More often than not, the expression ‘taken to its logical conclusion’ serves to point up the absurdity of a piece of reasoning we come across. But the phrase should be used carefully lest it should turn out the reasoning is fine but we are wrong. Some obvious examples:
The converse of a statement: All A are B, is not equivalent to: All B are A, although some B are A indeed.
If 2 is an integer and every integer is a rational number and every rational number is a real number, then 2 is a real number. It figures by the law of hypothetical syllogism. It doesn’t work in reverse, for the simple reason that the sentence: if it is a real number, then it is 2, will be true in one case and false in the infinity minus 1 cases.
But who can resist an occasional daft wish or piece of advice, such as the one about a buckled railway line I heard last week (further down)?
The golden standard I suppose is set by the rhyme: There is a hole in my bucket, dear Liza, dear Liza. Of course, fixing it requires the use of the bucket at some stage, and so the loop closes.
The all-time favourites are: I wish there were no Mondays, and I need a holiday to get over a holiday.
In the intense heat we are experiencing now, rail lines tend to buckle. A train I was on came to a halt before such a buckled rail, and various suggestions on how to fix it were bandied about as the passengers piled out. The mood among the crowd ranged from rage to light-hearted banter, and somewhere along this spectrum an idea was born: let’s unscrew a rail immediately behind the train to replace the buckled one immediately in front of the train. It can’t have been offered other than in jest, but the heat and the prospect of a long wait made it seem strangely viable at the time.
I have seen it done before in the old west movies, where the tracks were blown up by train robbers. But the thought hadn’t crossed my mind. Yet, taken to its logical conclusion, this is an answer to the ailing railway infrastructure. Consider the savings to be made by a railway company if they ran a service from one destination to another by having just enough track laid to extend a little in front of the train and a little behind the train! Twenty burly passengers get off, relay the track by removing the rails at the back and putting them down in front, the train moves on a few meters and stops again. It’s a DIY railway line! It could even take you to places where there are no railway stations! Not fast, but logical!
A logical variation of the idea involves the last carriage of a train being fitted with a spare rail, in the manner of cars carrying a spare wheel, so that a buckled rail could be replaced without leaving a gap in the line. The possibilities are endless.
There is a parallel there to The Tower of Hanoi problem, in terms of the time required if not the logic, with disks gradually increasing in size placed on one of three pegs. The monks must shift the disks from one peg to another by removing only one at a time and never placing a larger disk on a smaller one. The answer to how long it would take to move 64 disks can be found on Wikipedia.
The converse of a statement: All A are B, is not equivalent to: All B are A, although some B are A indeed.
If 2 is an integer and every integer is a rational number and every rational number is a real number, then 2 is a real number. It figures by the law of hypothetical syllogism. It doesn’t work in reverse, for the simple reason that the sentence: if it is a real number, then it is 2, will be true in one case and false in the infinity minus 1 cases.
But who can resist an occasional daft wish or piece of advice, such as the one about a buckled railway line I heard last week (further down)?
The golden standard I suppose is set by the rhyme: There is a hole in my bucket, dear Liza, dear Liza. Of course, fixing it requires the use of the bucket at some stage, and so the loop closes.
The all-time favourites are: I wish there were no Mondays, and I need a holiday to get over a holiday.
In the intense heat we are experiencing now, rail lines tend to buckle. A train I was on came to a halt before such a buckled rail, and various suggestions on how to fix it were bandied about as the passengers piled out. The mood among the crowd ranged from rage to light-hearted banter, and somewhere along this spectrum an idea was born: let’s unscrew a rail immediately behind the train to replace the buckled one immediately in front of the train. It can’t have been offered other than in jest, but the heat and the prospect of a long wait made it seem strangely viable at the time.
I have seen it done before in the old west movies, where the tracks were blown up by train robbers. But the thought hadn’t crossed my mind. Yet, taken to its logical conclusion, this is an answer to the ailing railway infrastructure. Consider the savings to be made by a railway company if they ran a service from one destination to another by having just enough track laid to extend a little in front of the train and a little behind the train! Twenty burly passengers get off, relay the track by removing the rails at the back and putting them down in front, the train moves on a few meters and stops again. It’s a DIY railway line! It could even take you to places where there are no railway stations! Not fast, but logical!
A logical variation of the idea involves the last carriage of a train being fitted with a spare rail, in the manner of cars carrying a spare wheel, so that a buckled rail could be replaced without leaving a gap in the line. The possibilities are endless.
There is a parallel there to The Tower of Hanoi problem, in terms of the time required if not the logic, with disks gradually increasing in size placed on one of three pegs. The monks must shift the disks from one peg to another by removing only one at a time and never placing a larger disk on a smaller one. The answer to how long it would take to move 64 disks can be found on Wikipedia.
As a child I remember not being able to see the point of the bucket 'story'. My mind said (and I wonder sometimes if there isn't a touch of Asperger's in me) 'well hang on, how much water do you need to wet a stone? Surely you just need to hold the bucket in such a way (on its side say) so that the hole is above the water line'. But that's not the point of farce, is it :)
ReplyDeleteTalking of which I don't remember ever coming across 'I wish there were no Mondays' however my favourite of all time is 'If God had intended for us to see the sunrise he would have scheduled it for later in the day' :)