Formal logic analysis technique is an important problem solving resource
Formal logic deals only with verifiable facts
In everyday life when you say, “What I said is logical”, usually you use the word 'logical' informally using your knowledge, perception and estimates regarding the subject of discussion.
For example, if you are discussing the suitability of a candidate for selection for job, you would have considered the academic performance, selection test performance, attitude to work, commitment, workplace experience, interview performance, problem solving capability and possibly a few other criteria of the candidate.
In most real life situations, evaluation of such criteria is subjective and specific to you. It may not be agreed by all. Your logic may not be the logic of another evaluator. Whether you would be able to establish your logic will depend on how strongly you make your arguments to the person, and also whether the person is able or willing to see your point of view.
It is a judgmental, subjective and uncertain situation.
In contrast, formal logic has no place for judgmental uncertain outcomes of any statement or conclusion. Every statement in formal logic can have either of the two values only – “Yes” or “No” (or “True” or “False”).
In formal logic, a statement, “Naina lives in Kolkata” can either be true or false. There is no place for an uncertain or third outcome of ‘maybe’ or ‘perhaps’. This truth or falsity is supported by the actual facts and when known, the statement will be evaluated to one of the two values of ‘yes’ or ‘no’. Thus the fact based verifiable statement “Naina lives in Kolkata” is a valid statement in formal logic.
But, “Renu is honest” cannot be considered valid in formal logic as, honesty has many shades and definitions and is not verifiable with total certainty.
The core concept of this approach of limiting the number of possible outcomes to only two makes this world of yes/no outcomes a certain world.
The power and shortcoming of formal logic analysis
To appreciate the power of this on/off or yes/no mechanism, you just have to consider that the whole of the digital world is built upon the foundation of two valued Boolean variables. We have achieved, among many other gains, a tremendous leap in accuracy in the digital world compared to the analogue world where a continuously changing parameter such as voltage can have any number of possible values in a continuum.
One might be tempted to assume that answers to all earthly questions can be obtained using pure logic. In the past some thinkers actually took up this attractive path.
Unfortunately, most of the important aspects in real life decision making do not adhere to only two possibilities and thus cannot be analyzed or predicted using pure logic. For real life problem solving, one needs to use whatever problem solving resources are suitable and available.
Logic analysis technique is a problem solving resource
Logic analysis is only one resource among a host of powerful problem solving tools, techniques, principles and approaches.
We define Deductive Reasoning as the overall chaining mechanism that binds all the fact gathering, analyses, and decisions from start to end of a problem solving process using one or more than one of the suitable problem solving resources.
Though logic analysis by itself may not seem to have much applicability in real life problem solving, we consider it very important because of three major reasons.
- Practice of logic analysis assuredly increases your precision of thinking.
- Furthermore, practicing logic analysis increases your ability to interconnect conditions and events in a long chain of deductive reasoning.
- More importantly, using yes/no mechanism, sometimes complex interdependent cause-effect relationships or evaluation criteria are simplified to independent ones for effective solution. Amongst others, Root cause analysis is a powerful tool that use this approach.
Because of its inherent precision and logic connection, logic analysis is considered an important component under the topic Reasoning and is used in many leading competitive tests and interviews.
Let us now delve a little deeper into this interesting area and solve a few logic problems to get a feel of it.
Case example 1 of logic analysis – problem of pairing
Question:
Four married couples belong to a theatre club. The wives’ are Puja, Sonia, Kaveri and Monica. The husbands’ are Naveen, Farukh, Tapan and Manav. Who is married to whom? Use the following clues.
- Naveen is Monica’s brother.
- Monica and Tapan were once engaged, but broke up when Monica met her present husband.
- Kaveri has a sister but her husband is an only child.
- Puja is married to Manav.
Before you go ahead through the solution please try to solve the problem yourself. It is not a difficult problem and will give you an idea of purely logic based problem solving.
Solution:
This is a logic problem as all statements and implications are fact based with outcomes true and false only.
