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Strenge Arithmetic: Rejection Of Modality Fallacies And Metavalues.pdf

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Preview of Strenge Arithmetic: Rejection of Modality Fallacies and Metavalues

🔗 Source: consequently.org

📊 Size: 159 KB

📄 Pages: 11 pages

⬇️ Downloads: 27

Summary

In Entailment, Anderson and Belnap modified Ackermann's strenge Implikation Π' as a logic of relevance and necessity. The kindred system R was seen as relevant but not as modal. Our systems of Peano arithmetic R# and omega arithmetic R## were based on R to avoid fallacies of relevance. However, problems arose as to which arithmetic sentences were (relevantly) true. Here, we base analogous systems on E to solve those problems. Central to motivating E is the rejection of fallacies of modality. Our slogan is, “No diamonds entail any boxes.”

We formulate the strenge Peano arithmetic E# like R#, adding appropriate forms of the Peano axioms to Ackermann’s E∀x. We extend E# to the strenge omega arithmetic E## by adding the ω-rule A(0), A(1), … ⇒ ∀xA(x). E# and E## make explicit a rejection of “fallacies of modality” implicit in R#, where already “equations” work like boxes and “unequations” like diamonds. (And no unequations relevantly imply any equations.) The R# theory of secondary formulas extends straightforwardly to our strenge arithmetics. Finally, metavaluing E## yields the strenge true arithmetic TE#. TE# treats truth-functions and quantifiers truth-functionally, settling sentences like 0=2 → 0=1 by affirming their negations.

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Description

Researchers Robert K. Meyer and Greg Restall developed a system of strenge arithmetic, E#, based on Anderson and Belnap's modified Ackermann's implication Π', to address relevance and necessity in arithmetic.

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File Format: PDF
File Size: 159 KB
Pages: 11
Language: EN
Total Downloads: 27
Last Updated: 6 days ago

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