Simple Logic analysis

Logic analysis technique

Logic analysis technique

We frequently use the term ‘logic’. People who use logic to back up their comments, decisions or actions are considered as logical.

Sometimes this term is interchangeably used with the term ‘rational’. People who do not act or behave according to commonly accepted norms are termed as irrational.

Human beings do not always follow stringent logic. Nevertheless ‘logic’ is a technical term that is well-defined and precise – nothing vague in ‘logic’ at all.

In logic domain, there is truth or falsity of propositions, nothing in between. It is a kind of black and white Boolean world. One says, the statement or the proposition – “Rainee lives in Kolkata” can either be true or false. Thus this statement or proposition is a valid proposition in logic.

“Rainee is a good girl” in contrast is not a valid proposition because 'good' is highly subjective introducing uncertainty and it is difficult to determine whether the statement is purely true or surely false.

Though real life cannot really be represented or explained using logic, still logic analysis as a technique in real life problem solving undoubtedly is valuable. Let us look at this concept domain more closely by solving a problem from Martin Gardner’s South Sea adventure stories.

Logic Problem: Which path would you take? (version 1)

A logician while visiting a South Seas island is at a fork facing two paths, one leading to the village he wants to go. The other path might lead to an unpleasant experience.

He needed to know which of the two paths in front of him leads to the village. Perchance he has two willing natives available near at hand. But one of them is from a tribe of invariable truth-tellers, and the other from a tribe of invariable liars.

He knows this predicament but does not know which one is the liar or the truth-teller. The natives though know each other well.

The problem is: the logician can ask only one question to any one of the natives, answer to which maybe yes or no only.

Can he know the right path to the village?

To remind you, Logic Analysis is a discipline where there are statements or propositions which can only be true or false (yes or no) and logicians are experts in dealing with problems in this domain.

As you are perhaps not a logician, you are not an expert in the process of solving logic problems like this. In spite of this shortcoming, at least you are a problem solver who is supposed to proceed solving any problem systematically.

So you need to proceed using deductive reasoning systematically, not in any random or chaotic manner. We would suggest you to devote a few minutes trying to pose as the successful logician.

Reasoning

The logician can ask the all important single question to only one of the natives without knowing whether the native is the liar or the truth teller. Depending on the answer he will know which the right path is.

If the answer is different from the two natives he can never be certain. So a necessary condition for the question to surely lead you to the right path, the answer must be same whether the native to whom the question is asked is the liar or the truth teller. Being a good deductive reasoner, you analyze and arrive at this conclusion.

Now the problem is transformed to  - how to form such a question with the same answer from the both?

Consider that a habitual truth teller can be symbolized as a logic transformation box that outputs the same logic variable state it takes as inputs. The following figure depicts the situation.

 

Truth teller input transformation

 

Similarly, the logic transformation box for the habitual liar will be to reverse the state of input. In other words, if the input is YES, it will output with answer NO and vice versa. The following figure depicts the situation,

 

liar transformation

 

So you understand that a single question will surely produce two values as answer from the two natives. This cannot be the solution then. It implies surely that a single simple question will not be the solution - you need then to ask not a simple question but a compound question comprising of two simple questions. You arrive at this conclusion by your deductive reasoning mechanism again.

Knowing that reversing twice restores the original state, we may think of asking the habitual liar two questions in conjunction, “Will you answer YES if I ask you whether the path on the right is the path to the village?”

The second part of the compound question goes as the first input to the logic transformation box of the habitual liar reversing the state of the input. In terms of our problem it means, if in reality the path on the right were the path to the village, the TRUE answer to this part of the question would be YES, but as the liar must reverse the state, he must answer with NO. This is hypothetical, but still the habitual liar has no other option than to imagine that his answer to the second part of the question would be a NO.

This NO then goes as input to the logic transformation box of the liar for answering the first part of the question and completing his whole task. Being a logic state reversing mechanism, the liar would have to reverse this NO to YES and actually arriving at the TRUE answer to the original core statement.

The truth teller poses no problem as he does not do any transformation in each of the two transformation stages. An input of YES passes straight through as YES in the output.

Note that, the question could have been “path on the left” instead of “path on the right” and also “NO” instead of “YES”.

Try these out. The figure below depicts the transformation sequence for the habitual liar.

 

liar transformation two stage

 

Even though you are not a logician, being a problem solver, you proceeded by adopting a systematic approach and finally arrived at the solution.

Logic analysis is an important problem solving skill and like most other skills you can boost your level of this skill by following systematic methods. As you can guess, practice or drill is an inseparable element in any such method.

 

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