9.5 Deductive Reasoning, Syllogisms, and Venn Diagrams

9.5 Deductive Reasoning, Syllogisms, and Venn Diagrams

The logical-reasoning items in the AdUCET aptitude test ask one narrow question: does the conclusion have to follow from the statements given? Not whether it sounds sensible, not whether it is true of the real Philippines, only whether the structure of the argument forces it. Nearly every mark lost on these items is lost the same way, by importing outside knowledge instead of reading the form.

Deduction versus induction

Deductive reasoning runs from a general rule down to a particular case, and it delivers certainty: if the premises are true, the conclusion cannot be false. Inductive reasoning runs the opposite way, from observed cases up to a general rule, and it delivers only probability.

'Every jeepney I have ridden along España Boulevard was painted red, therefore all jeepneys are red' is inductive. The evidence supports the claim without guaranteeing it, and one blue jeepney ends the argument.

Validity is not truth

An argument is valid when its form guarantees the conclusion whatever the facts happen to be. 'All jeepneys are submarines; all submarines are purple; therefore all jeepneys are purple' is valid and entirely false. An argument is sound only when it is valid and every premise is actually true. Examiners plant deliberately silly premises precisely to see whether you reason from form, so never reject a conclusion just because a premise offends common sense.

The four categorical propositions

Every categorical statement reduces to one of four types, labelled A, E, I and O. A term is distributed when the statement makes a claim about every member of that class.

Memory hook: universals distribute the subject; negatives distribute the predicate. 'All Adamson freshmen are enrolled students' tells you something about every freshman, so the subject is distributed, but it tells you nothing about every enrolled student in Manila, so the predicate is not.

Anatomy of a categorical syllogism

Three statements, three terms, each term used exactly twice.

• Major term (P) — the predicate of the conclusion; it appears in the major premise.
• Minor term (S) — the subject of the conclusion; it appears in the minor premise.
• Middle term (M) — appears in both premises and never in the conclusion; it is the bridge that links S to P.

Five working rules of validity

1. The middle term must be distributed at least once. If it is not, the argument commits the undistributed middle.
2. Any term distributed in the conclusion must already be distributed in its own premise. Breaking this for the major term is an illicit major; for the minor term, an illicit minor.
3. Two negative premises yield nothing. If both premises are E or O, no conclusion whatsoever follows.
4. If exactly one premise is negative, the conclusion must be negative; and a negative conclusion demands a negative premise.
5. If either premise is particular (I or O), the conclusion must be particular. Two universal premises never license a particular conclusion under the strict reading, because a universal statement never promises that anything exists.

Worked determination 1 — a valid syllogism

Major premise: All engineering freshmen take Calculus 1. Minor premise: Some AdUCET passers are engineering freshmen. Conclusion: Some AdUCET passers take Calculus 1.

Label the terms first. P = people who take Calculus 1 (predicate of the conclusion). S = AdUCET passers (subject of the conclusion). M = engineering freshmen (in both premises, in neither conclusion slot). The premise pattern is A then I.

1. M is the subject of 'All M are P', an A proposition, so M is distributed. Rule 1 passes.
2. The conclusion is an I proposition, which distributes neither term, so no term can be illicitly distributed. Rule 2 passes.
3. Neither premise is negative, so rules 3 and 4 pass and the conclusion is correctly affirmative.
4. One premise is particular and the conclusion is particular. Rule 5 passes.

Verdict: valid. The Venn check agrees. Shade the part of M lying outside P (that is what 'All M are P' means), then place the X for 'Some S are M' in the S-and-M overlap. The only part of that overlap still unshaded is the region inside all three circles, so the X lands inside P automatically — and 'Some S are P' is read off without adding a single new mark.

Worked determination 2 — an invalid syllogism

Premise 1: All varsity athletes receive a tuition discount. Premise 2: All Dean's listers receive a tuition discount. Conclusion: All Dean's listers are varsity athletes.

The middle term is 'people who receive a tuition discount'. It is the predicate of two A propositions, so it is undistributed both times and rule 1 fails at once. Verdict: invalid — undistributed middle.

The counter-model makes the failure concrete. A university may grant tuition discounts to varsity athletes, to Dean's listers and to employees' children, with not one athlete on the Dean's list. Both premises are true and the conclusion is false, which is exactly what invalidity means. On the diagram, shading the athlete region outside 'discount' and the Dean's-list region outside 'discount' leaves the 'Dean's lister but not athlete' region blank — neither shaded nor marked — so nothing is forced.

Named formal fallacies

Conditional statements and their relatives

Take the statement 'If Marisol passes the AdUCET, then she may enrol at Adamson', written p implies q.

Only the contrapositive is logically equivalent. The converse and inverse are equivalent to each other but not to the original, since each is the contrapositive of the other. Two argument forms are valid: modus ponens (p implies q, p is true, therefore q) and modus tollens (p implies q, q is false, therefore p is false). The two invalid forms are exactly the converse and inverse read as arguments — affirming the consequent and denying the antecedent.

Testing with Venn diagrams

Two conventions carry everything: shading means the region is empty, and an X means at least one thing lives there.

  (a) All S are P                (b) No S are P                (c) Some S are P
   +------+------+------+         +------+------+------+         +------+------+------+
   | #### |      |      |         |      | #### |      |         |      |      |      |
   | #### | S&P  |  P   |         |  S   | #### |  P   |         |  S   |  X   |  P   |
   | #### |      | only |         | only | #### | only |         | only |      | only |
   +------+------+------+         +------+------+------+         +------+------+------+
    S only

   #### = shaded, meaning that region is EMPTY
   X    = at least one member exists in that region

Those two-circle pictures also settle immediate inferences. 'No S are P' may be converted to 'No P are S', and 'Some S are P' to 'Some P are S', because both diagrams are symmetric. 'All S are P' may not be converted to 'All P are S' — the shading sits on one side only.

For a full syllogism, draw three overlapping circles and name the seven regions before marking anything: S only, P only, M only, S-and-P without M, S-and-M without P, P-and-M without S, and the central region inside all three. Then follow two procedural rules:

1. Diagram the universal premises (A and E) first, because their shading decides where a later X can legally go.
2. Read, never add. The syllogism is valid only if the conclusion is already visible on the finished diagram without one extra stroke. If you would have to invent a placement, it is invalid.

When a premise pushes an X toward two still-open regions and shading has not decided between them, place the X on the boundary line between them. A boundary X supports no definite conclusion about either region.

Either-or and complementary conclusions

Some items give two candidate conclusions and ask which one follows. Occasionally neither follows on its own, yet the two are complementary, meaning logic guarantees that one of them is true. The answer is then 'either I or II follows'.

Worked example. Statements: all Vincentian scholars are Adamson students; some Adamson students are varsity athletes. Conclusion I: some Vincentian scholars are varsity athletes. Conclusion II: no Vincentian scholars are varsity athletes. The middle term 'Adamson students' is the predicate of an A and the subject of an I, undistributed twice, so neither conclusion follows by itself. But I and II are an E–I contradictory pair, so exactly one of them must be true. The correct response is either I or II follows. Examiners apply the convention that every named class has at least one member, which is what makes the I–O row usable.

What examiners test

• Rejecting a valid argument because a premise is factually silly, or accepting an invalid one because the conclusion is factually true.
• Reading 'some' as 'some but not all'. In logic, 'some' means at least one, and it is perfectly compatible with 'all'.
• Confusing the converse with the contrapositive, which is the single most common conditional error.
• Forgetting that two negative premises license nothing at all, and that one negative premise forces a negative conclusion.
• Drawing the particular premise before the universal one, which lets an X land in a region that later shading proves empty.