# Socrates is Mortal

## 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.