# 21.1 Syllogisms in Prolog: Question

## Introduction to syllogisms

For those who need to refresh their memories on what a syllogism is, here is a little reminder.

Syllogisms make use of four kinds of two-place predicates:

 Kind Syntax Example Universal affirmative all(S,P) All whales are mammals Universal negative no(S,P) No bird is a mammal Particular affirmative some(S,P) Some rhino is a wall-paper eater Particular negative some_not(S,P) Some rhino is not grey

A syllogism consists of two premises and a conclusion. Both of the premises, as well as the conclusion, are predicates of the kind described above.

There are four figures to choose from:

 ```1st figure: y(M,P) x(S,M) Therefore: z(S,P) ``` ```3rd figure: y(M,P) x(M,S) Therefore: z(S,P) ``` ```2nd figure: y(P,M) x(S,M) Therefore: z(S,P) ``` ```4th figure: y(P,M) x(M,S) Therefore: z(S,P) ```

Here, x, y, and z are predicates of the kind described. S, P, and M are arguments such as "mammals", "birds", "wallpaper-eaters".

Syllogisms are named by Latin words, each of which has three of the vowels a, e, i, and o. Each of the vovels corresponds to one of the four kinds of predicates:

• a: Univeral affirmative
• e: Universal negative
• i: Particular affirmative
• o: Particular negative

The vowels 'a' and 'i' come from the Latin word 'affirmo', which means 'I affirm', while the vowels 'e' and 'o' come from the Latin word 'nego', which means 'I deny'.

For example, the syllogism "Darii" is a syllogism from the 1st figure, and contains a universal affirmative ('a') as the first premise, and a particular affirmative ('i') for its psecond premise as well as its conclusion:

 Darii Premise 1: all(M,P) All mammals are breast-feeders Premise 2: some(S,M) Some whales are mammals Conclusion: some(S,P) Some whales are breast-feeders

Note that it does not matter in which order Premise 1 and Premise 2 come. Both of them are assertions that must be true simultaneously, so the order does not matter.

Here is a complete list of the twenty four Aristotelian syllogisms:

• 1st figure: barbara, celarent, darii, ferio, barbarix, feraxo.
• 2nd figure: cesare, camestres, festino, baroco, camestrop, cesarox.
• 3rd figure: darapti, disamis, datisi, felapton, bocardo, ferison.
• 4th figure: bramantip, camenes, dimaris, fesapo, fresison, camenop.

## The beginning of a Prolog program to solve syllogisms

Here is the beginning of a Prolog program to deduct using syllogisms:

```syll(   all(M,P),
all(S,M),
all(S,P)).
// This is barbara in the 1st figure.

syll(   no(M,P),
all(S,M),
no(S,P)).
// This is celarent in the 1st figure.

syll(   all(M,P),
some(S,M),
some(S,P)).
// This is darii in the 1st figure.

syll(   no(M,P),
some(S,M),
some_not(S,P)).
// This is ferio in the 1st figure.

deduction(P1,P2,C):- syll(P1,P2,C).
deduction(P1,P2,C):- syll(P2,P1,C).
```

Copy-paste this into a new Prolog document (first fire up the Prolog system, then remember to press the 'New' button on the toolbar, or choose 'New' from the 'File' menu).

This can be used as follows:

```?- deduction(all(mammals, breast_feeders), some(whales, mammals), C).

{C = some(whales, breast_feeders)}
{C = some(breast_feeders, whales)}
```

Or, with another example:

```?- deduction(no(rhino, wall_paper_eater), some(Otto, rhino), C).

{C = some_not(Otto, wall_paper_eater)}
```

Then use the program to solve the following syllogisms:

• Some dogs are not cat-haters
• All dogs are mammals
• Therefore: ?
• All dogs are canines
• All canines are mammals
• Therefore: ?
• All dogs are dependent on people
• No cats are dependent on people
• Therefore: ?
• Some bears are animals of very little brain
• All animals of very little brain are humble
• Therefore: ?