# Computational Aristotelian Term Logic: Introduction

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# Formal Aristotelian logic

## Goal of this text

In the following we give a very short description of the fundamentals of Aristotelian logic and the use of the program. It will help the reader to explore the features of the program and to begin with experiments of his/her own.

## A The classical subjects of Aristotelian logic

The classical Aristotelian logic has been divided into three main subjects:

1. The doctrine of categorical terms
2. The doctrine of categorical propositions
3. The doctrine of categorical syllogisms

### 1. Categorical terms

Categorical terms (short: terms; gr. horoi; lt. termini) stand for universal concepts like "man", "animal", "living being", "gold", "metal" etc. We will denote terms by small latin letters x, y, z, etc.

### 2. Categorical propositions

Categorical propositions (short: propositions; gr. protasis; lt. propositio). This is a particular kind of sentence, combining two different terms x and y. There are four different types of categorical propositions:

• A(x,y): All x are y (All penguins are birds)
• E(x,y): No x are y (No birds are mammals)
• I(x,y): Some x are y (Some Greeks are philosophers)
• O(x,y): Some x are not y (Some animals are not carnivores)

### 3. Categorical syllogisms

Syllogisms are rules which allow to deduce, starting from a set of given propositions (the premises), additional propositions called conclusions. If, for example, the following premises are given:

• All penguins are birds: A(p,b)
• No birds are mammals: E(b,m)

then the rules of the Aristotelian system generate, as conclusion, the additional proposition

• No penguins are mammals: E(p,m)

We will denote this special rule (which bears the classical name "Celarent") as

Axy,Eyz->Exz

Aristotle has stated 14 different rules of this type which he called "sullogismoi". He found these rules by taking two of them as basic and deriving the others by a formal method called "reduction". Here is a reference to an concise introduction into the theory of the syllogism:

faculty.washington.edu/smcohen/433/Syllogistic.pdf

## B The rules of deduction

### 1 The syllogistic system

The syllogistic system contains 24 syllogistic rules. Each of these rules has got an individual name already many centuries ago : "Barbara", "Celarent", "Dario", "Ferio" , etc. There exists a classical subdivision of this set of syllogism into four groups, called "figures". These figures have been defined by means of the structure of the distribution of the terms occuring in the premises.

 First Figure A(x,y),A(y,z)-->A(x,z) Barbara A(x,y),E(y,z)-->E(x,z) Celarent I(x,y) A(x,z) --> I(x,z) Darii I(x,y) E(y,z) --> O(x,z) Ferio A(x,y) A(y,z) --> I(x,z) Barbari A(x,y) E(y,z) --> O(x,z) Celaront Second Figure A(x,y),E(z,y)-->E(x,z) Cesare E(x,y),A(z,y)-->E(x,z) Camestres I(x,y) E(z,y) --> O(x,z) Festino O(x,y) A(z,y) --> O(x,z) Baroco A(x,y) E(z,y) --> O(x,z) Cesaro E(x,y) A(z,y) --> O(x,z) Camestrop Third Figure A(y,x) A(y,z) --> I(x,z) Darapti A(y,x) I(y,z) --> I(x,z) Disamis I(y,x) A(y,z) --> I(x,z) Datisi A(y,x) E(y,z) --> O(x,z) Felapton A(y,x) O(y,z) --> O(x,z) Bocardo I(y,x) E(y,z) --> O(x,y) Ferison Fourth Figure A(y,x) A(z,y) --> I(x,z) Bamalip E(y,x) A(z,y) --> E(x,z) Camenes A(y,x) I(z,y) --> I(x,z) Dimatis A(y,x) E(z,y) --> O(x,z) Fesapo I(y,x) E(z,y) --> O(x,z) Fresison E(y,x) A(z,y) --> O(x,z) Camenop

In addition to the syllogisms there are 5 further rules, called "conversion" and "subalternation". We present these rules together with their historical names:

• E(x,y)-->E(y,x) E-conversion
• I(x,y)-->I(y,x) I-conversion
• A(x,y)-->I(x,y) A-subalternation
• E(x,y)-->O(x,y) E-subalternation
• A(x,y)-->I(y,x) A-partial conversion
• E(x,y)-->O(y,x) E-partial conversion

## C Description of an example

The program computes, starting from a set of propositions (premises) which are supplied as input, ALL conclusions which result by applying ALL the rules of the Aristotelian system. The set of the premises together with all conclusions generated by the syllogistic rules is called "Aristotelian closure" of the set of premises. The program has as its output a protocol of the rules which have been applied in the process of the computation of the Aristotelian closure.

If, for example, the premises

A(x,y) E(y,z)

are inserted into the input - box, then - after the button "Execute" has been pressed, the program responds by outputting three colums of data.

The leftmost column contains the input propositions; the column in the middle shows the output of the program (i.e., all conclusions drawn by means of the rules of the system), and the rigthmost column displays which rules have been applied during the computation of the conclusions 3 - 11:

 Input: 1: A(x,y) 2: E(y,z) Output 3: E(x,z) 4: E(z,y) 5: E(z,x) 6: O(y,z) 7: O(x,z) 8: O(z,y) 9: O(z,x) 10: I(x,y) 11: I(y,x) Protocol 1 , 2 --> Celarent --> 3 2 --> E-conversion --> 4 3 --> E-conversion --> 5 2 --> E-subalternation --> 6 3 --> E-subalternation --> 7 4 --> E-subalternation --> 8 5 --> E-subalternation --> 9 1 --> A-subalternation --> 10 10 --> I-conversion --> 11