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THE EQUIVALENT TRANSFORMATION BETWEEN NON-TRUTH-FUNCTION AND TRUTH-FUNCTION
XIAO-LONG WAN* and MING-YI CHEN†
*Prof. & Chair in Unit of Logic and Philosophy of Science and Tech.,
Department of Philosophy, Huazhong University of Science and Technology,
P. O. Box 430074, Wuhan, Hubei, P. R. of China.
†PhD Candidate in Logic and Philosophy of Science,
Department of Philosophy, Huazhong University of Science and Technology,
Unary Operator Theory (UOT) systematically researches unary operators by analyzing the unification between the formal universality of mathematical logic and the adaptability of the natural language. At first, it focuses on Special Theory of Relativity of Function (STRF) which is only useful for equivalent transformation between the non-truth-valued-function formed by non-functional unary operators and truth-valued-function formed by functional binary connectives in two-truth-valued classical propositional logic. Moreover, according to the STRF, there are new interpretations about problems both in philosophical logic and in philosophy of science which could not be solved only by the logic of truth-function in the past.
Keywords: Equivalence Non-Truth-Function Truth-Function STRF.
1. Introduction
The modern mathematical logic starting from Frege and Russell, which develops a new technology in the field of logic, has become a common theoretical foundation for information science and modern analytic philosophy. However, the classical part of modern logic seems not very well to deal with some complex problems, such as “the failure of equivalent substitution principle” which often appears in natural language, traditional philosophy, quantum theory and computer language. For example:
“I am very hungry but I have serious liver disease. I eat food and drink alcohol. I ought to eat food, and I ought not to drink alcohol”.
The Logical semantics in the last several sentences is: “I eat food” is true; “I drink alcohol” is true. “I ought to eat food” is true; “I ought to drink alcohol” is false.
Therefore, there is a formal problem (T): the two compound sentences are constructed by two component sentences with the same connective “ought to”. The two component sentences have the same truth-value, but their compound sentences have different truth-value.
Such problem (T) labeled as “the failure of equivalent substitution principle” had puzzled the founder of modern logic, Frege, and his followers, for a long time. Frege, and most logicians later thought that we had to use intensional logic to resolve it. It is well-known that modal logic is a typical intensional logic.
The modern modal logic, as one of the most typical and mature kinds of nonclassical logics, not only has a large and complex systems in foundation, but also is widely applied in computer science, modern analytic philosophy, the interpretation of quantum mechanics and even many social sciences and humanities. It seems that modal logic can solve the problem (T). Moreover, Quine’s questioning the validity of early modern modal logic stimulated the development of modal logical theories, such as the addition of the necessitation rule in syntax and the invention of possible world semantics. However, the possible world semantics does not make a modal formula correspond to a first-order formula one by one, and vice versa. Whats more, morally generalized modal logic processing will produce logical paradoxes which are difficult to solve. The friends in information science, economics, and mathematics tell me, they generally use statistical methods to solve the problem (T), because they worry about the non-total unity between modal logic and classical logic in information science and artificial intelligence.
Furthermore, both the intensional logic and extensional logic are formal logics, so any intensional logic is a formal reasoning of intensions. Thus, the further clear technology distinguishing extension logic and intensional logic is that the formal is about truth-function, while the later such as modal propositional logic is about non-truth-function. Equivalent substitution principle holds with truth-function but no longer holds with non truth-function. It is seemly why modern classical logic (or called standard logic) up to now can not effectively deal with the problem in the previous section of this article, for “ought to” is obviously not a truth-valued functional connective. That is to say, classical logic is just a truth-function logic.
A few papers are dedicated to systematically investigate into the nature of the non-truth-functions or of the non-truth-functional connectives (Quine, W. V. O. 1982, p. 8; Schnieder, B. 2008, p. 64; Hill, D. J. and Mcleod, S. K. 2010, p. 629). Even though they had studied in detail the differences between non-truth-function and truth-function or differences between non-truth-functional connective and truth-functional connective (Marcos, J. 2008, p. 215; Rescher, N. 1961, p. 1–10, Stephen, P. 2005, p. 93–105) no one recognizes the equivalent transformation between them, nor do more refer to give such transformation a tightly clear definition. Moreover, can the classical (propositional) logic truly not handle the problems with “non-truth-function operates ”?
There are still some philosophers faithfully defend only effective position of classical logic and refused to acknowledge the non-classical logic correction to the classical logic. In this respect, University of Oxford philosopher Tim of West Williamson (Timothy Williamson) is the most representative of the defense, He said:“the classical logic is normally used in logic, mathematics and science, in these areas it has proven to be extremely successful.’if, bad workman blames his tools’, then the classical logic and semantics is the philosophical craftsmen poor tool”1 There are many bottom-up ways to explore the relationship between classical logic and non-classical logic, but the characteristic of this article is along a top-down approach. We intend to pursue a unified method or comprehensive theory to understand the relationship between the non-classical logics and classical logics, on the basis of formal university of classical propositional logic and its proper compliance with natural language. Since classical propositional logic is the most mature and the simple modern logic, special modal operators are usually viewed as non-truth-functional and unary operators, so the axiomatic systems of the most non-classical logics are a set of axioms of truth-functions in classical logic plus the additional axioms of non-truth-functions formed by modal operators or other non-truth-functional connectives (Kripke, A. S. 1965, p. 208; Ming Xu 2008, p. 492), we concentrate on the relations between truth-valued function connective in classical propositional logic and non-truth-valued function operate in modal propositional logic with two truth-valued in this article. The Special Theory of Relativity of Function (STRF) is invented to deal with this relationship, and it is considered as the first step of more general one, “Unary Operator Theory”. Perhaps, classical logics and non-classical logics are to some extent unified in very basic respect. According to STRF, at least, we can further paraphrase modal propositional logic (and even some many-valued logics) as an “fragment”, or even another sight of classical logic.
2. STRF
The summary of aforementioned section is: classical logic vs. non — classicallogic → extension logic vs. intension logic → truth — function vs. non — truth — function → classical propositional logic vs. modal propositional logic. STRF studies the equivalent transformation between the two-valued non-truth-functions formed by non-functional unary operators (connectives) and truth-functions formed by functional binary connectives in classic propositional logic only.
2.1. Preliminary consideration
Generally speaking, a function is a corresponding relation of two variables (or of two sets or of two formulas). For a variable x and dependent variable y both in their valued-domain, if x has a value corresponding y having a and only a value, then this y is a function of x. Truth-function is the function in which the domain of the variable and the dependent variable are all the truth valued-domain only. Non-truth-function is not that the function in which the domain of the variable and of the dependent variable is all the truth valued-domain only. It includes the non-truth-valued function and the truth-value of non-function. That is to say, we study the non-truth-function is actually two-truth-valued non-function. However, to have the philosophy of logic corresponding to the text, we still call it non-truth-valued function.
In Table 1, D2 = p ∨ p1 is composed of p and p1, these two variables together form a truth-valued function, but D2 is not a truth-valued function by variable p only (when p takes a definite truth-value 0, p ∨ p1 is not only a corresponding definite truth-value). Therefore, D2 is a non-truth-valued function of p. This shows that at least some of the non-truth-function is actually equivalent to a truth-function in another sight. Among 16 truth-functions in Table 1, though they are all the truth-f...
