## Critical Reasoning Basics- 3: Syllogisms

This chapter deals with reasoning through premises. It is called syllogisms.

Broadly, you need to understand two types of arguments. We have already come to see what arguments look like. Now it is time to comprehend some ‘categories’ of arguments.

Analyse this, (P1= Premise 1, P2= Premise 2, C= Conclusion)

P1 – All men are buffoons.

P2 – Ravi (poor chap) is a man.

C – Ravi is a buffoon.

This kind of argumentation is known as deductive reasoning. Here, the conclusion arrived at, is a logical ‘necessity’, which you will find me referring to henceforth as an LN. The structure of the deductive argumentation is simple. We picked a set, gave it a characteristic (P1), picked an element from the set (P2), and with certainty, arrived at the conclusion that the element shall show the same characteristic.

P.S. I hope you understand that my sympathies with Ravi have nothing to do with the argument.

Now, the second type,

P1 – Ravi is an engineer.

P2 – Ravi is a fool.

C – All engineers are fools.

While many of you may express surprise, nay, even disdain for such argumentation, it is still deemed a valid form of argumentation. So much so, that we would not have had the evolutionary history if mankind had refused to allow room for such argumentation. Appalled? Do not be. All knowledge has been attained and transferred through this form of reasoning for thousands of years now in the evolution of ‘life’. This form of argumentation is known as ‘Inductive logic’. Here, the conclusion arrived at is not an LN but a logical ‘possibility’ (LP). The conclusion that we derived here ‘may or may not’ be true. And hence we call it an LP.

I hope you understand the structural difference between the two types of reasoning. Inductive reasoning suggests that if some (read ‘one’) elements of a set show a characteristic, others will too. In fact, you SHALL find yourself arguing many a time with exactly the same structure. Stereotypes, such as ‘women cannot drive’, ‘engineers are an intelligent species’, ‘politicians are corrupt’, ‘people with work experience have a better chance of getting into a B-School’ etc. are born of the same category of argumentation. While it is easy to refute such an argument as having a conclusion that you do NOT agree with, do understand that it MAY just be true! It is just that we do not possess information about the rest of the elements of the set and hence cannot say for certainty whether the conclusion will be correct or incorrect.

Also, do not be emotional with the variables- ‘engineers’, ‘fools’, ‘men’ etc. They are just representative and should be deemed as X’s and Y’s.

All said and done, when you are attempting a question, you must always try to look for a ‘logical necessity’ as an answer, not a ‘logical possibility’. We mark LP as an answer only if the answer choices do not HAVE a necessity answer choice in the first place.

One more thing before we move on to the next topic. DO NOT include anything external to the premises in the conclusion. For example, if

P1 – All women are intelligent.

P2 – Sita is a woman.

C – Sita is an intelligent woman.

This is specious reasoning. Our P1 does not state ‘intelligent women’, but simply ‘intelligent’.

Takeaways

- In questions, we are looking for LN’s, not LP’s. We shall mark LP as an answer choice only in the absence of an LN.
- Do not take the variables of the questions to heart, treat them as X’s and Y’s.
- Do not add anything external to the premises in the conclusion.

** There are four basic premises to understand in syllogisms**.

- a) All X are/is Y.
- b) No X are/is Y.
- c) Some X are/is Y.
- d) Some X are/is not Y.

Let us deal with each in totality.

- a)
__All X is/are Y.__

This statement comes under the ‘universal positive’ category. But that is elementary and not worth keeping in mind. What you really need to understand in this statement is that the usage of ‘is/are’ is not important; whatever verb appears here will be independent of interpretation. The simple translation of this statement is that all the elements of set X will also be elements of set Y.

*Another important thing to note is that, even if the statement does not have the prefix ‘all’, (e.g. x is y) it will have the same interpretation*.

Let us also try and understand this with Venn diagrams.

The second diagram shows a possibility that exists, in that the two sets X and Y are overlapping.

While solving questions, you should use the first diagram. And, as I have stated earlier, do not get emotionally involved in trying to picturise the verb. The trick is- find out the verb, then recognize the ‘doer’ of the verb (i.e. the ‘subject’ of the sentence) and put the subject in the inner circle, while the object/rest occupies the outer circle.

