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.


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.

$\blacksquare$


Sources