Understanding First-order Logic 115 a special kind of a two-player game, called (formula) evaluation game9 These games go back to Lorenzen’s work in the 1950s (if not much earlier), but were first introduced explicitly for first-order logic by Henkin and Hintikka The two players are called the Verifier and the Falsifier10 The game is played on a given first-order structure S, containing a variable assignment v , and a formula A, the truth of which is to be evaluated in the structure S for the assignment v As suggested by the names of the players, the objective of Verifier is to defend and demonstrate the claim that S , v |= A, while the objective of Falsifier is to attack and refute that claim The game goes in rounds and in each round exactly one of the players, depending on the current “game configuration”, has to make a move according to rules specified below until the game ends The current game configuration (S , w, C ) consists of the structure S, an assignment w in S, and a formula C (the truth of which is to be evaluated in S for the assignment w) The initial configuration is (S , v, A) We identify every such game with its initial configuration At every round, the player to make a move as well as the possible move are determined by the main connective of the formula in the current configuration (S , w, C ), by rules that closely resemble the truth definitions for the logical connectives The rules are as follows • If the formula C is atomic, the game ends If S , w |= C then Verifier wins, otherwise Falsifier wins • If C = ¬B then Verifier and Falsifier swap their roles and the game continues with the configuration (S , w, B ) Swapping the roles means that Verifier wins the game (S , w, ¬B ) iff Falsifier wins the game (S , w, B ), and Falsifier wins the game (S , w, ¬B ) iff Verifier wins the game (S , w, B ) Intuition: verifying ¬B is equivalent to falsifying B and vice versa • If C = C1 ∧ C2 then Falsifier chooses i ∈ {1, 2} and the game continues with the configuration (S , w, Ci ) Intuition: for Verifier to defend the truth of C1 ∧ C2 he should be able to defend the truth of any of the two conjuncts, so it is up to Falsifier to question the truth of either of them • If C = C1 ∨ C2 then Verifier chooses i ∈ {1, 2} and the game continues with the configuration (S , w, Ci ) Intuition: for Verifier to defend the truth of C1 ∨ C2 , it is sufficient to be able to defend the truth of at least one of the two disjuncts; Verifier can choose which one • If C = C1 → C2 then Verifier chooses i ∈ {1, 2} and, depending on that choice, the game continues with the configuration (S , w, ¬C1 ) or (S , w, C2 ) Intuition: C1 C2 ơC1 C2 ã If C = ∃xB then Verifier chooses an element a ∈ S and the game continues with the configuration (S , w[x := a], B ) Intuition: by the truth definition of ∃xB , verifying that S , w |= ∃xB amounts to verifying that S , w[x := a] |= B for some suitable element a ∈ S Also known as model checking game Also known by various other names, for example Proponent and Opponent, Eloise and Abelard, Eve and Adam For the sake of convenience, and without prejudice, here we will assume that both players are male 10