Syllogisms will be used in the exercises, so just to recapitulate, we here give examples of the four most common syllogism-patterns.
The four kinds of statements
In syllogisms, there are four kinds of statements:
There are 256 possible syllogisms, but only about 15 of them are valid. In the middle ages, the valid syllogisms were remembered by names that contained three of the letters above, for example ferio.
There are four figures, or basic patterns. The figure we will be using is the following (called figure 1):
Where x, y, and z are As, Is, Es, or Os. S, M, and P are classes of objects, for example "Mammal" (in the case of A and E) or individuals (in the case of I and O).
The four syllogism patterns which we will make use of in the exercises are:
They all conform to figure 1 (above). These are exemplified below.
Encoding syllogisms in CGs
You can encode syllogisms in CGs in a number of ways. Here, we present one of the ways.
Pattern - the overall structure
In order to be able to write syllogisms in a uniform manner, one needs a pattern to follow. The following could be such a pattern:
[If: [ ... Premise 1 ... (a CG) ] [ ... Premise 2 ... (a CG) ] [Then: [ ... Conclusion ... (a CG) ] ] ]
Pattern - Premises and conclusions
While the overall pattern is an IF-THEN construction with two premises and a conclusion, each premise and conclusion is either an IF-THEN construction or a conjunction of two graphs.
The patterns to follow for the four types of statement are:
Difference between Universal and Particular
The difference between a Universal and a Particular is that a Universal is an IF-THEN construction, whereas a Particular is a conjunction of two concepts.
Difference between Affirmative and Negative
The difference between an Affirmative and a Negative is the presence of a ("not") sign in front of the "B"-concept in the Negative.
Constructing a syllogism
Thus, in order to construct a syllogism, we will have an outer IF-THEN, inside which we find three graphs.
For example, below is an encoding of the following Barbara syllogism:
And one encoding could look like this:
[If: [If: [Rhino: *x] [Then: [Mammal: ?x] ] ] [If: [Mammal: *y] [Then: [Animal: ?y] ] ] [Then: [If: [Rhino: *z] [Then: [Animal: ?z] ] ] ] ]
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