Syllogism

This is the core question of the chapter: two statements, two conclusions, and four options, Only I, Only II, Both, Neither. In the latest SSC and Railway papers it is regularly asked, so make it your first strength.

Why do two statements join at all? Take Some pens are books and Some books are files. The books that are pens and the books that are files may be different books, so pens and files are not linked. Now take Some pens are books and All books are files. Every book is a file, so the books that are pens are files too, and the link is real. The middle class (the one common to both statements) must be spoken of as a whole, that is Full, in at least one statement. Otherwise the statements talk about different members and do not join.

Method: (1) underline the middle class; (2) tag it in both statements; (3) read the polarity: both positive give positive, one negative gives negative, both negative give nothing; (4) tag the two end classes from their own statements; (5) test each conclusion: All X are Y needs X Full, No X is Y needs both Full, Some X are Y needs nothing special, Some X are not Y needs Y Full.

Take: Some pens are books. All books are registers. The middle class is books, because it appears in both statements. In the first statement books is Part. In the second it is Full. One Full tag is enough, so the statements connect.

Both statements are positive, so the result is positive. Now the end classes: pens is Part (from the first statement) and registers is Part (from the second). A positive result with two Part ends gives Some pens are registers and its reverse. It cannot give "All pens are registers", because that needs pens = Full.

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If the middle class is Part in both statements, nothing connects. Then the answer is "Neither follows", with one exception: a conclusion that is just the reverse of a single statement. From "Some A are B. Some B are C", the conclusion "Some C are B" follows because it reverses the second statement alone.

Same first class. Sometimes both statements begin with the same class: All mangoes are fruits. No mango is a vegetable. Mangoes is Full in both, so the statements join. The result is negative, with fruits Part and vegetables Full. So Some fruits are not vegetables follows, but "Some vegetables are fruits" and "All fruits are vegetables" do not.

Letters and facts that clash. The classes may be letters ("All P are Q") or may clash with real life ("All stones are birds"). Accept every statement as true, and never use outside knowledge to accept or reject a conclusion. If the names distract you, cover them with letters in your rough work, and run the same tags and the same joiner.

When three or four statements are given, do not panic. A long set is only a chain of two-statement joins.

Write the chain on one line and put the Full / Part tag under every class. Join the neighbours first. A joined result is a new statement, so join it again with the next one. And remember: every pair you join can give conclusions, not only the first and last classes. Collect all the definite conclusions in one list, then match the options.

Take: All pens are books. All books are files. No file is a card.

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Pens and files: books is Part in the first statement and Full in the second, so All pens are files. Now join this with "No file is a card": files is Part in the result and Full in the No-statement, so they connect. The result is negative, and pens = Full, cards = Full. So No pen is a card, and therefore also "No card is a pen", "Some pens are not cards" and "Some cards are not pens".

Do not stop there. Books and cards: files is Part in "All books are files" and Full in the No-statement, so No book is a card, and Some books are not cards. These middle conclusions are just as valid.

Two things end a chain. First, a negative result joined to another negative statement gives nothing. Second, a "Some" end stays "Some" all the way: a Part class can never become Full later in the chain, so no "All" conclusion can be built on it.

"If ... then" statements and named items [New Pattern]. A statement such as "If a product is defective, it fails quality control" is simply All defective products fail quality control. A named item such as "Product X" is a class with one member, and it is Full when it stands first. So "Product X failed quality control" says that Product X lies wholly inside the products that failed. Join as usual: if failing the check leads to recall, Product X is recalled. But "Product X is defective" does not follow, because "fails quality control" is only Part in the first statement. "Product X may or may not be defective" is a fair statement, since neither "is" nor "is not" is forced.

Join neighbours, keep every result, and stop the moment two negatives meet.

A newer form of question writes its conclusions like this: "All gases being elements is a possibility" or "Some keys being doors is a possibility". Such a conclusion is not asking whether something is certain. It asks whether it can be true.

The test is simple. A possibility conclusion is true if at least one diagram that obeys the statements makes it true. It is false only when every such diagram rules it out.

So the method has two steps. Step 1: find the definite relation between the two classes in the conclusion, using the joiner. Step 2: look up the asked relation in the conversion lists. If it is in the "possible" list, or is the definite relation itself, it is a possibility. If it is in the "false" list, it is not.

The same question hides behind other words: "may be", "can be", "is possible", "might". "Some cars may be eco-friendly" is a possibility conclusion too. Note that a definite relation always passes the possibility test, because something true in every diagram is true in at least one.

Take: All keys are locks. No lock is a door. Keys lie inside locks, and locks are apart from doors, so keys are apart from doors. "Some keys are doors" is therefore not possible. But the locks circle can be bigger than the keys circle, so "Some locks are not keys" is possible.

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The picture shows both. In diagram 1 locks are larger than keys. In diagram 2 they are the same. The door circle stays apart from the keys in both, because the "No" statement is a wall.

