Heard in Parliament: "Will the minister not now discontinue her proposal to ban the protest?"
The minister replied "Yes I will."
To what had the minister committed herself logically, and why might that not have been her intention?
In a cricket tournament an umpire might be required to decide whether or not a batsman is out 'lbw', ie 'leg before wicket'. The lbw law for the tournament refers to parts of the cricket pitch as shown in the diagram (assuming a right-handed batsman):
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The umpire has to make a number of judgements:
A Would the ball have hit the wicket?
B Did the ball hit the batsman, or part of his equipment other than the bat, without hitting the bat?
C Did the ball hit the batsman, or part of his equipment other than the bat, before hitting the bat?
D Was the part of the batsman or his equipment which was hit by the ball, between the wickets when it was hit?
E Was the part of the batsman or his equipment which was hit by the ball, outside of the wicket on the off side when it was hit?
F Was the batsman attempting to play a stroke?
The law can be interpreted as saying that the batsman is out lbw if \([ ( \mathrm { A } \wedge \mathrm { B } ) \vee ( \mathrm { A } \wedge \mathrm { C } ) ] \wedge [ \mathrm { D } \vee ( \mathrm { E } \wedge \sim \mathrm { F } ) ]\).
The tournament's umpiring manual, in attempting to simplify the law, states that the batsman is out lbw if \(\mathrm { A } \wedge ( \mathrm { B } \vee \mathrm { C } ) \wedge ( \mathrm { D } \vee \mathrm { E } ) \wedge ( \mathrm { D } \vee \sim \mathrm { F } )\).
For an lbw decision this requires 4 conditions each to be true.
Use the rules of Boolean algebra to show that the manual's rule is logically equivalent to the law as stated above, naming the rules used at each step.
A trainee umpire, using the manual, considers each condition in turn and judges that the following are true: A; B; E; D.
What is her decision and why?
What is odd about her judgement, and does this make the logic invalid?