Propositional logic.html (2453B)
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="created" content="2018-02-06 14:46:07 +0000"/><meta name="updated" content="2018-03-27 15:54:45 +0000"/><title>Propositional logic</title></head><body><div><div>Declarative sentence — true or false</div></div><div>Argument abstraction — If p and not g, then r. Not r. p. Therefore q.</div><div>Argument formalisation: [ ( (p ∧ ¬ q) ➝ r ) ∧ (¬ r ∧ p) ) ] ➝ q</div><div>Symbols: ∧ (and), ∨ (or), ⨁ (xor), ¬ (not), ➝ (implication)</div><div><br/></div><div>Constructing formulae</div><div><ul><li>every propositional variable is a formula</li><li>so is its negation</li><li>so are constructors wit operators</li></ul></div><div><br/></div><div>Symbol priority: negation, then conjunction/disjunction, then implication</div><div><br/></div><div>Types of proposition:</div><div><ul><li>Tautology (p ∨ ¬ p) is always true</li><li>Contradiction (p ∧ ¬ p) is always false</li><li>Contingency is neither a tautology nor a contradiction</li></ul></div><div><br/></div><div>Rules of propositional logic</div><div><ul><li>Implication ϕ ➝ Ψ is </li><ul><li>false if ϕ true and Ψ false</li><li>true otherwise</li></ul><li>Bi-implication ϕ ⟷ Ψ (“ϕ if and only if Ψ”) is </li><ul><li>true if ϕ and Ψ have same truth value</li><li>false otherwise</li></ul></ul></div><div><ul><li>conjunction/disjunction (with conjunction as example)</li><ul><li>p ∧ q ⟷ q ∧ p</li><li>p ∧ (q ∧ r) ⟷ (p ∧ q) ∧ r</li><li>p ∧ (q ∨ r) ⟷ (p ∧ q) ∨ (p ∧ r)</li><li>p ∧ p ⟷ p</li><li>p ∧ (p ∨ q) ⟷ p</li></ul><li>negation</li><ul><li>p ∧ ¬ p ⟷ F</li><li>p ∨ ¬ p ⟷ T</li><li>¬ ¬ p ⟷ p</li></ul><li>demorgan</li><ul><li>¬ p ∧ ¬ q ⟷ ¬ (p ∨ q)</li><li>¬ (p ∧ q) ⟷ ¬ p ∨ ¬ q</li></ul><li>identity</li><ul><li>disjunction</li><ul><li>p ∨ T ⟷ T</li><li>p ∨ F ⟷ p</li></ul><li>conjunction</li><ul><li>p ∧ T ⟷ p</li><li>p ∧ F ⟷ F</li></ul></ul><li>implication</li><ul><li>p ➝ q == ¬ p ∨ q</li></ul></ul></div><div><br/></div></body></html>