Semantic Tableaux

We know that the theorems and the logically valid formulae of 1st order logic are identical - by soundness and completeness. We have no very useful method of directly discovering (trying to discover) whether a wff is a theorem. Exhaustion is not practical. We do have a reasonable method of trying to test for logical validity: the method of semantic tableaux or truth trees.

Note that because of the undecidability of 1st order logic we know that there is no effective procedure, and that truth trees aren't effective.

Motivation behind the construction of truth tres based on a set of formulae is to try to show that these formulae are not satisfiable, or to show how they may be satisfied.

Think of the construction of a tree from a formula A as criticizing a story told by A. Trees develop branches as we identify alternative ways that the story can be true. When a branch tells an unbelievable story we chop it off (close it.) Any open branch then is a believable version and we can construct a model for A based on it.