MANTEQ-PAZHUHI (LOGICAL STUDIES)
Year:2015 | Volume:6 | Issue:1 (11)|Papers Count:6

In 1899, David Hilbert offers an articulated axiomatic system for Euclidean geometry and, demonstrating conditionally the meta-theorems of compatibility and independence for this system, proposes a solution to one of the enduring problems of mathematics (known as the problem of parallel lines). Gottlob Frege, the founder of new formal logic, fundamentally disagreed with Hilbert’s formalistic approach and his proofs for the meta-theorems of compatibility and independence. The reasons for the opposition show that Frege’s view on formality of logic and meta-theorems of compatibility and independence is very different from today's point of view. In this paper, after briefly discussing Hilbert’s method in demonstrating meta-theorems of compatibility and independence, and also the main Frege’s objections toward it, I will indicate to Frege’s own method dealing with these issues, and then discuss why eventually mathematicians and logicians, following Hilbert, ignored Frege’s remarks and modern logic, proposing a model theory, stepped on a road which was for Frege a wrong way.

In this article, at first, I have explained correspondence theory of truth and I have dealt with its problems. Then I have explained the truth maker principle in the light of correspondence theory of truth. At the end, I have discussed how this theory can support negative proposition.

According to Smith, Organon for Aristotle is a special means to demonstrate that first philosophy is possible. For him, it’s incorrect to think that the aim of Organon is the introduction of Aristotle’s methodological theories. In Posterior Analytics, Aristotle by a demonstration tries to show that an undemonstrative knowledge is necessary and sufficient condition for demonstrative knowledge (episteme). The aim of undemonstrative knowledge is the aim of first philosophy, that is, coming to the first principles of any science. Smith considers the demonstration in question depends on other Aristotle’s logical theories in Organon involving theories of definition, predication, dialectic, categories, and deduction. Furthermore, in the last section of Posterior Analytic, Aristotle expressing a process called Nous shows that coming to the undemonstrative knowledge is possible.

In traditional logic, definitions are divided into essence oriented and accident oriented definitions, which the first one is considered inaccessible for some reasons and thus, the second is seen as the only feasible kind of definition. However explaining the nature and the conditions of the accident oriented definition has created some problems. As a result, accident oriented definition ended up with the fate of essence oriented definition and lost its logical and epistemological value. It seems that one might in the Avicenna's logic tradition reconstructs accident oriented definition, which of course requires reconsidering ‘definition’ and its conditions.

According to the common interpretation, Aristotle, in the fourth chapter of “De Interpretatione”, defines statement based on truth and falsehood. But in the end of the fifth chapter, he offers another ‘account’ of ‘simple’ statement. In this article, I will argue, that the account is in fact the definition of statement without qualification based on belonging and not belonging as elements of the realm of reality. Accordingly, the interpretations of commentators about the last part of the fourth and the fifth chapter will be presented. Then it is discussed that the definition of statement based on truth and falsehood is incorrect and has some incompatibilities with the Aristotle's text. After that, my interpretation of Aristotle's text will be explained and it is indicated that dividing concepts of signifiers is defined in a different manner. Finally, the chosen definition of statement and its similarities with Porphyry's interpretation are explained.

Gentzen divides rules of logical system into ‘operational rules’ and ‘structural rules’. By operational rules she means the rules of introduction and elimination of a logical constant. Structural rules represent the fundamental (structural) characteristics of an argument in such a way that any change in them causes changing in the whole system. In his works, Gentzen mentions that the meaning of logical constants can be achieved only through operational rules. This point is the infrastructure of inferentialism approach on meaningfulness of logical constants. Christopher Peacocke criticizes the basis of inferentialism approach. He believes that all structural and operational rules should be considered as the definition of logical constants. In response to this claim, Ian Hacking argued that accepting Peacock’s idea, will lead to a lack of conservation. Lack of conservation causes system incompatibility. In this paper, after careful examination of structural rules and expressing its difference with operational rules, Hacking argument will be evaluated and criticized and finally a solution to the problem will be presented.