Aristotle, General Subjects: Aesthetics | Aristotle, General Subjects: Metaphysics | Aristotle, General Topics: Rhetoric | Aristotle, Special Subjects: Mathematics | Aristotle, Special Topics: On Coherence | Chrysippus | Diodorus Cronus | Future quotas | Logic: Old | Logic: relevance| Megaric School| Place of the opposition| Modern modal logic treats necessity and possibility as interdefinable: “necessarily P” equals “maybe not P” and “maybe P” equals “not necessarily not P.” Aristotle gives the same equivalences in the interpretation of On. In Prior Analytics, however, he distinguishes two ideas of possibilities. On the first, which he adopts as his preferred term, “possibly P” is synonymous with “not necessarily P and not necessarily not P”. He then recognizes an alternative definition of possibility according to modern equivalence, which plays only a subordinate role in his system. The non-logical terminology of language consists of its individual constants and predicate letters. The symbol “(=)” for identity is not an illogical symbol. By considering identity as logic, we deal with it explicitly in the deductive system and in the semantics of model theory. Most authors do the same, but there is some controversy on the subject (Quine (1986, chapter 5)). If (K) is a set of constants and predicate letters, then we are laying the groundwork for a (LKe) language based on this set of non-logical terminology. It can be called the first-order language with identity on (K).

A similar language that does not have the identity symbol (or that considers identity illogical) can be called (mathcal{L}1K), the first-order language without an identity on (K). There are three fundamental laws of logic. Suppose P is an indicative sentence, say, “It`s raining. From mid to late 1800, these expressions were used to refer to the Boolean algebra theorems on classes: (ID) each class contains itself; (NC) each class is such that its intersection (“product”) with its own complement is the null class; (ME) Each class is such that its union (“sum”) with its own complement is the universal class. More recently, the last two of the three expressions have been used in the context of classical propositional logic and so-called protothetic or quantified propositional logic; in both cases, the law of non-contradiction involves the negation of the conjunction (“and”) of something with its own negation, ¬(A∧¬A), and the law of the excluded middle includes the disjunction (“or”) of something with its own negation, A∨¬A. In the case of propositional logic, the “something” is a schematic letter that serves as a placeholder, while in the case of protothetic logic, the “something” is a real variable. The terms “law of non-contradiction” and “law of the excluded middle” are also used for the semantic principles of model theory in relation to propositions and interpretations: (NC) under no interpretation is a given sentence both true and false, (EM) under no interpretation is a given sentence true or false. This raises questions about the philosophical relevance of the different mathematical aspects of logic. How are deductibility and validity as properties of formal languages – strings on a fixed alphabet – related to correct thinking? What do the mathematical results reported below have to do with the original philosophical questions about valid argumentation? This is an example of the philosophical problem of explaining how mathematics is applied to non-mathematical reality. Im 18. In the nineteenth century, Immanuel Kant argued that logic should be understood as the science of judgment, so that the valid conclusions of logic derive from the structural characteristics of judgments, although he always claimed that Aristotle essentially said everything there was to say about logic as a discipline.

This question of how such a priori knowledge can exist leads Russell to an examination of Immanuel Kant`s philosophy, which he rejects after careful consideration as follows: Aristotle`s proofs can be divided into two categories, based on a distinction he makes between “perfect” or “complete” deductions (teleios) and “imperfect” or “incomplete” deductions (atelês). A deduction is perfect if it “does not need an external term to show the necessary result” (24b23-24), and it is imperfect if it “requires one or more additional terms necessary by the assumption but not assumed by the premises” (24b24-25). The exact interpretation of this distinction is disputed, but it is clear that Aristotle does not see perfect conclusions as requiring proof in a sense. For imperfect conclusions, Aristotle gives proofs that always depend on perfect conclusions. Therefore, with some reservations, we could compare perfect derivatives with the axioms or primitive rules of a deductive system.