Syllogism

Aristotelian Syllogistic Logic

    Aristotle argued for a distinction between what could be learned by observation of the natural world, and what could be validly inferred about its underlying principles. This he discussed in his two works on formal logic in the Analytics. He invented the syllogism as a means for valid inference of truths about specific observations from general principles. Famously,

        A: All men are mortal: the Major Premise, derived in this case from Observation
        B: Socrates is a Man: the Minor Premise, an assertion about the matter being investigated
        C: Therefore, Socrates is Mortal: Conclusion, a deduction from the two premises

Stated formally: If A then B: A, therefore B., or in the notation of mathematical logic, A ⊃ B, A ∴ B

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    Major Premises can also derive from Axioms, which are truths that are (assumed to be) self-evident, but which cannot be proved. For example, a fundamental axiom of Euclidean geometry is that any given line has only one line parallel to it through a particular point. This axiom reflects everyday observation, but cannot be proved. However, it is possible to construct coherent non-Euclidean geometries that allow multiple parallel lines to pass through the point

Errors may arise in several ways. First, the Major Premise may not be true. For example,
        A: All mammals are live-bearing (viviparous).
        B: Platypuses are egg-bearing (oviparous).
        C: Therefore, Platypuses are not mammals.

    This may lead to new knowledge, in this case that other properties of Platypuses group them with mammals, and therefore the definition of mammal must be modified. Aristotle himself recognized live-bearing dolphins as mammals rather than fish, despite their lack of the typical mammalian character of hair.

Second, the logical system may be misapplied. Again, famously,

        A: Witches float in water.
        B: Ducks also float in water.
        C: Therefore, if she weighs the same as a duck, she's a witch.
        [It's a fair cop].

This is a variant of a typical logical error, the fallacy of assuming the consequent:

        All witches float in water.
        She floats in water.
        Therefore, she's a witch.

The incorrect argument stated formally is, If W then F: F, therefore W. Granted the premise that there are witches, the fallacy is that the truth of the major premise does not establish that anything that floats (for example, a duck) is a witch. That is, the truth of the consequent (F) does not prove the antecedent (W). In the previous example, given the validity of the major premise that All Men are Mortal, it would be incorrect to infer that, if Socrates is Mortal, then Socrates is a Man. The counterexample is that Ducks are also mortal, but are not men.

Other forms of logical inference derived from the syllogism include:

    If A then B; not B, therefore not A. Formally, A ⊃ B, ~B ∴ ~A

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    A if and only if B. This can be shown to be equivalent to [( A ⊃ B) & ( B ⊃ A)], which can be shown to be equivalent to [(~A ⊃ ~B) V (~B ⊃ ~A)]
        For example, Class is cancelled, if and only if I am sick. [where C = cancelled, S = Sick]
        From this it follows
            If class is cancelled, then I am sick. [C ⊃ S]
            If I am sick, then class is cancelled. [S ⊃ C]
            If I am not sick, then class is not cancelled. [~S ⊃ ~C]
            If class is not cancelled, then I am not sick. [~C ⊃ ~S]