نشریه منطق پژوهی
سال دهم شماره 2 (پاییز و زمستان 1398)

Cohen’s method of forcing is one of the main tools in set theory for constructing models of ZFC. In this paper, we consider different methods of introducing forcing, and show that they are all equivalent. First we introduce the method of forcing using partial orders and state some of its basic properties. Then we consider the method of Boolean-valued models and show that it is equivalent to the first approach using partial orders. We do this by showing that each forcing notion can densely be embedded into a complete Boolean algebra. Then we introduce the topological approach to forcing and compare it with the partial order approach to forcing. We show that the forcing relation defined in a topological manner is the same as the forcing relation defined using partial orders and hence these two methods are essentially identical. Finally we consider the categorical approach to forcing and compare it with the method of Boolean-valued models. We show that for a given complete Boolean algebra, the category of sheaves over it is essentially the same as the Boolean-valued model constructed using that Boolean algebra.

classical logic has had some problems in explaining issues such as semantic paradoxes, vagueness problem, and quantum phenomena and have led logicians to seek non-classical logical formulations in which such problems do not arise. However, the undeniable growth of mathematics and its widespread influence in other disciplines has often led non-classical logicians to emphasize adherence to mathematical reasoning with the principles of classical logic by separating mathematical reasoning from non-mathematical. Against this approach, Williamson shows that the strategy of separating mathematics from non-mathematics and adhering to non-classical logic in non-mathematical fields disrupts the applicability of mathematics, and non-classical logicians need to think about solving this problem. In this essay, while expressing Williamson's arguments on the tension between advocating non-classical logic and the applicability of mathematics and emphasizing some of them, we show that, unlike Williamson, scientific activity based on deductive inference does not follow classical logic completely and therefor the tension sometimes subsides.

The uninorm logic UL is a fuzzy, substructural and semi-relevant logic. The Gentzen-style system for UL is obtained by removing the contraction rules and weakening from the Gentzen-style system of Godel fuzzy logic. The UL lacks "excluded middle", "positive paradox" and "negative paradox". The truth function of uninorm is a relevance weakening of the t-norm function. In this article, we introduce the new logic ULΔ. ULΔ is obtained by adding Δ to UL. ULΔ, an expansion of classical logic, is a normal semilinear modal logic; i.e. it is strongly sound and complete w.r.t. a linearly ordered algebra. And with the theorem of (p→q)∨Δ(q→p) it is distinguished from other standard systems of modal logic. Δφ is intuitively interpreted as "true that φ" or more precisely "classically true that φ". In this paper, we introduce the semi-classical logic ULΔ with four approaches, axiomatizations, hypersequent calculi, algebraic semantics and standard semantics. metatheorems we are considering include Delta deduction, strong soundness, strong standard completeness and definability of classical logic.

The structure of Aristotle’s Science is deductive. It needs the premises that cannot be ‎deduced. Therefore, knowledge of the premises of science is an important stage in ‎scientific research. Aristotle, in Analytics and Topics, suggests induction and ‎dialectic for this stage. Aristotle's commentators disagree about this. For example, ‎Bolton prefers induction and Irwin prefers dialectic. Aristotle, according to Bolton’s ‎interpretation, is an empiricist; he starts his researches from particular sense data ‎and then discovers the general ‎principles of science by induction. Irwin believes that ‎scientific researches of Aristotle, as a rationalist philosopher, begins from special ‎kind of reputable opinions (ενδοξα) and then the principles of science are known by ‎a specific dialectic. This paper shows the differences between two interpretation of ‎Bolton and Irwin; observing the issue of the methodology for recognizing the ‎principles of science. Then, we investigate some important difficulties of their ‎interpretations and, finally, suggest some ideas for a better interpretation.‎

Self-evident propositions as the foundation of Philosophy possess a significant position, but due to various reasons, they have been discussed only in margins. To explain the validity and inerrancy of these propositions, philosophers come up with different ideas. One of these Ideas was suggested by Misbah Yazdi in which he refers them to immediate(presential) knowledge in order to benefit from the unerring feature of this kind of knowledge. Among all self-evident propositions in formal logic, he only considers the inner experience and prime knowledge(Awwaliyat) as two true and eligible instances of self-evident, because among all only these two meet the criteria of being self-evident, and because He believes that sensory knowledge is the source of the mistake, thus he tries to separate self-evident propositions from sensory perception and tie them up to immediate knowledge in order to guarantee the truth of these propositions. But this theory seems to have some errors, in this article we try to explain his theory and then elaborate on the features of immediate knowledge to discover some objections to this novel but apparently incomplete theory.

Before offering his own theory of the logical predication, Sadr al-Din Dashtaki first criticizes two alternative theories; the first interprets logical predication as the attribution of something (predicate) to something (subject), and the second takes it as expressing the conceptual difference between the subject and the predicate and their essential identity. He then proposes the “conventional difference and existential identity” as the criterion of the logical predication. This theory is constituted of several elements including his peculiar analysis of the derivative words. Dashtaki discovers that both theories of simplicity and complexity of the derivative words are inconsistent with his theory of predication and thus, introduces a new analysis of the derivative words. This analysis pays way to differentiating between attribution of the root to the subject and the identity of subject and predicate. This complex net of principles helps Dashtaki to provide a plausible analysis of special sorts of propositions like existential propositions and hypothetical propositions (la Batti).

