Solving Propositional Logic Word Problem
Propositional logic is a formal language that treats propositions as atomic units. A typical propositional logic word problem is as follows:
A, B, C, D are quarreling quadruplets. If A goes to the party, then B will not go. If C goes to the party, then B will not go. What is the largest possible number that will go to the party?
Logic is the study of valid reasoning. It is applied not only in studies but also in our day-to-day lives. Using simple logical reasoning and deduction, we are able to obtain information from a certain premise. Likewise, we can also verify or disprove statements.
We will be discussing ways of identifying common mistakes and how to avoid them. We will also talk about different proof techniques, such as using Venn diagrams and analogies so that you have a toolkit for solving logic word problems.
What's wrong with this proof?
In this section, we will use familiar notations used in propositional logic. You might want to familiarize yourself with Propositional Logic first.
Like solving any other questions, we should always ask ourselves what we can and can't do when writing out our reasoning. The first step to learn how to solve propositional logic problems is to list out what can't be done or what is not a possibility so we can narrow down what the possible scenarios are. Remember that it is very easy to fall into an erroneous conclusion based on faulty reasoning. Take the statements below as an example, if the first statement is true, is the second statement also true?
\[\begin{array} &\text{"If it's raining, then I can't play soccer".} &\text{"If I can't play soccer, then it's raining."} \end{array}\]
It is pretty clear that the issue here is: there can be other reasons why I can't play soccer, which doesn't necessarily depend on the weather. If we make such simple errors in reasoning when the context is very clear, just imagine what will happen when you are less certain about statements that are more vague. In the next paragraph, we will be introduced to these errors.
Converse and Inverse Errors
As a beginner, the most common mistake you can make is to assume that the converse and/or inverse of the original statement is also true. Take a look at the two sections below:
Introduction to Converse Error with erroneous reasoning:
Premise: If it's raining, then I can't play soccer.
Conclusion: If I can't play soccer, then it's raining.
Explanation: From the first statement, we are given a condition and a result: "raining" as a condition and "I can't play soccer" as a result. The entire premise is phrased in such a way that if the condition is fulfilled, then the result will occur. However, the conclusion shows that if the result is fulfilled, then the condition will occur. This does not make sense because it is not necessary for the condition to take place if the result occurs first. This is known as a converse error.
In a general form, the argument for a converse error is as follows:
- If P occurs, then Q occurs.
- Q occurs.
- Therefore, P also occurs.
Introduction to Inverse Error with erroneous reasoning:
Premise: If it's raining, then I can't play soccer.
Conclusion: If it's not raining, then I can play soccer.
Explanation: From the first statement, we are given a condition and a result: "raining" as a condition and "I can't play soccer" as a result. The entire premise is phrased in such a way that if the condition is fulfilled, then the result will occur. However, the conclusion shows that if the condition does not occur, then the result does not occur either. This does not make sense because there can be other reasons/factors such that the result does occur. This is known as an inverse error.
In a general form, the argument for an inverse error is as follows:
- If P occurs, then Q occurs.
- P does not occur.
- Therefore, Q also does not occur.
It may now be abundantly clear that it is easy to identify we've made an erroneous reasoning. However, what if the statements given appear more vague? This is the reason why we introduce the two errors above (converse error and inverse error) to show that not all wrong statements are easily identifiable. Simply put, the relationship between two events do not necessarily imply that one causes the other. In short, we are pointing out the common fact that "correlation does not imply causation".
Now that we have seen these mistakes first hand, let's do another example to remind ourselves that they are mistakes and we can hopefully avoid them in the future. Keep in mind that some of the converse/inverse statements can appear ridiculous but some do not.
We are given the following statement: If today is Sunday, then the weather is sunny. \[\]
\(\qquad \text{ (i)}\) Write the inverse and converse of this statement.
\(\qquad \text{(ii)}\) Identify which of these statements you have made is not logical and explain why.
\(\text{(i)}\) Inverse and Converse
- Inverse: If today is not Sunday, then the weather is not sunny.
- Converse: If the weather is sunny, then today is Sunday.
\(\text{(ii)}\) Logical or Illogical
Though they are the inverse and converse of the original statement, we must keep in mind that they might not necessarily be an error. However, there is no harm in checking whether they are correct or not.
The inverse statement implies that the day has a direct relation on the weather being sunny or not, which is ludicrous because there can also be non-sunny days which do not fall on a Sunday.
The converse statement implies that only if the weather is sunny then the day is Sunday, which is also ludicrous because they can also have a sunny weather on days not falling on a Sunday. \(_\square\)
Pinpoint the exact error
Now that we can identify how the errors occur, let's take a step further and apply these techniques so that we can pinpoint exactly where the error occurs. Note that the easiest way to identify where the error arose is to convert logical statements into symbolic forms (like P implies Q). Let's try the following example.
