Logic Tutorial 2
1. An Exercise taken from Year 1 Logic – with my thanks to Prof. Ian Hodkinson
Consider a set of objects labelled A, B, C, … placed on a 3*3 grid, and the following
atomic formulas talking about the objects:
[x next-to y] means (that is, it is true if) x and y are adjacent (horizontally or
vertically, but not diagonally);
[x sees y] means x and y are in the same row or the same column;
[x left-of y] means x is in a column to the left of the column of y;
[x above y] means x is in a row above the row of y.
(a) For the placements shown in figure below, which of the following evaluate to true, and
why?
i. [A sees B] [B sees C]
ii. [B next-to D] [B next-to E]
iii. ¬ ([A left-of F] [F above A])
iv. ¬ ([E left-of D] → ¬[D next-to C]) → ¬[A sees E]
v. ([E sees D] [F sees E]) → ¬([B above E] [B next-to C])
A D
C F B
E
(b) Place the 6 objects A, … , F on the grid so that all the formulas above are true.
2. For each of the following determine if it is a tautology, inconsistency or contingency by drawing the
truth table.
a. P (P Q) d. (P (Q P)) P
b. (P Q) (P Q) e. (P Q) ( P Q)
c. Q P (P (Q P)) f. ((P Q) (R S) (P R)) (Q S)