For example,

All men are blue.

Here, the verb is ‘are’, and the subject ‘all men’. Hence the set of ‘men’ will be represented by the inner circle, and the set of ‘blue’ by the outer circle.

Sometimes, if one becomes paranoid about being able to picturise stuff, things can get tricky. For example, if the statement were “monkeys have brains”, one would be tempted to draw the outer circle to represent the monkeys. Do NOT be tricked by the verb. Follow the same rule that I have mentioned earlier. The verb here is “have”, the doer of which is “monkeys”. Hence, make the inner circle to represent monkeys and the outer to represent brains.

On a parting note, do remember

- The presence or absence of the prefix “all” does not matter. The statement shall still be treated as mentioned above. Hence “all X is Y”, is the same as “X is Y”.
- Put the subject of the sentence in the ‘inner’ circle.
- For solving a question, use the first diagram. The second diagram is a possibility to be kept in mind for solving CR/ RC questions.

- b)
__No X is/are Y.__

This is a rather simpler statement to understand. It means that no elements of set X are elements of set Y. Simply put, the elements of the two have nothing in common. These can be easily represented as disjoint sets, i.e. two circles, not touching each other anywhere.

However, there are some other important things to learn here. Please understand that this premise (which, incidentally, comes under the ‘universal negative’ category) has some misrepresentations as well. Many people try to represent the opposite of ‘all X is Y’ as ‘all X is NOT y’. Now, this is fallacious, since such a negation becomes dubious to interpret and hence ambiguous. Premises in logic cannot afford to be ambiguous, since it is they who set the stage for the conclusion to follow. You just have to try different emphasis points in this kind of negation to understand what I mean.

*All engineers* are *not fools*. (Implies that no engineer is a fool.)

*All* engineers are not fools. (Implies that only some of them are.)

Since it is semantics at play here, such a negation is considered illogical.

Similarly, a negation of the nature “Not all X are Y” has comparable problems, and hence is not deemed a valid negation of “All X are Y”.

Final takeaways

- The negative of “All X is Y” is “No X is Y”.
- Can be represented by disjoint sets.
- “All X are not Y” / “Not all X are Y” are invalid premises.

- c)
__Some X are/ is Y.__

Unlike the universals we have been looking at so far, where it was either an all or none case, thereby justifying the usage of the word ‘universal’, we now shift our focus to ‘particular’ premises. These premises have prefixes that look like- some, many, a lot of, most et al. Understand that these words have little representative or absolute value, until pitted against their respective ‘whole’ numbers. Hence our comprehension of the same will have to be careful.

Let us then agree to interpret these two statements by concurring that

- If some X are Y, then some Y must definitely be X.
- The interpretation of the prefix “some, many etc.” will be “AT LEAST ONE”.
- If some X are Y, it does not imply that some X are then definitely NOT Y.

The first interpretation is fairly simple to understand. If some elements of set X are also elements of set Y, those same elements are both X and Y. Hence some elements of Y automatically become elements of set X.

The second point, when elaborated, means that in logic, the prefix ‘some’ in itself means nothing except “at least one”. Even if the prefix is ‘a lot, many, most, several’ etc. our interpretation of the same shall remain ‘at least one’.

Now, for the third point, some logic books state that if the premise states ‘some X are y’, then it definitely means that some X are NOT Y. This is bad reasoning. Just as we saw in inductive reasoning erstwhile, if some elements of a set *do* show a certain trait, then we cannot for certainty say EITHER that the rest *will* *not* show the same trait OR that they *will*. Hence, to conclude from ‘Some X are Y’, as a necessity, that ‘Some X are not Y’, is simply not correct. And henceforth you and I shall not indulge in such fallacies.

Time for a Venn interpretation.

The first diagram that I have presented below is what we shall use for solving questions. The rest are just indicative of ‘possibilities’ that may exist, and with which we must familiarise ourselves, for they will help us understand things better when we finally arrive at long CR questions.

