Categorical syllogism

A categorical syllogism is a valid argument of the following form:

a\in b

.

b\in c

.

Therefore,

a\in c

.

In other words, this kind of argument states that if all a's are b's, and if all b's are c's, then all a's are c's.

A categorical syllogism contains two premises followed by a conclusion, both of which are generically known as propositions. There can be only three terms predicated through out the entirety of the syllogism. Each proposition must be in the following form: A subject linked to a predicate by a copula. A premise might go as follows: All S is P, where S is the subject, 'is' is the copula, and P is the predicate.

Categorical propositions can be categorized on the basis of their quality quality, quanity and distribution qualities. Quality refers to whether the proposition affirms or denies the inclusion of a subject to the class of the predicate. The two qualities are affirmative and negative. On the other hand, quantity refers to the amount of subjects in one class are included in the other class. The first quantifier is the universal, all. This means that every subject of one class has membership in the predicated class. The other quanitfier is called a particular. It is an indifinative number, which could mean five, twenty or, perhaps, all, but always at least one. From quality and quantity are 4 types of categorical propositions designated alphanumerically:

A proposition is a universial affirmative: All S is P

E proposition is a universial negative: No S is P

I proposition is a particular affirmative: Some S is P

O proposition is a particular negative: Some S is not P

A, E, I and O propositions have different distribution properties. Distribution referes to what can be infered from the proposition. An A proposition distributes the subject to the predicate, but not the reverse however. Consider the following categorical proposition: All dogs are mammals. All dogs are indeed mammals but it would be false to say all mammals are dogs. E propositions do distrubute bidirectionally between the subject and predicate. From the categorical proposition--No beatles are mammals--we can infere that no beatles are animals and likewise, no mammals are beatles. Both terms in a I pproposition are undistributed. For example, some Americans are conservatives. Neither term in tthe proposition can be entirely distributed to the other term. From this proposition its not possible to say that all Americans are conservatives or all conservatives are Americans. In an O proposition only the predicate term is distributed. Consider the following: Some hardware are nails. By knowing screws are considered hardware it can stated that there exists some members outside the class of nails that are members of the class of hardware. However, it can be infered that all nails are hardware. Thus, only the predicate term is distributed in an O proposition.