Socrates is Mortal

Theorem

$(1): \quad$ All humans are mortal.

$(2): \quad$ Socrates is human.

$(3): \quad$ Therefore Socrates is mortal.

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

$(2): \quad$ Socrates is a man.

$(3): \quad$ Therefore Socrates is mortal.

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 \ \ $

$\ds \map H S$

$\implies$

$\ds \map M S$

$\text {(2)}: \quad$

$\ds \map H S$

$\ds $

$\text {(3)}: \quad$

$\ds \therefore \ \ $

$\ds \map M S$

$\ds $

That is:

Socrates is mortal.

$\blacksquare$

The syllogism is often seen presented with a variety of subjects.

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

The syllogism appears first to have been presented by Aristotle.

1951: Willard Van Orman Quine: Mathematical Logic (revised ed.) ... (previous) ... (next): Introduction
1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $1$ Introduction: Logic and Language: $1.2$: The Nature of Argument
1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $4$: Propositional Functions and Quantifiers: $4.2$: Proving Validity: Preliminary Quantification Rules
1993: Richard J. Trudeau: Introduction to Graph Theory ... (previous) ... (next): $1$. Pure Mathematics: Games
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): logic
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): logic