In this diagram the shaded portion represents the area in which our ‘at least one X’ and ‘at least one Y’ lie. This is the diagram we shall use for solving questions.

Here, the portion of X that coincides with the portion of Y is our area of concern. Also, please understand that one line of argument may state that here ‘‘aren’t all Y’s, X’s too?’’ To this, a logical response is that our premise concerns itself with some of the X’s being Y’s, not Y’s being X’s. In the process, if all Y’s turn out to be X’s, it is just a possibility, and of course not our primary concern. We had started with trying to prove that at least one X ought to be Y, and the diagram does justice to that. (Remember, we are dealing with all the ‘possibilities’ here.)

Here again, one might point out that all of the X’s are Y’s. However, by now, you and I understand that we had set out to prove that at least one X should be Y, and in the process, if all X’s DO happen to be Y’s, so be it. Our one X is still safely within Set Y, and our diagram, yet again, does full justice to that.

Well, if you understood the previous diagram, you would find it easy enough to understand that this too is a possibility that exists. And again, our one X is still ensconced firmly within Set Y.

Synopsis

- If some X are Y, then some Y must definitely be X.
- The interpretation of the prefix “some, many etc.” will be “AT LEAST ONE”.
- If some X are Y, it does not imply that some X are then definitely NOT Y.
- For solving a question, we shall use the first diagram.

- d)
__Some X are/is not Y.__

This statement has several interpretations across the globe. But we shall treat it as a logically inconsistent premise. Although the statement “Some X are not Y” CAN hold true as a conclusion, it falls flat as a premise. (Hope you remember the distinction between ‘premises’ and ‘conclusion’ well enough by now!)

For instance, let us try with

P1 – Some boys are not mature.

Immediately with this premise you will have to go with three possible diagrams simultaneously, i.e.

P2 – Some mature are fools/ All matures are fools/ No mature is a fool.

You understand that any of these three premises will have different impacts on the three possible diagrams that we have made. With such a scenario we shall NOT be able to arrive at a sustainable conclusion at the third stage. In syllogisms, as you must have noticed earlier, we do arrive at a conclusion at the third stage. Hence, this statement, we shall treat as an ‘illogical’ premise.

However, this statement DOES have validity as a **conclusion**.

For instance,

P1 – Some buckets are trees.

P2 – No tree is a fool

Now, in all of the three possible diagrams you can see that as an LN conclusion, we can safely say that,

Some buckets are not fools. (i.e. the buckets that lie in intersection with trees.)

Takeaways

- Some X are/is not Y is a logically inconsistent premise.
- Some X are/is not Y has an absolutely logical existence as a conclusion.

Finally,

__Only X are/is Y.__

This is the only remaining premise we need to get hold of, so far as syllogisms are concerned.

To begin with, if you encounter a statement such as “Only X are Y”, quickly convert it into “All Y are X”. The diagram should be simple now- Y inside, X outside. For solving a question, this much of dope should be enough.

For the sceptics, however, an explanation is just what the doctor ordered!

So here we go! Let us see if the formula works or not.

P1 – Only boys wear trousers.

If this be our premise, isn’t it easy to figure out that the moment I see someone wearing a pair of trousers, without even looking further, I should be safely able to conclude that the person is a ‘boy’? What I mean is that since the premise explicitly states that only boys can wear trousers, then nobody else can wear them. Therefore if someone is wearing trousers, the person OUGHT to be a boy, else our premise falls. Hence, is it not easy to figure out that ‘All trousers can be worn by boys only’? Well, you’ve got it now!

If, “only boys wear trousers”, then “all trousers are worn by boys”! Simple!

Takeaway

- Convert “only X are Y” to “All Y are X”, and then work with what you have learnt from the “all” prefix statements, i.e. make Y the inner circle and X the outer circle.

**One final word** – While solving questions in syllogisms, do remember that the conclusion should be derived using *both* of the previous *two* premises, and *not one premise alone*. For example,

- All babies are black.
- My baby is cute.

Conclusion- My baby is black.

This is incorrect since the conclusion can be derived using the first premise itself.

Time for you to jump into deeper waters!!

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