Syllogisms — categorical, hypothetical, and disjunctive patterns

Based on https://www.taoke.de/ke/Logic/Syllogisms/index.html.

Any valid pattern of inference was coined by Aristotle with the word syllogism ([Gand2002], page 52). The kinds of syllogisms developed in antiquity were categorical syllogisms for reasoning about categories, hypothetical syllogisms based on if–then patterns and disjunctive syllogisms based on inclusive and exclusive or. Boethius (6th century) introduced the letters A, I, E and O as mnemonics for naming the four different sentence patterns used, where S and P represent terms in the subject and predicate:

• A for universal affirmative like: every S is P e.g. “Every animal is a living thing.”
• I for particular affirmative like: some S is P, e.g. “Some animal is a beast.”
• E for universal negative like: no S is P, e.g. “No plant is a rational living thing.”
• O for particular negative like: some S is not P, e.g. “Some animal is not a mammal.”

From the 256 possible three-sentence syllogisms only 19 are logically valid. From the 19 patterns, 17 can be derived from the patterns Barbara and Celarent, which Aristotle chose as the two basic patterns. With S being the term in the subject of the conclusion, and P being the term in the predicate of the conclusion and M being the middle term appearing in premises but not in conclusions, the four figures for patterns are:

1. M − P ∧ S − M ∴ S − P
2. P − M ∧ S − M ∴ S − P
3. M − P ∧ M − S ∴ S − P
4. P − M ∧ M − S ∴ S − P

The following example (M − P ∧ S − M ∴ S − P) is a categorical syllogism based on the Barbara pattern represented in the OntoGraph of Figure HisLT:

• (A) “Every animal is a living thing”
• (A) “Every human is an animal”

Therefore

• (A) “Every human is a living thing”

Abbreviating the subclass relation, this can be modelled by a relator without asserting the base triples directly into the knowledge base (see τ-mappings on the canonical page). The Barbara pattern in general maps using internal identifiers for syllogism, premises, and conclusion.

The OntoGraph in Figure logic-hia represents the result of the syllogism. This is a typical example for entailment by inheritance as described in section Type Propagation Rule. [BaMc2003] gives DL background for related reasoning tasks.