The first statement implies: “Naveen is not married to Monica” with certainty. Similarly, the second statement implies, “Monica and Tapan are not married to each other.” The last statement expressing a direct relationship, “Puja is married to Manav” not only puts one relationship out of any doubt, it also implies that neither Puja nor Manav are married to any of the other six.
From the first two implications we find Monica common to both and is not married to either Tapan or Naveen. As Manav is already married, we conclude, “Monica must then be married to the fourth man, Farukh.
Now two pairings are left out of Naveen, Tapan, Sonia and Kaveri.
From the third statement we conclude, "Kaveri’s husband is a single child". Out of the two men left now, as "Naveen is not a single child" from the first statement, then we conclude, “Kaveri must be married to Tapan” and so “Sonia must be married to Naveen.”
The whole chain of implications, condition analysis and conclusions finally arriving at the result step by step is Logic analysis in action. No guesses, no estimations. Clear verifiable conclusions. All logic statements, implications and conclusions, judiciously connected to each other.
But nevertheless to analyse properly we needed to select the proper pairings of statements and conclusions leading to the desired result. That is your logic analysis capability.
Instead of this purely statement based approach, use of Decision table makes the analysis easier and more transparent.
Using decision table to solve pairing problem
The decision table that we will use here is a four column by four row table as shown.
| Sonia | Kaveri | Puja | Monica | |
|---|---|---|---|---|
| Manav | X | X | Y | X |
| Tapan | X | Y | X | X |
| Naveen | Y | X | X | X |
| Farukh | X | X | X | Y |
There are sixteen cells between four possible couples out of which only four cells will have a Y indicating married relation. Rest 12 will have Xs indicating no-spouse relationship.
With a Y in Puja-Manav cell from the fourth statement, we cross out all cells in Manav row and Puja column.
From first and second statement, we cross out Tapan and Naveen cells of Monica column. Remaining Farukh cell automatically is then paired with Monica. But this crosses out the two remaining cells of Farukh also.
At the last step, the third statement along with the first statement implication that Naveen is not a single child decides a Y in Kaveri-Tapan cell. The last is then a Y in Sonia-Naveen cell.
When you use a decision table, you do not lose track of the results of your analysis at any stage. Pictorially and unambiguously you account for all the results of your analysis step by step. But in any case there is no going away from the fact that, you have to do the logic statement analysis in either method you adopt.
Case example 2 of logic analysis: Which path would you take? (version 2)
Question:
A logician while visiting the South Seas again is at a fork in an island. He needed to know which of the two paths in front of him leads to the village (this again is a problem from Martin Gardner's famous collection).
In this case, he has three willing natives available nearby. But one of them is from a tribe of habitual truth-tellers, another from a tribe of habitual liars and the third from a tribe of habitual random answerers. The logician knows this but does not know who is which. The natives though know each other well.
The logician can ask only two yes-no questions, each time directed to just one of the natives.
Can he know the right path to the village?
Very hard? Well, it seems to be so in the beginning. But remember one golden principle: however difficult and complex a problem seems to be in the beginning, it is bound to get simpler if you can break it up intelligently into smaller pieces. This is the much used Problem breakdown principle.
With this hint we will leave you to solve this problem yourself. Please do make a serious attempt to solve this problem. Use your deductive reasoning. Analyze and formulate inviolable conclusions. Only then proceed further.
Solution:
The first thing that we notice is that the problem contains two stages. We assume here that we don't know about the simple logic analysis problem we discussed earlier.
Under this situation, what can we conclude of the nature of question and whom to ask the question at each stage? At the first stage you have no choice - you have to ask the first question to any of the three. what about the second stage?
Second stage requirement analysis
Let's think about second stage first - this is kind of end state analysis. Unless we create a favorable last stage, we would fail in our quest to find the right path without any doubt. How many answerers should be there at the second stage - three or two? Just by commonsense reasoning we can conclude, unless we reduce the number of natives to ask the last question to two, we won't be able to reduce the complexity of the first stage.
