Lesson 14 · The syllogism III: Figures 2, 3 and 4

Sources: Book pp. 45–56; S5 slides 11–13, 15.

Figure 1's conclusions are self-evident. The other figures are valid too, but each needs to be shown valid, usually by turning it into Figure 1. There are three ways to do that:

Conversion (ʿaks): convert a premise (Lesson 11), sometimes also swap the premises or convert the conclusion. This is what S5 slide 15 means by making a proposition "opposite".
Khalf (refuting the contradictory): assume the conclusion is false, take its contradictory, join it to one premise in Figure 1, and derive something that contradicts the other premise.
Iftirāḍ (supposition): give a name, D, to the "some" in a particular premise, and reason about D with universal premises.

1. Figure 2: the middle is the predicate of both

Form: S – M, P – M.

Conditions

The premises must differ in quality (one affirmative, one negative). Two things can share a predicate (human and horse are both animals) and still be distinct, or be the same (human and rational are both animals).
The major premise must be universal.

So Figure 2 always concludes negatively.

Mood	Form	Example	Into Figure 1
1. A + E → E	Every S is M; no P is M; ∴ no S is P	Every ruminant has hooves; no bird has hooves; ∴ no ruminant is a bird.	convert the major (no M is P) → Fig. 1 mood 2
2. E + A → E	No S is M; every P is M; ∴ no S is P	No contingent thing is eternal; every necessary being is eternal; ∴ no contingent thing is a necessary being.	convert the minor, swap the premises, then convert the conclusion
3. I + E → O	Some S is M; no P is M; ∴ not every S is P	Some metal is gold; no silver is gold; ∴ not every metal is silver.	convert the major → Fig. 1 mood 4
4. O + A → O	Not every S is M; every P is M; ∴ not every S is P	Not every object is metal; all gold is metal; ∴ not every object is gold.	khalf (see below)

Mood 4 by khalf. Suppose the conclusion is false; then its contradictory is true: "Every object is gold." Join it to the major in Figure 1: "Every object is gold; all gold is metal; ∴ every object is metal." But the minor premise says "not every object is metal". The supposition led to a contradiction, so it is false, and the conclusion stands.

S5 slide 11 (Figure 2, mood 1): "Everything between the navel and the thigh is ʿawra. Nothing that may be looked at is ʿawra. ∴ Nothing between the navel and the thigh may be looked at."

ʿAqīda link: al-Naḥl 16:17. "Is He who creates like one who does not create? Will you not then remember?" Laid out as Figure 2, mood 1:

Allah creates. None of the idols creates. ∴ Allah is not one of the idols (nor like them).

Book slips: p. 48, mood 2's conclusion is printed "No potentiality is eternal" (read: "...is a Truth"), and mood 3's is printed "Some metals are not gold" (read: "Not every metal is silver").

2. Figure 3: the middle is the subject of both

Form: M – S, M – P.

Conditions

The minor premise must be affirmative.
At least one premise must be universal.

Figure 3 always concludes particularly, even from two universals, because the minor term may be wider than the middle.

Mood	Form	Example	Into Figure 1
1. A + A → I	Every M is S; every M is P; ∴ some S is P	All gold is metal; all gold is precious; ∴ some metal is precious.	convert the minor → mood 3
2. A + E → O	Every M is S; no M is P; ∴ not every S is P	All gold is metal; no gold is silver; ∴ not every metal is silver.	convert the minor → mood 4
3. I + A → I	Some M is S; every M is P; ∴ some S is P	Some birds are white; every bird is an animal; ∴ some white things are animals.	convert the minor → mood 3
4. A + I → I	Every M is S; some M is P; ∴ some S is P	Every bird is an animal; some birds are white; ∴ some animals are white.	convert the major, swap, convert the conclusion
5. A + O → O	Every M is S; not every M is P; ∴ not every S is P	Every animal senses; not every animal is human; ∴ not every sensing thing is human.	khalf or iftirāḍ
6. I + E → O	Some M is S; no M is P; ∴ not every S is P	Some animals are white; no animal is a stone; ∴ not every white thing is a stone.	convert the minor → mood 4

Mood 5 by iftirāḍ. Let D be the animals that are not human (dogs, for example). Then two things are certainly true: "Every D is an animal" and "No D is human". From "Every D is an animal; every animal senses" (Figure 1) we get "Every D senses". Then "Every D senses; no D is human" (Figure 3, mood 2) gives "Not every sensing thing is human".

S5 slide 12 (Figure 3, mood 1): "Every thief is a fraud. Every thief has his hand cut. ∴ Some frauds have their hands cut." Only "some", because not every fraud is a thief.

Book slips: p. 51, mood 2's reduction prints "No B are A" (read "No B is C"), and mood 3's conclusion is printed "Some birds are animals" (read "Some white things are animals"); p. 52, mood 6's conclusion is printed "Some metals are not gold" (read "Not every metal is iron").

