Socrates is Mortal

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Theorem

$(1): \quad$ All humans are mortal.
$(2): \quad$ Socrates is human.
$(3): \quad$ Therefore Socrates is mortal.


Variant

$(1): \quad$ If Socrates is a man then Socrates is mortal.
$(2): \quad$ Socrates is a man.
$(3): \quad$ Therefore Socrates is mortal.


Proof

Let $x$ be an object variable from the universe of rational beings.

Let $\map H x$ denote the propositional function $x$ is human.

Let $\map M x$ denote the propositional function $x$ is mortal.

Let $S$ be a proper name that denotes Socrates.

The argument can then be expressed as:

\(\text {(1)}: \quad\) \(\, \ds \forall x: \, \) \(\ds \map H x\) \(\implies\) \(\ds \map M x\)
\(\ds \therefore \ \ \) \(\ds \map H S\) \(\implies\) \(\ds \map M S\) Universal Instantiation
\(\text {(2)}: \quad\) \(\ds \map H S\) \(\) \(\ds \)
\(\text {(3)}: \quad\) \(\ds \therefore \ \ \) \(\ds \map M S\) \(\) \(\ds \) Modus Ponendo Ponens

That is:

Socrates is mortal.

$\blacksquare$


Also presented as

The subject of this syllogism varies.

For example, 1993: Richard J. Trudeau: Introduction to Graph Theory presents it as Plato.


Historical Note

The syllogism Socrates is Mortal appears first to have been presented by Aristotle.


Sources