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.

Sources

• 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