The induction is a kind of argument in logic, which, despite the degradation of its validity in comparison with the demonstrative proof, has considerable significance. One of the best method for motivating, learning, strengthening beliefs and also guidance to the general public, is the inductive method. Although the Qur'an has used arguments in specific cases for guidance purposes in various ways, the inductive method has a lot of abundance. Other methods of reasoning, such as the rational argument, are less in the Qur'an compared to the induction. The reason for this claim is the God's recommendation to look at the fate of the past generations. In fact, one of the arguments of the Holy Quran in some verses is that if you act like the previous generations, you will have the same fate. It seems that one of the God's purpose is to warn his audiences and guide them. Due to the high volume and number of subjects in the Quran about inductive method, this essay deals only with the research of the terms "Siru" (look for) and "unzoru" (look) and their derivatives. The result of this research suggests that induction is more applied than other kinds of arguments and its effect is undeniable.

This article studies the possibility of talking about ontology in Wittgenstein’s Tractatus Logico- Philosophicus. In the first step, we will consider various earlier readings of the first part of book, so called the "world" part and we will examine the theories of advocates and opponents of this idea that tractatus includes some ontological views. In the next step, by analyzing and comparing the sentences of the first part of the book, we will try to gain an explicit understanding of the concept of world and other ostensibly ontological categories. After studying two prominent concepts of objects and facts, the findings of the inquiry suggest that the ontological and realistic readings of book are wrong and what they recognize as ontological part of the book, is actually a discussion about logic. ThusThe world in Tractatus is the logical space and the territory of thought, not our actual and concrete world as it seems initially.

In Priori Analytics, II. 25, Aristotle proposes a sort of reasoning called apagpge. Scholars differ about its translation, definition, and formulation. Ross believes that it is a semi-demonstrative, semi-dialectical first-figure syllogism, with a probable conclusion derived from a more probable minor premise and apodictic major premise, which is applicable as a method of discovery. Peirce holds that it is same as Abduction, or an anticipation of it. But Farabi, without any discussion about apagoge, put the Arabic translation of epagoge, i.e. induction, in the place of apagoge, which inspire the hypothesis that apagoge in Priori Analytics, II. 25 is distortion of epagoge. On my analysis, Peirce and Farabi’s interpretations both are abductions that, assuming strangeness of Priori Analytics, II. 25, propose explanatory hypotheses apposite to economy and consistency, while Ross rejects the assumption at all. Peirce’s theory on the Aristotelian origin of abduction, with its problems and alternatives, would be questionable; and this result may be important for the Histories of Logic and science.

Dialetheism is the view that some (and not all) contradictions are true. Since in classical logic the principle of impossibility of contradiction (the Law of Non-Contradiction, i.e., LNC) is widely accepted, the challenge between dialetheism and classical logic surely occurs. In this paper we have tried to explain the main problem of dialetheim and also examined Graham Priest’s arguments against LNC and his advocating to paraconsistent logic. Accordingly, Priest’s examples of dialetheic propositions in some systems such as Law, Natual and Formal Languages and Actual world are examined and criticized. The result is that if Priest’s argument be sound dialetheism may be acceptable. Dialetheism is the view that some (and not all) contradictions are true. Since in classical logic the principle of impossibility of contradiction (the Law of Non-Contradiction, i.e., LNC) is widely accepted, the challenge between dialetheism and classical logic surely occurs. In this paper we have tried to explain the main problem of dialetheim and also examined Graham Priest’s arguments against LNC and his advocating to paraconsistent logic. Accordingly, Priest’s examples of dialetheic propositions in some systems such as Law, Natual and Formal Languages and Actual world are examined and criticized. The result is that if Priest’s argument be sound dialetheism may be acceptable.

In the Tractatus (remarks §3.02-3.031) Wittgenstein argues that an illogical world is inconceivable and it is impossible to talk about it. Ryle (1946), in contrast, thinks that one’s puzzlement with inconceivability of an illogical world and impossibility of talking about it, is resolvable. He believes that the puzzle is not real and the debate about meaningfulness of sentences about illogical world is provoked by categorical mistake. Ryle argues that the term “(il)logical” can only be applied to subjects who are capable of observing and breaching logical rules. The world and its facts, accordingly, cannot follow or breach the rules of logic. This means that the whole worry about predication of the concept “(il)logical” on the world, is groundless and absurd. The term “(il)logical” cannot be applied to the world and predication of it on the world is, indeed, an instance of committing category mistake. In this paper, after explaining Wittgenstein and Ryle’s positions in this debate, I shall try to show that Ryle’s argument against Wittgenstein is not sound. To this end, I appeal to Gilbert Harman’s distinction between logic and rationality, and implication and inference, respectively. I argue that Ryle has failed to observe this vital distinction.

The separation of ontological issues from ontology is one of the most important issues in epistemology. The rules of logic are responsible for analyzing and explaining the laws of detection Unknown conception and Unknown ratification. In traditional logic, the proposition, as the subject of the logic of assertion, is Logical Secondary intelligible and has a formal and anecdotal truth. Formal logic is responsible for providing the instruments of composition between the theorems. The theorem content, the type of its connection with outside world and its Semantic issues are outside from this domain. It seems that the modal discussions in traditional logic, in some cases, have provided the interference of matter and form and entry of some objective rules into the subjective proposition. Actuality, permanence and faculty are Different from the formal. These are related to the process of objective objects in the external world. Adding these modals to the theorem structure has caused confusion, complexity, discrepancy, and ambiguity in traditional modal logic. In this article, we have demonstrated by analytical way that traditional logic can be criticized for adding modals to the proposition structure and, and for limiting the modal logic in certain modals.