One point is settled then - by the first question we must reduce the number of natives to ask the second question to two. What about the nature of the second question? Again by commonsense reasoning it is rather obvious that unless we ask a question like, "Does the path on the right lead to the village?" we stand no chance of knowing the right path to the village. In other words, we must ask about the nature of paths as specifically and as suitably as needed at the second stage. But we need not go deeper into the second question because that stage is yet to come. Just assessing its nature is enough at the moment.
The third and most important requirement of the second stage though hinges on the question, whom to eliminate at the first stage?
If we examine the nature of the natives, we find both the liar and the truth-teller following fixed predictable pattern of answering. But the random answerer is totally unpredictable - he may answer "yes" or "no" to any question regardless of the truth or falsity of the answer. His answer has no relation to the question.
Using your commonsense reasoning, you may again conclude that unless you eliminate the random answerer with the first question, getting to the right path at the second stage will be very difficult if not impossible. This is a reasonable plan of action for a practical problem solver.
First stage requirements
So at the first stage you would ask such a question as to identify two prospective natives for the second stage in which random answerer won't be there.
What about the nature of question? As you must ask the second question about the nature of path, you won't repeat it at the first stage. Also, at this stage you have to eliminate a native. Thus it follows that the question must be on the nature of answering pattern of the natives.
When you ask the question, all three natives are equivalent to you without any distinction. But distinction automatically happens as you ask the question. The natives are divided into two parts - the answerer whom you have asked the question and the other two natives. Again by commonsense reasoning you conclude that you may get more information about the natives if your question is on nature of answering of the other two natives and not about the single answerer himself.
Would you ask about their nature by pointing individually to them? Here we introduce an important rule of how you get maximum information about two objects by a single operation.
A comparison operation on two objects will generate maximum information about the two objects.
It stands to reason, as by comparison only you would be able to combine both of them in a single question.
Key information discovery and enumeration
Now it is time to actually compare the nature of answering of the three natives. The state of facts are summarized in the following table.
| Truth-teller | Random answerer | Liar |
|---|---|---|
| answers truthfully always | answers randomly | answers falsely always |
| answers truthfully 100% of the time | answers truthfully 50% and falsely 50% of the time | answers falsely 100% of the time |
| more likely to answer truthfully compared to the random answerer | more likely to answer truthfully compared to the liar | less likely to answer truthfully compared to the random answerer |
Till now you have used your deductive reasoning and ultimately resorted to data or fact testing by forming this table. Unless we actually analyse the nature of the data, how can we determine what course of action is to be adopted. This is the step of Data or Fact analysis that is vital in most cases of real life problem solving. We have used the most convenient tool for this task as Table tool. A table form is a versatile form of data representation that we already have used for representing the relationship status in our earlier problem. A table is a very useful problem solving tool and is conveniently represented in a spreadsheet type program.
As we went on putting the facts in the table columns we found finally that the phrase "more likely" is the key phrase for our task at hand. As we are primarily after knowing who is the truth teller, our question to a native should then be like, "Is the native here (say native B) more likely to tell the truth compared to the native there (say native C)?". By deductive reasoning this is our conjecture.
Now we would put this conjecture to test using the technique of enumeration. We would assume first B as the random-answerer and C as the liar and take A as the first native whom you ask the first question. Naturally A is the truth-teller in this case. This is the first scenario.
First scenario: Ask the first question to Native A truth-teller
From the table above you would find that answer to the question would be "yes". As the first native A is the truth-teller he would also answer "Yes" and you being focused on eliminating the random-answerer you would choose C for the second question. Again, if the nature of B and C are reversed the answer will be "no" and you would then choose B for the second question.
The following table depicts the situation.
| Native A truth-teller answers | Native B random answerer Native C Liar | Native C random answerer Native B Liar |
| Yes | Choose C | |
| No | Choose B |
Second scenario: Ask the first question to native A the liar
Doing the same exercise in the second scenario, we ask first the Liar who is A in this scenario. The following table depicts the situation in this scenario.
| Native A liar answers | Native B random answerer Native C truth-teller | Native C random answerer Native B truth-teller |
| Yes | Choose C | |
| No | Choose B |
When the liar A is asked the question, "Is the native B more likely to tell the truth compared to the native C?", where B is the random answerer and C is the truth-teller, liar finds the correct answer to be "no". So he must answer "yes". Knowing everything in this scenario (assumptions), you would choose C without any hesitation. If the nature of B and C are reversed, Liar will answer "no" and you would choose now B, the truth teller.
