Illogical, Captain!

A brief introduction to Logic Reasoning

This section explains four important logical constructions that you may meet in critical reasoning. I have given a brief explanation for each of them, together with examples showing how they can be misused and a mathematical-type representation that might make them easier to understand and learn.

Modus ponens

This simply means: If X is true, then Y must be true. We know that X is in fact true. So Y must be true as well. Writing it symbolically, X ® Y. X true, \ Y true. (where ® means "implies" and \ means "therefore").

It is also possible to use modus ponens incorrectly. Here is the invalid form of the same thing:

If X is true, then Y must be true. We know Y is true. So X must be true as well." or X ® Y. Y true, \ X true.

This doesn't necessarily follow. We know that X being true makes Y true, but there may be other things apart from X which make Y true as well. Here are two examples of modus ponens, the first a correct one, the second incorrect:

Whenever it rains, I take an umbrella to work. It is raining, so I shall take an umbrella to work today.

That is perfectly logical. Don't be put off by the use of the future tense, or the fact that we assume you are going to work (it might be Sunday) - they are side points to the argument. If we write "It is raining" as X and "I take/will take an umbrella" as Y, then the statement fits the modus ponens pattern exactly. Tick! Well done!

The bells you can hear now are always rung during a funeral. Someone must have died!

This is a little harder to recognise. It doesn't use words like "implies" or "therefore", but it is still a version of modus ponens. We can rewrite it as follows: Whenever a person dies/has a funeral, those bells are rung. The bells are ringing now. Therefore someone has died/there is a funeral.

We are taking "a person dies" and "there is a funeral" to mean the same thing - a reasonable assumption. Taking that to be X, and "the bells are rung" to be Y, you see that it fits the pattern for the invalid form of modus ponens perfectly. It doesn't follow that the bells are only run when there is a funeral. They could be rung for weddings, baptisms, air raids etc.

Modus tollens

This is the flip-side of modus ponens: "If X is true, then Y must be true. We know that Y is not in fact true. So X can't be true either." Symbolically, X ® Y. Not Y, \ not X.

Since X automatically causes Y, if Y is not true, then X can't be either. Again, there is an invalid form of modus tollens:

If X is true, then Y must be true. We know that X is not in fact true. So Y can't be true either." Symbolically, X ® Y. Not X, \ not Y.

Hey! Wait a moment. We know X causes Y, but that doesn't mean that there aren't other things that cause Y. Just because X isn't true, Y could be caused by something else. Here are two examples of modus tollens, again, one goodie and one baddie!

Whenever the weather is fine, I am always in a good mood. I am rather out of sorts today, so the weather can't be that good!

Yes, this is logical. X here becomes "The weather is fine/good," and Y becomes "I am in a good mood." You have to know that "out of sorts" means "in a rather bad mood," in which case Y is not true, so X can't be either.

Paul gets a win on his premium bonds quite often. A premium bond win causes him to come in to work with cigars for everyone. He did not win on the premium bonds today, so I don't think he will be handing round the cigars when he gets in!

There may be many reasons why Paul hands round cigars. He did it on his birthday, and again when he got engaged to Sarah. Just because he hasn't had that little envelope from Ernie (the computer that picks the premium bond winners) doesn't necessarily mean that he won't be giving out cigars at work.

Careful of the wording!

You have to check the wording of a question very carefully. If it includes phrases such as "if and only if" then you may find that the invalid forms of modus ponens and modus tollens suddenly become valid. Let's take a look at that example about the bells again:

The bells you can hear now are only rung during a funeral. Someone must have died!

Notice the subtle change? Now the bells are rung during a funeral and on no other occasion. What you see above is now a valid logical deduction. If you go through it with a fine tooth comb, you will find that it is a version of modus ponens, with X now being "The bells are ringing" and Y being "There is a funeral going on." You will notice that X and Y are now the opposite way round from the previous version. All that from changing just one word!

Discount your own views

Remember, you should only follow the logic of the argument, not try to include your own opinions or things that you think you know. Even if the statement includes "facts" which directly contradict your own opinions, you should put that on one side. Here's an example:

All snakes have feathers. I have a pet snake. I must have at least one pet which has feathers.

Huh?! Snakes don't have feathers. That's not the point. The fact is, that statement is a perfectly logical version of modus ponens. You just have to suspend your disbelief about snakes and feathers for a moment. A less flippant example:

Jesus did not state that smoking was a sin. Therefore smoking is not a sin.

This is based on an assumption that whatever Jesus said was a sin, is actually a sin. Obviously, whether you accept this or not depends on your own personal religious views, not to mention any views you may have on smoking! However, whether you accept it or not, you should still be able to argue it logically. Writing X for "Jesus stated that smoking is a sin" and Y for "Smoking is a sin" we get X ® Y, not X, \ not Y, which you should be able to see is an invalid form of modus tollens.