3. Figure 4: subject of the minor, predicate of the major

Form: M – S, P – M. It is Figure 1 back to front, which makes it the hardest to see.

Conditions (besides the general rules)

Neither premise may be an O.
If both premises are affirmative, the minor must be universal.

Mood	Form	Example	Into Figure 1
1. A + A → I	Every M is S; every P is M; ∴ some S is P	Every human is an animal; every rational being is human; ∴ some animals are rational.	swap the premises, convert the conclusion
2. A + I → I	Every M is S; some P is M; ∴ some S is P	Every human is mortal; some Arabs are human; ∴ some mortals are Arabs.	swap, convert the conclusion
3. E + A → E	No M is S; every P is M; ∴ no S is P	No stone is alive; every pebble is a stone; ∴ nothing alive is a pebble.	swap, convert the conclusion
4. A + E → O	Every M is S; no P is M; ∴ not every S is P	Every liquid evaporates; no metal is a liquid; ∴ not everything that evaporates is a metal.	convert both premises → mood 4
5. I + E → O	Some M is S; no P is M; ∴ not every S is P	Some liquids evaporate; no metal is a liquid; ∴ not everything that evaporates is a metal.	convert both premises → mood 4

(Moods 4 and 5 cannot be fixed by swapping, because the swapped Figure 1 would have a negative minor premise.)

S5 slide 13 (Figure 4, mood 4): "Every intentional eating breaks the fast. Nothing inhaled is intentional eating. ∴ Not everything that breaks the fast is inhaled." (The slide words the conclusion as "Some that cancels fasting is nothing inhaled".)

4. Summary of all four figures

Figure	Middle term	Conditions	Moods	Conclusions
1	predicate of minor, subject of major	minor affirmative; major universal	4	A, E, I, O
2	predicate of both	differ in quality; major universal	4	E, O (always negative)
3	subject of both	minor affirmative; one premise universal	6	I, O (always particular)
4	subject of minor, predicate of major	no O premise; if both affirmative, minor universal	5	I, E, O

That is 19 productive moods in all. S5 slide 15's summary of how to reach Figure 1: Figure 2, convert the major premise; Figure 3, convert the minor premise; Figure 4, swap the premises, or convert both of them. (A few moods need khalf instead, as shown above.)

Quiz 14

Name the figure, check the conditions, and give the conclusion (if any):
- a) "Every prophet is truthful; no liar is truthful."
- b) "Every martyr is alive with his Lord; every martyr died in Allah's path."
- c) "Every believer loves the Prophet ﷺ; some who love the Prophet ﷺ fall short."
- d) "No idol hears; every god hears."
- e) "Some animals are not human; every scholar is human."
Why must Figure 2's premises differ in quality? Give a pair of same-quality premises that fails.
Why does Figure 3 never give a universal conclusion?
Reduce Figure 2, mood 1 ("Every S is M; no P is M") to Figure 1. Which mood do you get?
Prove Figure 2, mood 4 ("Not every object is metal; all gold is metal; ∴ not every object is gold") by khalf.
Which mood of which figure is S5 slide 13's fasting example? Reduce it to Figure 1.
Review: "Every intoxicant is forbidden. Nabīdh intoxicates." Which figure and mood, and what is the conclusion?

Show answers

a) Figure 2, mood 1 (A + E): "No prophet is a liar." · b) Figure 3, mood 1 (A + A): "Some who are alive with their Lord died in Allah's path." · c) The middle term "loves the Prophet ﷺ" is the predicate of the minor and the subject of the major: Figure 1. But the major is particular, breaking Figure 1's condition: no conclusion. · d) Figure 2, mood 2 (E + A): "No idol is a god." · e) The middle ("human") is the predicate of both: Figure 2. But the negative premise is the minor and the affirmative major is universal, so mood 4 (O + A): "Not every animal is a scholar."
Because two different things can share a predicate (or share its absence) whether or not they are the same. "Every human is an animal; every horse is an animal" gives nothing.
Because the minor term may be wider than the middle: "Every M is S" leaves room for S's that are not M, so we can only speak of some S.
Convert the major "No P is M" to "No M is P": "Every S is M; no M is P; ∴ no S is P", Figure 1 mood 2.
Suppose "Every object is gold". With "all gold is metal" (Figure 1) this gives "Every object is metal", which contradicts the premise "Not every object is metal". So the supposition is false, and "Not every object is gold" is true.
Figure 4, mood 4 (A + E → O). Convert both premises: "Some things that break the fast are intentional eating; no intentional eating is inhaled; ∴ not everything that breaks the fast is inhaled": Figure 1, mood 4.
Figure 1, mood 1 (the minor, "nabīdh intoxicates", is singular in effect): "Nabīdh is forbidden."