64 2.3.3 Logic as a Tool Unsigned version of the system of Semantic Tableaux Unlike the signed version presented above, in the unsigned version of Semantic Tableaux all formulae appearing on the branches are assumed to be true; there is therefore no need to assign truth values For that purpose, every signed formula A : T is replaced simply by A, whereas A : F is transformed to ¬A The rules for the unsigned version can be produced by a simple modification of the rules for the signed version, by using the basic equivalences used for importing the negation inside the other propositional connectives They are listed in Table 2.1 Non-branching rules (α-rules) Branching rules (β-rules) A ∧B (∧) A, B ¬ (A ∧ B ) (¬∧) ¬A ¬ (A ∨ B ) (¬∨) (∨) ¬ A, ¬ B A ∨B A ¬ (A → B ) (¬ →) (→) A, ¬ B (¬¬) ¬¬ A A ¬B B A → B ¬A B A ↔ B (↔) A, B ¬A, ¬B ¬ (A ↔ B ) (¬ ↔) A, ¬B Figure 2.1 ¬A, B Rules for Semantic Tableaux in propositional logic: unsigned version A branch of the unsigned tableau is closed if a “complementary pair” of formulae A, ¬A appears on it; otherwise, it remains open The rest is the same as for the signed version The two versions are essentially equivalent and it is only a matter of preference or convenience when deciding which to use References for further reading D’Agostino et al (1999) is a comprehensive handbook on tableaux methods, not only in classical logic Fore more details on theory and examples of derivations in Semantic Tableaux see Nerode and Shore (1993), Jeffrey (1994, who used the expression “analytic trees”), Smullyan (1995, one of the first versions of Semantic Tableaux), Fitting (1996), Smith (2003, who called them “trees”), Ben-Ari (2012), and van Benthem et al (2014)