15 Syllogisms

Introduction

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:

  • A: Universal affirmative. All A are B
  • I: Particular affirmative. Some A are B
  • E: Universal negative. No A is B
  • O: Particular negative. Some A are not B
These are rememberered by the letters A, I, E, and O:
  • "A" and "I" are from the Latin "affirmo", "I affirm", while
  • "E" and "O" are from the Latin "Nego", "I deny".

Syllogism patterns

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):

  • y(M,P)
  • x(S,M)
  • Therefore: z(S,P)

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:

  • Barbara
  • Celarent
  • Darii
  • Ferio

They all conform to figure 1 (above). These are exemplified below.

Barbara

  • A:  All birds are egg-layers.
  • A:  All sea-gulls are birds.
  • A:  Therefore, all sea-gulls are egg-layers.

Celarent

  • E:  No mammals are birds.
  • A:  All whales are mammals.
  • E:  Therefore, no whales are birds.

Darii

  • A:  All swans are white.
  • I:  Some birds are swans.
  • I:  Therefore, some birds are white.

Ferio

  • E:  No university student is a toddler.
  • I:  Some skaters are university students.
  • O:  Therefore, some skaters are not toddlers.

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:

A: Universal affirmative. All A are B

  [If:
     [A:  *x]
     [Then:   
        [B: ?x]     
     ]
  ]

I: Particular affirmative. Some A are B

   [Proposition:
      [A: *x]
      [B: ?x]     
   ]

E: Universal negative. No A is B

  [If:
     [A:  *x]
     [Then:   
       [B: ?x]     
     ]
  ]

O: Particular negative. Some A are not B

 
   [Proposition:
      [A: *x]
     [B: ?x]     
   ]

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.

Each subgraph should use a different coreference link (e.g., *x, *y, *z) for clarity, although because of the scope rules, they need not be different.

Example: Barbara

For example, below is an encoding of the following Barbara syllogism:

  • A:   All rhinos are mammals.
  • A:   All mammals are animals.
  • A:   Therefore, all rhinos are animals.

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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