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Halldén, Sören

On the semantic non-completeness of certain Lewis calculi

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  • Media type: E-Article
  • Title: On the semantic non-completeness of certain Lewis calculi
  • Contributor: Halldén, Sören
  • imprint: Cambridge University Press (CUP), 1951
  • Published in: Journal of Symbolic Logic
  • Language: English
  • DOI: 10.2307/2266686
  • ISSN: 0022-4812; 1943-5886
  • Origination:
  • Footnote:
  • Description: <jats:p>I intend to show that for <jats:italic>S</jats:italic>1, <jats:italic>S</jats:italic>2, and <jats:italic>S</jats:italic>3 the class of true formulas cannot coincide with the class of theorems. This is to hold irrespective of the meaning assigned to “◊”. when only “∼”, “▪”, and the variables are interpreted in the customary manner.</jats:p><jats:p>The idea of the proof is extremely simple and can be illustrated by the following argument concerning <jats:italic>S</jats:italic>3. The formula ~(◊(<jats:italic>p</jats:italic>▪~<jats:italic>p</jats:italic>)⥽▪<jats:italic>p</jats:italic>▪~<jats:italic>p</jats:italic>) ∨ (◊(<jats:italic>q</jats:italic>▪~<jats:italic>q</jats:italic>)⥽▪<jats:italic>q</jats:italic>▪~<jats:italic>q</jats:italic>) is a theorem of <jats:italic>S</jats:italic>3. Suppose now that all <jats:italic>S</jats:italic>3-theorems are true. Then either ~(◊(<jats:italic>p</jats:italic>▪~<jats:italic>p</jats:italic>)⥽▪<jats:italic>p</jats:italic>▪~<jats:italic>p</jats:italic>) is true (for every <jats:italic>p</jats:italic>) or (◊(<jats:italic>q</jats:italic>▪~<jats:italic>q</jats:italic>)⥽▪<jats:italic>q</jats:italic>▪~<jats:italic>q</jats:italic>) is true (for every <jats:italic>q</jats:italic>). But none of these formulas is an <jats:italic>S</jats:italic>3-theorem. Then some true <jats:italic>S</jats:italic>3-formula is not an <jats:italic>S</jats:italic>3-theorem. Hence, if all <jats:italic>S</jats:italic>3-theorems are true, some true <jats:italic>S</jats:italic>3-formula is not an <jats:italic>S</jats:italic>3-theorem. Then the class of <jats:italic>S</jats:italic>3-theorems cannot be identical with the class of true <jats:italic>S</jats:italic>3-formulas.</jats:p>