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Blog / 2007-10-26b-SemanticTableaux

Semantic Tableaux

I wasn't very happy with the explanation of semantic tableaux given in Graham Priest's Non-Classical Logic. Here's a preferred one:

  • A semantic tableau is a tree where each node contains a proposition.
  • Put a (compound) proposition at the root node of the tree. You are about to test the truth of this proposition.
  • Each node of the tree branches into the minimum number of nodes for at least one of the nodes to hold, given that its parent node holds.
  • Whenever you find an obvious contradiction on a branch (ie, the branch requires A and not-A to simultaneously hold) close the branch.
  • Keep applying this procedure until you've closed all the branches, or you're down to atomic propositions (ie, you've completed the branch). If all the branches close, the root proposition is false, if none of them do then it's a tautology, if some of them are closed and some open then it's conditionally true.

Each branch of the tree represents an intepretation of the root proposition. An interpretation is a way of assigning truth values to the atoms of the root proposition.

Closed branches are those branches for which the assigned truth values of the atoms result in the root proposition being false.

Completed, open branches are those branches for which the assigned truth values of the atoms result in the root proposition being true.

Uncompleted branches need to be completed, because they could close further down, and until you're finished you don't know if they do or not.

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Page last modified on October 26, 2007, at 12:14 PM