Exercise 1

In this exercise, each of the deductions is either modus ponens or modus tollens, and it may be valid or invalid. Identify each deduction appropriately. For each question, write down in words what is represented by X and Y and also put down the symbolic form of the deduction in that question.

  1. There is no evidence for UFOs being of extraterrestrial origin. Therefore, UFOs do not come from another planet.
  2. Skunks always produce a powerful unpleasant smell. There isn't a skunk in this room. Therefore, there isn't an unpleasant smell in the room either.
  3. I despise people who read and believe horoscopes. Paul reads his horoscope every day and accepts every word, and as a result, I hold him in utter contempt!
  4. Lead sulphide is a yellow compound which melts at less than 200oC. I have a sample of a chemical in front of me which has not melted, even though I have heated it to several hundred degrees with a bunsen burner on its hottest setting. I conclude that this sample is not lead sulphide.
  5. All bings are bangs and all bangs are bongs. I have an item in front of me which I am sure is a bing. Therefore I have at least one bong in front of me. (Another whisky please, bartender!)
  6. My dog Dylan loves being brushed. At the moment, he is not a happy doggie, so I can't have just brushed him. True?
  7. Dylan barks loudly when he is alarmed or frightened. One night I woke up when he barked fiercely. I concluded that he must have been either alarmed or frightened, so I tiptoed down the stairs expecting to find a burglar in the house. Was my conclusion a logical one?

Exercise 2

This exercise is more general. There are no answers to it - it is just for you to think about and discuss.

  1. It is known that an influenza sufferer always has a runny nose and a high temperature. What can we reasonably conclude from the fact that Henry does not have a runny nose?
    1. He must have a high temperature.
    2. He must have influenza.
    3. He may have influenza but it is not certain.
    4. He cannot have a high temperature.
    5. He cannot have influenza.
  2. Tom placed a bet on Greased Lightening at the greyhound stadium last night. After the race he went to collect his winnings from the bookies with whom he bets regularly. Which one of the following MUST be true?
    1. Greased Lightening won the race.
    2. Tom did not win any money on Greased Lightening.
    3. Greased Lightening may have won the race, but it is not absolutely certain.
    4. Tom must have bet upon another dog running in the same race as Greased Lightening.
    5. Tom placed a bet on at least two dogs running at the stadium that night.
  3. If a creditor has a distraining order against him and a constraining order against him, then he is legally declared extincta crapta. Up in front of the judge this morning is a man who has been declared extincta crapta. The judge must conclude that
    1. The man may have either a constraining order or a distraining order against him.
    2. The man may have either a constraining order or a distraining order but not both against him.
    3. The man must have both a constraining order and a distraining order against him.
    4. The man can have neither a constraining order nor a distraining order against him.
    5. The man must have a constraining order against him, but nothing is known about whether he has a distraining order against him.
  4. If a number is a multiple of 6, then it must also be a multiple of 3 and it must be an even number. I have a number X which is a multiple of 3. I may assume that:
    1. It must be an even number.
    2. It may be an even number but doesn't have to be.
    3. It cannot be an even number.
    4. It cannot be a multiple of 6.
    5. It can be a multiple of 6 or an even number, but not both.

The Alternative Syllogism

You what? Don't worry. This is really rather simple: "We know either X or Y is true. X isn't true, so Y must be" Symbolically, X or Y. Not X, \ Y.

If at least one of the things must be true, then if the first one isn't true, the second one must be. The same thing applies the other way round: If the second item isn't true, then the first one must be! In fact, we can write the alternative syllogism the other way round: "We know either X or Y is true. Y isn't true, so X must be" Symbolically, X or Y. Not Y, \ X.

You want an example? All right then!

I am thinking of a whole number. It is not an even number. Therefore it must be an odd number.

What I haven't stated is that there are only two types of whole numbers, even and odd. However, I think that you might be expected to know that already. Since the number must be either even or odd, and since it is not even, it must therefore be odd. Entirely logical!

You notice that the alternative syllogism doesn't state that only one of the statements is true. They can both be true. Indeed the invalid form of the alternative syllogism is to assume that because one of the statements is true, the other one can't be: We know either X or Y is true. X is true, so Y must be false (or Y is true, so X must be false.) Symbolically: X or Y. X, \ not Y., or alternatively X or Y. Y, \ not X.

Again you have to be very careful about the wording. In the example above, the number was either odd or even. We knew it couldn't be both at the same time. You have to read the questions carefully to determine whether both the possible options could happen at the same time.

When it rains, as it is doing today, I always take an umbrella or go to work by car in order to avoid getting wet. I decided to go to work by car today, so it follows that I didn't take an umbrella with me.

No it doesn't! You could have taken the umbrella with you as well. Generally speaking, look out for the words either ... or in the question. This will invariably indicate that they can't both happen together, and the assumption that "X happened so Y couldn't have happened" is a valid one.

