## Logic – Aristotle & Gotama

Aristotle in ancient Greece and Gotama in ancient India examined structures of arguments, and each came up with similar conclusions.  Aristotle’s famous example of the primary form of deductive argument is:

All men are mortal.

Socrates is a man.

Therefore Socrates is mortal.

It should be pointed out that Socrates was mortal, as he is now no more.  Similarly, in the Nyaya Sutra, Gotama’s example of the primary form of deductive argument is:

Wherever there is smoke, there is fire, as in a kitchen.

Because there is smoke on the hill, there is fire on the hill.

Both of these share the same structure, though we must rearrange each argument to show what this structure is.  First, we must recognize that both are making statements about objects and categories.  Just as nouns can be persons, places or things, individual objects can be either living individuals, such as Socrates or Gotama, places, such as a hill or kitchen, or things, such as a hammer or textbook.  Categories are groups of objects that are similar in some way, such as men, mortals, places there are smoke, and places there are fire.  We can think of objects as categories that contain one single person, place or thing.

Aristotle realized that we can symbolize each object or category with a single alphabetical letter, such as A, B or C.  Later, inspired by Aristotle and Indian mathematicians, who used a dot to stand for an unknown value, Islamic logicians and mathematicians invented algebra, which uses alphabetical letters to stand for unknown values.  Let us rearrange Aristotle’s argument as such:

Socrates is a man.

All men are mortal.

Therefore, Socrates is mortal.

If we symbolize the individual Socrates as A, the category of men as B, and the category of objects that are mortal as C, we have:

A is B.

All B is C.

Therefore, A is C.

What we are arguing is that A is in category B, category B is contained in category C, so A must also be contained in category C.  If we use solid dots for objects, and circles for categories, this can be illustrated as such.

Similarly, if we symbolize Gotama’s hill as A, the category of places where there is smoke as B, and the category of places where there is fire as C, this can be illustrated as such.

Notice that Gotama adds the example of a kitchen to strengthen his argument inductively, providing another example as evidence, but this is in addition to his deductive argument.  We can arrange Gotama’s argument to read:

The hill is a place where there is smoke.

Wherever there is smoke, there is fire.

Therefore, the hill is a place where there is fire.

Recall, from the previous chapter, these two examples of argument:

Whoever can use logic can think rationally.

All people can use logic.

Therefore, all people can think rationally.

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All puppies are green.

Everything that is green is evil.

Therefore, puppies are evil.

Both of these arguments have the same structure as Aristotle and Gotama’s primary arguments.  Each is arguing that because A is contained in B, and B is contained in C, A must be contained in C.  Notice that for Aristotle and Gotama’s primary examples, A is an individual object, but for the two examples above, A is a category of objects.  The structure works the same either way.  Using categories, which can be single objects, the argument’s structure is:

All A is/are B

All B is/are C

All A is/are C

This is illustrated similarly as before, but now A is a category that can contain objects, rather than a single object itself.

Medieval European logicians who studied Aristotle called this form Barbara.  This is the first of what they called the four perfect forms of Aristotle, which are Barbara, Celarent, Darii, and Ferio.  Notice that these four start with the first four consonants (as opposed to vowels such as A and E) of the alphabet.

Exercise 2.1: Create four Barbara arguments, as creative or bizarre as you like.

Aristotle argued that all statements fall into four categories:

1) Positive Universal: All A is/are B (ex: All pears are fruit)

2) Negative Universal: No A is/are B (ex: No cows are aliens)

3) Positive Particular: Some A is/are B (ex: Some cars are blue)

4) Negative Particular: Some A is/are not B (ex: Some cars are not blue)

Medieval European logicians illustrated the differences and similarities of these four types of statements with a diagram known as the Square of Opposition:

Notice that Barbara is composed of three positive universal statements.

Celarent, like Barbara, is entirely composed of universal statements, but unlike Barbara, Celarent contains two negative universal statements.  The form of Celarent is:

All A is/are B

No B is/are C

No A is/are C

Aristotle gives us the example, “If all humans are animals, and no animals are made of stone, then no humans are made of stone”.  Similarly, we could reason that if all cats fly spaceships, and everything that flies a spaceship is a rock star, then all cats are rock stars.

Note that if A is entirely contained in B, and if B and C are mutually exclusive, not overlapping in any part, then it is impossible for any part of A to overlap with any part of C, and so A and C must also be mutually exclusive.

Exercise 2.2: Create four Celarent arguments, as creative or bizarre as you like.

Unlike Barbara and Celarent, Darii and Ferio contain particular statements.  Darii begins and ends with positive particular statements, joined together by a positive universal statement.  The form of Darii is:

Some A is/are B

All B is/are C

Some A is/are C

For example, we can reason that if some movies are funny, and if all things that are funny make us laugh, then some movies make us laugh.  We can also reason that if some gorillas play the kazoo, and if all who play the kazoo can speak German, then some gorillas can speak German.

Notice that A and B overlap, such that some A is some B.  Because all of B is contained in C, this means that the part of B that overlaps with A must also overlap with C.

Exercise 2.3: Create four Darii arguments, as creative or bizarre as you like.

Ferio is the only one of the four perfect forms that contains a negative universal statement, which is also its conclusion.  Like Darii, Ferio begins and ends with particular statements joined together by a universal statement, but unlike Darii, Ferio contains a negative universal statement rather than a positive universal statement, and so even though it begins with a positive particular statement it has a negative particular statement as its conclusion.  The form of Ferio is:

Some A is/are B

No B is/are C

Some A is/are not C

For example, we can reason that if some animals are humans, and no humans are reptiles, then some animals are not reptiles.  We could also reason that if some robots are Japanese, and if nothing that is Japanese is made of armadillos, then we can conclude that at least some robots are not made of armadillos.

Notice that A and B overlap, such that some A is some B, but because B and C are mutually exclusive, and do not overlap in any part, then there must be some part of A that does not overlap with some part of C.  A and C are not necessarily mutually exclusive, as it is possible that they overlap in some part other than the part of A and B that overlap, but these two parts could not possibly overlap with each other.  To use our first example, it is true that some animals are reptiles even though some animals, such as humans, are not reptiles.  To use our second example, though we know that Japanese robots are not made of armadillos, this does not mean that no robots are made of armadillos, as Spanish robots could possibly be made of armadillos.

Exercise 2.4: Create four Ferio arguments, as creative or bizarre as you like.

There is another form of argument that is central for Gotama and the Nyaya school that involves negative statements, but we will cover it in the next chapter with conditionals.