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The square of opposition, under this Boolean set of assumptions, is often called the modern Square of opposition. In the modern square of opposition, A and O claims are contradictories, as are E and I, but all other forms of opposition cease to hold; there are no contraries, subcontraries, subalternations, and superalternations. Thus, from a ...
Classical logic is a 19th and 20th-century innovation. The name does not refer to classical antiquity, which used the term logic of Aristotle. Classical logic was the reconciliation of Aristotle's logic, which dominated most of the last 2000 years, with the propositional Stoic logic. The two were sometimes seen as irreconcilable.
Classical propositional calculus is the standard propositional logic. Its intended semantics is bivalent and its main property is that it is strongly complete, otherwise said that whenever a formula semantically follows from a set of premises, it also follows from that set syntactically. Many different equivalent complete axiom systems have ...
In classical logic, particularly in propositional and first-order logic, a proposition is a contradiction if and only if. Since for contradictory φ {\displaystyle \varphi } it is true that ⊢ φ → ψ {\displaystyle \vdash \varphi \rightarrow \psi } for all ψ {\displaystyle \psi } (because ⊥ ⊢ ψ {\displaystyle \bot \vdash \psi } ), one ...
Subalternation [1] [2] is an immediate inference which is only made between A (All S are P) and I (Some S are P) categorical propositions and between E (No S are P or originally, No S is P) and O (Some S are not P or originally, Not every S is P) categorical propositions of the traditional square of opposition and the original square of opposition. [3]
The logical hexagon extends the square of opposition to six statements. In philosophical logic, the logical hexagon (also called the hexagon of opposition) is a conceptual model of the relationships between the truth values of six statements. It is an extension of Aristotle's square of opposition.
In modern logic only the contradictories in the square of opposition apply, because domains may be empty. (Black areas are empty, red areas are nonempty.) In first-order logic, the empty domain is the empty set having no members. In traditional and classical logic domains are restrictedly non-empty in order that certain theorems be valid.
In the system of Aristotelian logic, the triangle of opposition is a diagram [which?] representing the different ways in which each of the three propositions of the system is logically related ('opposed') to each of the others. The system is also useful in the analysis of syllogistic logic, serving to identify the allowed logical conversions ...