If the statements give no link at all between two classes, every form involving them is possible. And remember, a possibility must fit with all the statements together, so in longer sets check the whole diagram.

Possible = true in at least one diagram. A "No" link between the classes kills every positive possibility.

Sometimes neither of two conclusions follows on its own, yet one of them must be true. This is the either-or case. Think of a coin hidden in one of two closed hands. You cannot say which hand holds it, but you are sure that one does.

Three conditions must hold together.

• Neither conclusion follows on its own.
• Both are about the same two classes.
• They are a complementary pair, which means that in every possible diagram exactly one of them is true.

There are only two complementary pairs: (All A are B, Some A are not B) and (Some A are B, No A is B).

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Look at the picture. From "Some X are M. Some M are Y" nothing connects, because M is Part in both. Diagram 1 has X and Y overlapping, so "Some X are Y" is true. Diagram 2 keeps X and Y apart, so "No X is Y" is true. Every diagram makes one of the two true and the other false. That is why the pair is either-or.

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Now see why the other two pairs fail. All and No can both be false: if X and Y overlap partly, neither "All X are Y" nor "No X is Y" holds. Some and Some-not can both be true: partial overlap makes both true.

Keep the order of the classes. "All A are B" pairs with "Some A are not B", not with "Some B are not A". With "Some" and "No" the order does not matter, because both read the same either way round.

If a third conclusion is present, the correct option reads like "Only III and either I or II follow".

Either-or = same two classes + neither follows alone + (All, Some-not) or (Some, No).

In this form there is no list of conclusions with "Only I follows". The four options are the conclusions themselves, and you pick one. The question words tell you what to pick: "which must be true", "which can be concluded", "which does not follow", "which is definitely false". It works with one statement ("All kites are toys"), with two ("All novels are books, some books are boring"), and sometimes with a named person ("All men are mortal. Socrates is a man.").

Label every option with one of three letters. D (definite) means true in every diagram. P (possible) means true in some diagrams only. X (impossible) means false in every diagram. Then obey the question word: "must be true" picks D; "does not follow" picks P or X; "definitely false" or "cannot be true" picks X only.

Take: Some pens are books. All books are files. No file is a card. Joining gives Some pens are files, and No book is a card.

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Test three conclusions. "Some pens are books" is given, so it is definite (tick). "Some pens are cards" is only possible, because the pens that are not books are not described at all, and they may be cards (question mark). "Some books are cards" is impossible, because books lie inside files and files are apart from cards (cross).

So if the question says "does not follow", both of the last two qualify. If it says "definitely false", only the last one does. A well-set question has exactly one option that fits its wording, so read the wording first.

Reverse questions (rare) print the conclusions first and make the four options pairs of statements. Test every option on its own, and test every conclusion in it. Never mix a statement from one option with a statement from another. Work backwards to save time: a negative conclusion needs exactly one negative statement; a positive one needs both positive; "All X are Y" needs X to be the subject of an All; "No X is Y" needs both ends Full; and a pair whose middle class is Part in both, or which shares no class, is out at once.

Take the conclusions: I. No hat is a scarf. II. Some scarves are not hats. Suppose option (a) is "Some hats are caps; No cap is a scarf" and option (b) is "All hats are caps; No cap is a scarf".

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Recent papers hide the four forms inside special wording: "Most of the engineering students are adults", "25% Hit are Hop", "Only Car are Vehicle", "Only a few Youth is Young", or a conclusion such as "All Flat can never be Truck". Do not solve these as they stand. Convert first, then use the joiner.

Why do "most" and "25%" differ from "few" and "several"? An exact percentage strictly between 0% and 100% (25%, 40%, 75%) names a part of the class that is neither none nor all, so both "Some A are B" and "Some A are not B" are certain. (0% means No A is B; 100% means All A are B.) "Most" only says that more than half are, so it gives Some A are B; "Some A are not B" follows from "most" only when the statement says so, as in "most but not all". "A few", "several" and "a fraction of" only say that some are, so keep them plain Some. In every case Some never becomes All: no wording of this kind gives an "All" conclusion.

"Only A are B" means that every B is an A: All B are A. "Only teachers are poets" does not mean that every teacher is a poet. "Only a few A are B" says two things at once: Some A are B and Some A are not B (and by reversal Some B are A). Plain "a few" is only Some.

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After converting, treat each part as its own statement. The Some part joins with the other statements as usual. The Some-not part is a ready conclusion about its own two classes; it reaches a third class only when an All-statement points into the class after "not". Take: Only a few lamps are bulbs. All bulbs are tubes. Some lamps are bulbs and all bulbs are tubes, so Some lamps are tubes. But "Some lamps are not tubes" does not follow, because the lamps that are not bulbs are not described by the second statement.

A conclusion worded "All A can never be B" is not "No A is B". It says that it is impossible for all A to be B, which is Some A are not B. It follows only when "Some A are not B" follows.

Convert first: a percentage and Only a few give Some and Some not, while Most gives Some; Only A are B turns around into All B are A.