Taking the long view on your education, you go to the Prestige Corporation and ask what you should do in college to be hired when you graduate. The personnel director replies that you will be hired only if you major in mathematics or computer science, get a \(\text{B}^\text{+}\) average or better, and take accounting. You do, in fact, become a math major, get a \(\text{B}\) average and take accounting. You return to Prestige Corporation, make a formal application, and are turned down. Did the personnel lie to you?
Let's list down the requirements to be hired:
\(\begin{array}{r r l} & \text{(i)} & \text{Major in mathematics or computer science}\\
& \text{(ii)} & \text{Get a } \text{B}^\text{+} \text{ average or better}\\
& \text{(iii)} & \text{Take accounting}\\
\end{array}\)Since you became a math major, criteria \(\text{(i)}\) is satisfied.
Since you got a \(\text{B}\) average instead of a \(\text{B}^\text{+}\) average, criteria \(\text{(ii)}\) is not satisfied.
Since you took accounting, criteria \(\text{(iii)}\) is satisfied.Since you did not satisfy all the criteria and were turned down, the personnel didn't lie to you. \(_\square\)
Now that you're familiar with writing out these statements and identifying possible errors, let's try another example that uses such a property!
A store has been raided by looter/s, who drove away in a car. Three well-known criminals Satvik, Krishna and Sharky are brought to the police station for questioning. Inspector Aditya of the police extracts the following facts:
\((1)\) None other than Satvik, Krishna and Sharky was involved in the robbery.
\((2)\) Sharky never does a job without using Satvik (and possibly others) as accomplices.
\((3)\) Krishna doesn't know how to drive.
Find the person who, in any case, is guilty.
This problem is the part of my set "Is This What You Call Logic?"
Formal terminologies
In the previous sections, we have learned the two most common errors that students will make when solving a logical reasoning problem. However, we did not formally touch on the terminologies for those terms: converse error and inverse error. Let's begin!
Contrapositive: A statement is logically equivalent to its contrapositive. The contrapositive negates both terms in an implication and switches their positions. For example, the contrapositive of "P implies Q" is the negation of Q implies the negation of P.
Converse: The converse switches the positions of the terms. The converse of "P implies Q" is "Q implies P".
"If and only if", sometimes written as iff and known as equivalence, is implication that works in both directions. "P if and only if Q" means that both "P implies Q" and "Q implies P".
Let's try a few examples that cover this area!
\(\text{ (i)}\) Write down the contrapositive statement for
\[\text{"If you are human, then you have DNA."}\]
\(\text{(ii)}\) Write down the two if-then statements for
\[\text{"A polygon is a quadrilateral if and only if the polygon has 4 sides."}\]
\(\text{ (i)}\) contrapositive
\(\qquad\) If you do not have DNA, then you are not human.\(\text{(ii)}\) if-then statements
\(\qquad\) If a polygon is a quadrilateral, then it has 4 sides.
\(\qquad\) If a polygon has 4 sides, then it is a quadrilateral. \(_\square\)
Simple, isn't it? Let's try some problems that apply the techniques we have learned above.
If Jeff spends 5 hours playing video games, then he cannot finish his math homework.
If Jeff finishes his math homework, then he will do well on his next math test.
Based on this information, which of the following is logically correct?
Selena, Jennifer and Miley wear a blue dress, yellow dress, and green dress in an unknown order. It is known that:
1) If Selena wears blue, then Jennifer wears green.
2) If Selena wears yellow, then Miley wears green.
3) If Jennifer does not wear yellow, then Miley wears blue.
What is the color of the dress Selena is wearing?
Now that we have mastered these techniques, let's move on to the following section for other cool proof techniques!
Proof by Venn Diagram
In this section, we will be applying some basic rules of set notations. You might want to familiarize yourself with sets and Venn diagram first.
In the previous section, we have learned the most common ways in identifying and pinpointing errors. In this section, we will apply the use of Venn Diagram as an alternative proof in solving logical reasoning problems. But what's the benefit of this? Well, it's simple: We do not need to verbalize these statements and we can use the visual aids to guide us to solve these problems.
Recap of set notations and Venn diagram \[\]
Let's do a brief recap for the application of Venn Diagrams by taking the following as an explicit example:
Consider \(W,X,Y,Z\) as sets, each with their own elements in them. Then by interpreting the Venn Diagram, we can obtain information like:
- All elements in set \(W\) is in set \(Y\).
- All elements in set \(X\) is in set \(Z\).
- Not all elements in set \(Y\) is in set \(W\).
- etc.
How to use Venn diagrams to solve a logical word problem \[\]
Let us consider the following statements and deduce whether the conclusion is true or false by Venn diagram:
True or false? \[\]
It is given that all birds have wings.
All chickens are birds.
Therefore, all chickens have wings.
Explanation: By Venn diagram, the statement "A chicken is a bird." implies that the set "all chickens" is a subset of "all birds." Thus we can say that all chickens have the same characteristics as a bird. Because it is given that all birds have wings (a characteristic), all chickens have wings too. Thus the conclusion is correct.
Note: We should keep in mind that this only works if the premise is true. For example, if we replace the word "wings" by "forearms" in the first statement, then the conclusion of "All chickens have forearms." will inevitably be true despite its ridiculous claim.
