Arguments {syllogism} can have general statement or assumption {major premise}, fact or information {minor premise}, and statement to prove {conclusion, syllogism}.
types
Syllogism types correspond to statement types used for premises. Traditional logic used different premise figures and moods to make 16 * 16 = 256 syllogisms. 24 are valid. 19 are strong, because conclusion is as strong as the premises. 15 have equal strength in premises and conclusion {fundamental syllogism}.
Allowing negations, the eight possible forms each can have eight different expressions, making 64 possibilities for each two-premise-one-conclusion combination. Therefore, 64 * 64 * 64 different syllogisms exist.
universals
There cannot be particular solution with two universal premises.
Syllogisms {alternative syllogism} can use OR. Alternative syllogism has the following forms. Either A or B is true, not A is true, so B is true. Either A or B is true, not B is true, so A is true. A and B do not have to be mutually exclusive.
Syllogisms {antilogism} can have two premises and conclusion negation. Statement pairs can lead to third-proposition negation.
Syllogisms {categorical syllogism} can have major premise, minor premise, and conclusion. Major premise and minor premise share term. For example, "all a are b" is major premise. "c is a" is minor premise. Therefore, "c is b" {conclusion}, because both premises share term a.
negative
Categorical syllogism can have only one negative premise, and then conclusion must be negative.
quantifier
Categorical syllogisms can have premises about all, some, or none, in four forms: "All A are B", "No A are B", "Some A are B", or "Some A are not B". Verb must be "to be".
First, statement states All, Some, or No {quantifier, syllogism}. Noun phrase {subject class} follows quantifier. Verb {copula, syllogism} follows subject class. Second noun phrase {predicate class} follows copula. Therefore, categorical syllogism has three terms: two noun phrases and copula.
types
Categorical syllogisms have four types {figure, syllogism}. All A are B, C is A, and so C is B {first figure of syllogism}. All B are A, C is A, and so C is B {second figure of syllogism}. All B are A, A is C, and so C is B {third figure of syllogism}. All A are B, A is C, and so C is B {fourth figure of syllogism}.
The four types can use "All", "Some", or "No", making twelve categorical syllogisms. With "Some" in conclusion {particular statement}, only one premise can have "All" or "No" {universal statement}.
errors
Errors can be in categorical syllogisms. Major premise can be untrue. Syllogism can reverse major premise. Terms can be ambiguous. Middle term can be ambiguous {ambiguous middle}. For example, A is B(1) is true, B(2) is C is true, so A is C is true.
Hexameter rhymes {Celarent Barbara} [1200] can be mnemonics for valid syllogisms.
Syllogisms {conditional syllogism} {hypothetical syllogism} can use IF ... THEN .... Conditional syllogism has the following forms. If A then B is true, A is true, so B is true. If A then B is true, not B is true, so not A is true. Unless A then B is true, not A is true, so B is true. If not A then B is true, not A is true, so B is true. If first statement implies second and second implies third, first implies third: if (p -> q) & (q -> r), then (p -> r).
Syllogisms {disjunctive syllogism} can use NOT BOTH. Disjunctive syllogism has the following forms. Not both A and B is true, A is true, and so not B is true. Not both A and B is true, B is true, and so not A is true.
Middle term must distribute at least once in categorical syllogism {distributed middle}|.
Statements can exclude or include all members {distribution, logic}. Distributed terms refer to all individuals. In "All A are B", A distributes, but B does not. In "No A are B", A and B distribute. In "Some A are B", A and B do not distribute. In "Some A are not B", A does not distribute, but B distributes. Categorical syllogisms have three terms. If one term distributes in premises and either of the other two terms does not distribute in premises, the other two terms cannot distribute in conclusion.
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Date Modified: 2022.0225