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We start with the modus ponens, a simple, intuitive rule of inference, which, from the validity of A and A ⇒ B, allows the derivation of B. We write this formally as A, A⇒B . B This notation means that we can derive the formula(s) below the line from the comma-separated formulas above the line. Modus ponens as a rule by itself, while sound, is not complete. If we add additional rules we can create a complete calculus, which, however, we do not wish to consider here. 1) as an alternative. The derived clause is called resolvent.

485 · 108 alternatives on the right side. In first-order predicate logic, we can define for this a predicate Position(number, xPosition, yPosition). The above relation must no longer be enumerated as a huge number of pairs, rather it is described abstractly with a rule of the form ∀u ∀v is_further_right(u, v) ⇔ ∃xu ∃yu ∃xv ∃yv position(u, xu , yu ) ∧ position(v, xv , yv ) ∧ xu > xv , Where ∀u is read as “for every u” and ∃v as “there exists v”. W. 1007/978-0-85729-299-5_3, © Springer-Verlag London Limited 2011 31 32 3 First-order Predicate Logic In this chapter we will define the syntax and semantics of first-order predicate logic (PL1), show that many applications can be modeled with this language, and show that there is a complete and sound calculus for this language.

Bn . This clause can be transformed in two simple steps into the equivalent form A1 ∧ · · · ∧ Am ⇒ B1 ∨ · · · ∨ Bn . This implication contains the premise, a conjunction of variables and the conclusion, a disjunction of variables. ” is a proposition of this form. The receiver of this message knows for certain that the sender is not going swimming. ”. The receiver now knows definitively. Thus we call clauses with at most one positive literal definite clauses. These clauses have the advantage that they only allow one conclusion and are thus distinctly simpler to interpret.