You would find to your joy that in both scenarios, where the first answerer is either the truth-teller or the liar, if he answers to your question, "Is the native B more likely to tell the truth compared to the native C?" as "yes" you would choose C for the second question eliminating the random-answerer. Conversely, if the answer is "no" you would choose B for your second question again eliminating the random answerer.
Third scenario: Ask the first question to native A the random answerer
This is a trivial case as in this scenario random answerer, being the answerer, is automatically eliminated and finally in all three scenarios you are left with B or C, one of them truth-teller and the other liar.
Second stage question
At this stage you have either B or C to question. One of them is a habitual truth-teller, and the other is a habitual liar. You have two paths in front of you one of which you have to choose.
You have already analysed that you would form the question somewhat like, "Does the path on the right lead to the village?". You know if this question by chance goes to the truth-teller and indeed if the path on the right is the path to the village, you would get answer "yes". But instead if the answerer is the liar he would answer "no". It means - with this question you may get an answer "yes" or "no". You won't be able to decide in that case then.
Being aware of this possibility what would you conclude? The first conclusion is, whatever be the question, both the liar and the truth-teller must give the same answer to it. Otherwise you won't be able to decide.
With the knowledge available till now, we can also conclude that the question should be something like, "Does the path on the right lead to the village?". Otherwise you won't know which one is the right path.
Finally everything boils down to the problem: how to make both of the natives to answer the same when the question is like,"Does the path on the right lead to the village?". With the truth-teller you don't have a problem. What about the liar?
An important rule in logic domain is,
Lying twice reverses the first lie to the original truth.
You may use this rule or further analyse to decide that the question can't be a simple one-part question. It follows immediately that the target question must be like "Does the path on the right lead to the village?" and assuming that the path on the right is indeed the correct path, the liar will say "no". If you put this answer as another question to liar he would have no other option than to reverse it to "yes".
Indeed if path on the right is the correct path and if you ask the liar whether it is the correct path, he would say "no". But combining, if you ask him,
"If I ask you whether the path on the right is the correct path what would you answer?"
he would have to reverse the answer of the first part when answering the second part and make it a "yes". The truth-teller would also tell you the truth with a "yes". In both cases you would take the path on the right. In case the answer is "no" you would follow the path on the left.
Effectively at the second stage, you have forced the liar to tell the truth.
Problem solving resources you have used
First you have used the end state analysis approach and analyzed the second stage.
In second stage analysis you have used your deductive reasoning to form main objective of first stage question as - elimination of the random answerer.
By this analysis, you had also been able to split the problem into two parts using - problem breakdown principle. Being a problem solver you knew,
Complex real life problems can often be broken down into a series of simpler and smaller problems.
Taking up the first stage analysis, using deductive reasoning, you identified the question content to be - nature of the other two natives and further decided that the single operation giving maximum information about the two natives must be - comparison. This again was concluded by deductive reasoning and if you knew, the maximum information about a pair in a single operation rule.
To get more idea about comparison of nature of answering patterns, you resorted to Testing and Key information discovery by using enumeration technique and table tool. Indeed you discovered the key phrase "more likely" to be included in the first question. You got ready with the next action by forming the test question as a promising conjecture. This is promising conjecture formation.
Having sufficient control over the problem internals, you now straightway resort to enumeration of scenarios in scenario analysis.
You had used deductive reasoning at each step of scenario analysis to accurately depict the results in each scenario.
Looking at the scenario results you find that indeed your conjecture was right. This is verification of conjecture.
With first stage over, you had isolated the liar and the truth-teller for your last question.
At the last stage, using deductive reasoning or the rule of double inversion restores original stage, you formulate the final question.
All through you have not thought and took action like a logician, but you analyzed and acted like a problem solver.