these existing notations will be used throughout the research paper. Also, a major hindrance, from the reader’s point of view, is one’s subjective opinions about logic. It is in this regard that I shall request that the reader temporarily suspend ones personal beliefs and opinions regarding the said research topic. For, if there is one thing that I would like to achieve, that would be an appreciation of the significance of logic. Despite the complexity of logic, I shall do my best to interpret the said topic in the simplest possible manner. In the final section of the paper, I shall go beyond the confines of logic by showing why it is important to account for both intension and extension, as opposed to a purely extensional reading. What is the significance of explicating the nature of identity? One major concept herein is ontological commitment. For, we are inevitably committed to our own belief systems, that it seems ridiculous to question fundamental concepts such as the logic of identity. Nevertheless, by providing an analysis of intensional logic, hopefully, the readers would change the way they perceive logic and appreciate its value. Finally, before I end my research paper, I shall return to the story that I began with in the introduction. Upon ending the story, I shall have ended my research as well. References Carnap, Rudolf. 1947. Meaning and Necessity. Chicago: University of Chicago Press. Copi, Irving M. and Carl Cohen. 2009. Introduction to Logic. New Jersey: Pearson Prentice Hall. Frege, Gottlob. “Sense and Reference”. In The Philosophical Review, Vol. 57, No. 3 (May, 1948): 209-230. Hughes, Christopher. 2004. Kripke: names, necessity and identity. New York: Oxford University Press. Kleane, William and Martha Kneale. 1962. The development of logic. New York: Oxford University Press. Kripke, Saul. 1980. Naming and Necessity. Harvard: Harvard University Press. Pappas, Theoni. 1989. The Joy of Mathematics. San Carlos: Wide World Publishing. Parsons, Charles. Intensional Logic in Extensional Language. In The Journal of Symbolic Logic, vol. 47, no. 2 (Jun., 1982) : 289-328. Peregrin, Jaroslav. 2006. Extensional vs. Intensional Logic. In The Handbook of Philosophy of Science, vol. 5: philosophy of logic. Netherlands: Elsevier B.V. Quine, Willard Van Orman. 1982. Methods of Logic. Harvard: Harvard University Press. Quine, Willard Van Orman. 1986. Philosophy of Logic: London: Harvard University Press. Quine, Willard Van Orman. 1962. Mathematical Logic. New York: Harper & Row. Van Benthem, Johan. 2009. Modal Logic for Open Minds. Stanford: CSLI Publications. Wittgenstein, Ludwig. 2001. Tractatus Logico Philosophicus. New York: Routledge. Wreen, Michael. Proper names and the necessity of identity statements. In Synthese, vol. 114, no. 2 (1998) : 319-335 Read More