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Proof theory for modal logic
Proof theory for modal logic

... sign). For S5 also negations of modal formulas are allowed among the assumptions. The necessitation rule is regarded as the introduction rule for 2, whereas the elimination rule is simply 2A/A. Although “natural,” the system is not normalizable, as there are non-eliminable detours. Prawitz introduce ...
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... construction of the usual cumulative hierarchy of sets generated by a collection of atoms can be carried out within a locally small complete topos E (in particular, any Grothendieck topos). This leads to a topos E*—in a certain sense a well-founded part of E—which is a model of IZF with atoms. By st ...
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pdf - at www.arxiv.org.

Frege, Boolos, and Logical Objects
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... Frege’s plan in the Grundlagen and Grundgesetze.4 Boolos notes that the explicit assertion of the existence of numbers embodied by Numbers is a way of making clear the commitment implicit in the use of the definite article in ‘the number of F s’.5 In his papers of [1986] and [1993], Boolos returned t ...
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A Computationally-Discovered Simplification of the Ontological

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A Computationally-Discovered Simplification of the Ontological

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Axiom

An axiom or postulate is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.' As used in modern logic, an axiom is simply a premise or starting point for reasoning. What it means for an axiom, or any mathematical statement, to be ""true"" is a central question in the philosophy of mathematics, with modern mathematicians holding a multitude of different opinions.In mathematics, the term axiom is used in two related but distinguishable senses: ""logical axioms"" and ""non-logical axioms"". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A and B) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). When used in the latter sense, ""axiom,"" ""postulate"", and ""assumption"" may be used interchangeably. In general, a non-logical axiom is not a self-evident truth, but rather a formal logical expression used in deduction to build a mathematical theory. As modern mathematics admits multiple, equally ""true"" systems of logic, precisely the same thing must be said for logical axioms - they both define and are specific to the particular system of logic that is being invoked. To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms). There are typically multiple ways to axiomatize a given mathematical domain.In both senses, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Within the system they define, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow otherwise they would be classified as theorems. However, an axiom in one system may be a theorem in another, and vice versa.
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