# Socrates is Mortal

## Theorem

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

## Proof

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

Let $H \left({x}\right)$ denote the propositional function $x$ is human.

Let $M \left({x}\right)$ denote the propositional function $x$ is mortal.

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

The argument can then be expressed as:

 $(1):\quad$ $\displaystyle \forall x: \ \$ $\displaystyle H \left({x}\right)$ $\implies$ $\displaystyle M \left({x}\right)$ $\displaystyle \therefore \ \$ $\displaystyle H \left({S}\right)$ $\implies$ $\displaystyle M \left({S}\right)$ Universal Instantiation $(2):\quad$ $\displaystyle H \left({S}\right)$  $\displaystyle$ $(3):\quad$ $\displaystyle \therefore \ \$ $\displaystyle M \left({S}\right)$  $\displaystyle$ Modus Ponendo Ponens

That is:

Socrates is mortal.

$\blacksquare$