Wednesday, 23 September 2009

The Deductive Syllogism

Aristotle believed and taught that the best means of orienting ourselves in the world was through thinking.

He did not reject Plato’s teaching of the world of Forms and Archetypes, but he had some trenchant criticisms to make of it. (E.g. How would the form donkey relate to the form horse? How would either of these relate to the form animal?)

Aristotle taught that there are two kinds of knowledge: that which is arrived at deductively ( i.e. from conditioned to cause; such as footprints in the sand, indicating that someone has walked on it ) and that which is arrived at inductively ( i.e. from consideration of particulars we arrive at the universal; such as if all the carrots you have ever seen are orange, this indicates that all carrots are orange.)

BUT! Can we really be sure that we have seen enough carrots to make the judgment that all carrots are always orange? In other words, inductive thinking is inferior because we can never be sure that we have enumerated all the cases.)

Aristotle gave us the categories which enable us to carry out the processes of thought. These are:
i) substance, ii) quality, iii) quantity, iv) relation, v) place, vi) time, vii) posture,
viii) possession, ix) action, x) passivity

If we have a hallucination or see a mirage, these things are present, but if we ascribe extra-mental reality to them, ( i.e. if we think they are real) it is our judgment that is at fault.

Aristotle and his students worked out a grammar of thinking called logic as a means of testing the reality of statements.

The deductive syllogism is a series of three statements connected in theme. The first statement is the Major Premiss. The second statement is the Minor Premiss. The third statement, the Conclusion, is the only statement that can be made as a logical deduction from the two previous statements.

There are four kinds of statement in logic: the Universal Affirmative (A) the Universal Negative (E) the Particular Affirmative (I) and the Particular Negative (O). These make up the Mood of a syllogism. (See below.)

 Valid Moods

The medieval scholars who developed formal logic used names with which they were familiar to apply to the Valid Moods for each figure.

A Mood is identified by the nature of each statement, whether they are Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative.

Universal Affirmative = A
Universal negative = E
Particular Affirmative = I
Particular Negative = O

All tortoises have shells. (All tortoises in the universe have shells.)                                  A
No fish ride bicycles. (No fish ever in the entire world ride bicycles.)                  E
This bus is red. (Other buses may be other colours, but this one is red.)                        I
This man is not a postman. (Other men may be postmen, but not this one.)                   O

Particular affirmatives might be indicated by e.g. Some, as in “Some women are ballerinas”; “Some guitarists are left-handed”.

There are three figures in logic. (Some logicians have found a fourth figure, though this has not been universally accepted.)

Fig. 1=             M – P             
                        S – M
                        S – P
The valid moods for Fig. 1 are:- BARBARA, DARII, CELARENT, FERIO

In Fig. 1 the Major Premiss is always Universal (an A or an E type) the Minor Premiss is always affirmative. (An A or an I type.)

Fig. 2=             P – M
                        S – M
                        S – P
The valid moods for Fig. 2 are:- BAROCO, CAMESTRES, CESARE, FESTINO

In Fig. 2 the Major Premiss is always Universal, (an A or an E type) and one premiss is always negative. (An E or an O type).

Fig. 3=             M – P
                        M – S
                        S – P
The valid moods for Fig. 3 are:- BOCADO, DARAPTI, DATISI, DISAMIS,         FELAPTON, FERISON

In Fig. 3 the Middle Term is the subject in both premises. The Minor Premiss is always affirmative, (A or I) and the Conclusion is always particular (I or O).


Fig. 1

All neap tides are ebb tides. (Major Premiss) Universal Affirmative.                       A
This is a neap tide. (Minor Premiss) Particular Affirmative.                         I
:. This is an ebb tide. (Conclusion) Particular Affirmative.                                       I

Fig. 2

All soldiers learn to shoot. (Major Premiss) Universal Affirmative.              A
No milkmen learn to shoot. (Minor Premiss) Universal Negative.                            E
:. No milkmen are soldiers. (Conclusion) Universal Negative.                                  E

Fig. 3

All trumpets are made of brass. (Major Premiss) Universal Affirmative.                 A
All trumpets are musical instruments. (Minor Premiss) Universal Affirmative.       A
:. Some musical instruments are made of brass. (Conclusion) Particular Affirmative.                                                                                                                    I

Analysing the Syllogism

We start with the Conclusion.
As in grammar, we identify the Subject  (S) and the Predicate. (P)
 - Here the resemblance to grammar ends! -

In the Minor Premiss, we try to identify where the Subject (S) occurs.
What is left is the Middle Term (M).

In the Major Premiss, we identify where the Middle Term and the Predicate occur.             
For example, in Fig. 1“This” is the Subject.
In Fig. 2 “soldiers” is the Predicate
In Fig. 3 “trumpets” is the Middle Term.

If the mood does not fit the figure, the syllogism is no good.

For instance:-

All lobsters are red.                                                   A
All communists are red.                                             A
:. All communists are lobsters.                                  A

Conclusion: “All communists” = Subject.
“are lobsters” = Predicate

Minor Premiss “All communists” =  Subject
“are red” = Middle Term

Major Premiss “All lobsters” = Predicate
“are red” =  Middle Term

Fig =    P – M
            S – M
            S – P

So this is Fig. 2           


The mood is AAA (BARBARA) and NOT a valid mood for Fig. 2. So the syllogism is no good.

Consider this:-

God is that than which no greater can be thought.

But that than which no greater can be thought must exist, not only mentally, in idea, but extra-mentally.

:. God must exist, not only mentally, in idea, but extra-mentally.

This was a proof given by St. Anselm in the 11th Century for the existence of God. Does it work out?

Lewis Carrol, the author of Alice in Wonderland, turned the deductive syllogism into a board game, and tried to explain it to the 11year-old Alice Liddell, who promptly fell asleep.

With the basics as you have them, can you work out how this might be turned into a board game with counters of different colours, etc? (N.B. There might be some money in this!)

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