Adequate systems of connectives.html (1522B)
1 <?xml version="1.0" encoding="UTF-8"?> 2 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> 3 <html><head><link rel="stylesheet" type="text/css" href="sitewide.css"><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/><meta name="exporter-version" content="Evernote Mac 7.0.3 (456341)"/><meta name="keywords" content="logic"/><meta name="altitude" content="-0.2514409720897675"/><meta name="author" content="Alex Balgavy"/><meta name="created" content="2018-02-20 14:47:19 +0000"/><meta name="latitude" content="52.33331960345897"/><meta name="longitude" content="4.865656661188186"/><meta name="source" content="desktop.mac"/><meta name="updated" content="2018-02-20 15:15:30 +0000"/><title>Adequate systems of connectives</title></head><body><div>a system C of connectives is adequate if every truth table can be expressed as formula with connectives C</div><div><br/></div><div>example:</div><div>express (p ∧ ¬ q) ∨ (¬ p ∧ q) in the system {¬, ∧}</div><div><ul><li>use: ϕ ∨ Ψ <span style="font-size: 14px;">≡ ¬ (¬ ϕ ∧ ¬ Ψ)</span></li><li>rewrite like hell</li><li>result: ¬ (¬ p ∨ ¬¬ q) ∨ ¬ (¬ ¬ p ∨ ¬ q)</li></ul><div><br/></div><div>Sheffer stroke is an adequate system — {|}</div></div><div>ϕ | Ψ means "not both ϕ and Ψ” (ϕ NAND Ψ)</div><div>ϕ | Ψ ≡ ¬ (ϕ ∧ Ψ)</div><div><br/></div><div>adequate systems are: {|}, {¬, ∧}, {¬, ∨}, {¬, ∧, ∨}, etc.</div></body></html>