Food for thought: If all phones have batteries and I have a phone, does it mean that my phone has a battery?
Careful! There are other ways of drawing out Venn diagrams!
Though it may appear very simple to set up a Venn diagram, the setup may not necessarily be unique. Let us consider a revised version of the statements above and deduce whether the conclusion is true or false by Venn diagram.
True or false? \[\]
It is given that all birds have wings.
All chickens are birds.
Therefore, all birds are chickens.
Explanation: By Venn diagram, the statement "A chicken is a bird" implies that the set "all chickens" is a subset of "all birds." Thus we can say that all chickens have the same characteristics as a bird. However, it is not necessarily true that all birds share the same characteristics of a chicken. (Sounds familiar? It's converse error.) So the claim "All birds are chicken" must be false.
Note: To fix the conclusion, you should say "Some birds are chickens" instead of "All birds are chickens."
Now that we know the fundamental applications of proof by Venn diagram, let's apply these knowledge we learned on the following examples:
Isn't the proof by Venn diagram fun? You don't need to use actual words to formalize these statements. Looks very unusual, right? But it works. Speaking of unusual, is it possible to solve these logical statements if we were to spice things up by dramatizing out the statements? Yes, we can! Proof by analogy is another proof technique to solve logical problems. See the following section:
Proof by Analogy
How do we solve texts that are seemingly hard to decipher?
All pangs are pings.
Some pings are pongs.
Therefore some pangs are pongs.
Consider the logical statements given above. Since we can't relate to or identify what pangs, pings, or pongs are, it will appear that these terms are vague or all too similar. How are we supposed to solve problems like this if we have little to no clue to what is going on? Well, proof by analogy will be useful here: this is when we dramatize or caricaturize the terms used.
For example, we may call pangs as humans, pings as apes, and pongs as gorillas. With these new terms, we are able to visualize what they are. Rewriting them into the original 3 statements shows that
\[\text{All humans are apes. Some apes are gorillas. Therefore some humans are gorillas.}\]
So the given conclusion is wrong because of the ridiculousness of the conclusion "Some humans are gorillas."
However, an important question to ask is why this works. This is too good to be true, right? Or, are we running into some wrong argument? Why does this work?
Explanation of how this works
The reason why proof by analogy works is because we make an inference that if the objects have multiple similar characteristics, and it is given that you know one of them have an extra characteristics (call it X), then it is not a bad inference to conclude that the other object shares that same characteristic X.
To put it short, the generalized/structured form for proof by analogy is:
- P and Q has similar properties \(x_1, x_2, x_3, \ldots, x_n\).
- We know that P has a further property \(y\).
- Therefore, Q probably has property \(y\) too.
Now let's try a modified version of the ping-pang-pong question from earlier!
True or false? \[\]
\(\qquad\) All yangs are yengs and yings.
\(\qquad\) Some yengs are yings.
\(\qquad\) Then, all yengs are yangs.
This is false.
Let "yangs" be defined as "pets", "yengs" as "tigers", and "yings" as "cats".
So it is true (or at least still reasonable) that all pets are cats and tigers and that some tigers are cats. But it is not true that all tigers are pets. \(_\square\)
Note: The reason why proof by analogy works best here is because we couldn't label or identify any characteristics for yangs, yengs, and yings. Therefore, a sensible approach is to prove by analogy.
Now that you're ready to solve logical problems by analogy, let's try to solve the following problem again, but this time by analogy!
Suitability of analogy
Notice from the previous section that we've mentioned that "Q probably has property \(y\) too." instead of "Q definitely has a property \(y\) too." This is because the argument may provide what appears to be the right evidence, but the conclusion does not always follow. This subsection explains why this proof (arguemnt) might not always work.
Though it is true that we highlight or amplify parallel characteristics, the differences between things can often overwhelm their similiarities. One might note that it is always possible to extend an analogy to the point of absurdity. For illustration, take the following famous "Information Argument":
DNA is a code.
A code requires an intelligence.
Therefore, DNA comes from an intelligence.
Yes, this will sound completely logical if we apply the implications approach. That is, it's P implies Q and Q implies R, so P implies R. However, the argument here is not valid because the statement "DNA is a code." is purely an analogy and thus it is not an entirely accurate statement to begin with. Thus, we have started with a wrong premise. So the merits of analogy do not hold. This is further explained in Analyzing arguments from analogy.
We can see that proof by analogy is very useful and can also be used to make incorrect conclusions. Thus, one must be careful in labeling certain characteristics when using this method. Let's see the following examples to see how the proof by analogy backfires:
True or false? \[\]
All squares and rectangless are convex, have four sides and form right angles at their vertices.
All squares have sides of the same length.
Therefore, all rectangles have sides of the same length.
This is obviously false because by definition, all rectangles do not have sides of the same length, but only squares have sides of the same length. We make the wrong conclusion that rectangles also have this characteristic because it is known previously that both share a number of characteristics. \(_\square\)
