March 7, 2017

An introduction to logic [Lecture notes] by Mark V. Lawson

By Mark V. Lawson

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ADEQUATE SETS OF CONNECTIVES 31 only logical connectives from that set. In these terms, we proved above that the connectives ¬, ∨, ∧ form an adequate set. We can if we want be even more miserly in the number of logical connectives we use. The following two logical equivalences can be proved using double negation and de Morgan. • p ∨ q ≡ ¬(¬p ∧ ¬q). • p ∧ q ≡ ¬(¬p ∨ ¬q). From these we can deduce the following. 2. 1. The connectives ¬ and ∧ together form an adequate set. 2. The connectives ¬ and ∨ together form an adequate set.

If B is a consequence of A1 , . . , An we write A1 , . . , An B and we say this is a valid argument. This definition encapsulates many examples of logical reasoning. It is the foundation of mathematics and the basis of trying to prove that programs do what we claim they do. We shall see later that there are examples of logical reasoning that cannot be captured by PL and this will lead us to the generalization of PL called first-order logic or FOL. 1. Here are some examples of valid arguments.

Then by construction A is satisfiable precisely when the original problem is satisfiable and a satisfying truth assignment to the atoms can be used to read off a solution as follows. Precisely the following atoms are true: c113 , c122 , c131 , c211 , c223 , c232 , c312 , c321 , c333 and all the remainder are false. It is now easy in principle to generalize our two examples above and show that a full-scale Sudoku puzzle can be solved in the same way. This consists of a grid with 9 × 9 cells and each cell can contain exactly one of the numbers 1, 2, .

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