Disjunctive syllogism

Oh, crumbs! He's off again with the gobbledy-gook. What's a disjunctive syllogism? Well, it's the flip-side of the alternative syllogism. It says that X and Y can't both be true. X is true, so Y can't be. Symbolically: Not (X and Y). X is true, \ not Y.

Just as the alternative syllogism can be written the other way round, so can the disjunctive syllogism: X and Y can't both be true. Y is true, so X can't be. Symbolically: Not (X and Y). Y is true, \ not X.

This is really the true identity of the either ... or case mentioned in the previous section. "Either ... or" implies one or the other but not both, which is the disjunctive syllogism.

Just as the alternative syllogism has an invalid form, so does the disjunctive syllogism: X and Y can't both be true. X is not true, so Y must be. Symbolically: Not (X and Y). not X, \ Y. (You can, of course, swap the Xs and Ys round in that). The reason that this form is invalid is, just because they can't both be true, it doesn't necessarily follow that one of them has to be.

No chemical compound of chromium is ever coloured white. I have a sample of material that is coloured white. Therefore, it is not a chromium compound.

This is logical. The two mutually exclusive alternatives are that either the compound contains chromium or that it is coloured white. Since it is white, then it can't contain chromium.

No chemical compound of chromium is ever coloured white. I have a sample of material that isn't coloured white. Therefore, it must be a chromium compound.

No, wait a minute! There are many non-white compounds that don't contain chromium (potassium permanganate, for example), so this is the invalid form of the disjunctive syllogism.

Exercise 3

This exercise is similar to Exercise 1, except that the deductions can be any of the four types discussed above, as well as being valid or invalid. Please determine what X and Y are in each case, what type of logic is being applied and whether the deduction is a valid or an invalid one.

  1. I through a die just once. It can't come up showing both a 1 and a 4. It has not come up showing a score of 1. Therefore it must be showing a score of 4.
  2. Whenever the red light is on and the green light is off, it means that the protection shields are no longer in place covering the uranium core. The protection shields are covering the uranium core, yet the green light is off. This means that the red light must be off also.
  3. All cannabis users inevitably graduate on to using heroin. Herbie is now using heroin, so he must have started off as a cannabis user.
  4. Our leisure centre had a budget of £100,000 last year to be spent on a swimming pool costing £60,000 or a gymnasium costing £55,000. We went ahead and ordered the swimming pool to be built. Therefore we did not spend any money on having a gymnasium built last year.
  5. Linda was told that if she was given a dog for her last birthday, she would have to walk it regularly. She does not go out regularly with a dog, therefore she was not given a dog for her last birthday.
  6. My biology text book tells me that no birds are mammals. I conclude that no mammals are birds.

Exercise 4

This exercise is similar to the preceding one, except all the deductions follow from this short passage of text. Again, you should identify what logical construction is being used and whether it is a valid one.

There are only two things on the menu at my local greasy-spoon cafe, i.e. the ploughman's lunch or Sammy's Surprise. If I order the ploughman's lunch, I will end up eating mashed potato. If I order Sammy's Surprise, I will find myself eating spicy sausage. I order one of these two meals.
  1. If I eat spicy sausage, then I must have ordered Sammy's Surprise.
  2. If I do not eat mashed potato, then I must have ordered Sammy's Surprise.
  3. If I eat mashed potato, then I must have ordered the ploughman's lunch.
  4. I will eat either mashed potato or spicy sausage, but not both.
  5. I will eat both mashed potato and spicy sausage.

An example question from a real GMAT test

Well, actually, this is from page 152 of the 2000 Edition of the ARCO GMAT CAT Guide, by Thomas Martinson (published by Macmillan, USA). I do hope they don't mind me quoting it!

When it rains, my car gets wet. Since it hasn't rained recently, my car can't be wet.

Which of the following is logically most similar to the argument above?

  1. Whenever critics give a play a favourable review, people go to see it; Pinter's new play did not receive favourable reviews, so I doubt that anyone will go to see it.
  2. Whenever people go to see a play, critics give it a favourable review; people did go to see Pinter's new play, so it did get a favourable review.
  3. Whenever critics give a play a favourable review, people go to see it; Pinter's new play got favourable reviews, so people will probably go to see it.
  4. Whenever a play is given favourable reviews by the critics, people go to see it; since people are going to see Pinter's new play, it will probably get favourable reviews.
  5. Whenever critics give a play a favourable review, people go to see it; people are not going to see Pinter's new play, so it did not get favourable reviews.

If you write "It rains" as X and "my car gets wet" as Y, the original statement becomes X ® Y, X is false, \ Y is false. You should by now instantly recognise this as the invalid form of modus tollens. What we need to do is to go through the five statements about Pinter and his play and see which one matches the same pattern.

I will leave you to do the logic. Fortunately, since all the possible options are based on the same two basic statements, we need only do the translation once: X = "Critics give a play favourable reviews," and Y = "People go to see the play." Be careful of options where the critics' response is dependent on whether people go to see the play - you may need to write these options the other way round.