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.
Contents
The classical Aristotelian logic has been divided into three main subjects:
- The doctrine of categorical terms
- The doctrine of categorical propositions
- 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
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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
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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
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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
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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
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2 Additional rules
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
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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:
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Input:
1: A(x,y)
2: E(y,z)
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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)
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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
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These are the main rules for feeding the program with input propositions:
- Propositions have the form A(x,y), E(x,y), I(x,y), O(x,y). They may be inserted anywhere into
the input box.
- The terms x, y , z ,.. in a proposition may be any string of characters. Thus, A(man,animal)
is a well formed proposition.
- Propositions are separated by space (one or more blanks). You may also separate propositions
by ";" or "+", but NOT BY COMMA.
- Text included in // (text) // will be taken as commentary.
- If both terms in a proposition are constituted of only one character each, for example,
A(x,y), then you may delete the brackets and simply write Axy. Thus, for example, AMN and IxZ are
well formed input propositions.
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