Converting Natural Language Statements to First-Order Logic: “All men are mortal. Socrates is a man.”

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Nov 4, 2026

To convert a natural-language sentence into first-order logic (FOL), you (1) choose a domain of discourse, (2) pick predicates for the relevant properties, and (3) use quantifiers (especially $\forall$ for “all”) to match the sentence structure.
For the statements:

“All men are mortal.”
“Socrates is a man.”

A standard FOL setup uses:

Domain: all individuals (e.g., people/objects of the relevant universe).
Predicates: $Man(x)$ and $Mortal(x)$ .
Constant: $s$ (or specifically “Socrates”) to denote the particular individual Socrates.

Then the translation is:

“All men are mortal.” $\;\;\;\Rightarrow\;\;\; \forall x\,(Man(x)\rightarrow Mortal(x))$
“Socrates is a man.” $\;\;\;\Rightarrow\;\;\; Man(s)$

This is the typical meaning-preserving formalization: any individual $x$ that is a man must also be mortal, and Socrates (a specific constant) is asserted to be a man.

Key keyword concepts: predicates, constants, quantifiers, implication.

First-Order Logic (FOL) Translation: Quantifiers and Predicates

Pro Tip: Map grammar to logic

For FOL translations, “all” usually means a universal quantifier $\u2200$ , while “is” often becomes either predicate application (e.g., $Man(s)$ ) or part of an implication (e.g., $Man(x)\u2192Mortal(x)$ ).

Step-by-step FOL conversion for the given statements

1
Step 1
Let the domain be all individuals you are talking about (commonly, all people).
2
Step 2
Use Man(x) for “x is a man” and Mortal(x) for “x is mortal.”
3
Step 3
Let $s$ be a constant symbol meaning “Socrates.” Then “Socrates is a man” becomes Man(s).
4
Step 4
Use $\u2200x$ to cover all individuals $x$ . Then “All men are mortal” becomes: $Man(x)\u2192Mortal(x)$ .
5
Step 5
The full set of translations is: $\u2200x\,(Man(x)\u2192Mortal(x))$ and $Man(s)$ .

Final First-Order Logic (FOL) representation

Let:

$Man(x)$ = “ $x$ is a man”
$Mortal(x)$ = “ $x$ is mortal”
$s$ = “Socrates” (a constant)

Then:

All men are mortal $\forall x\,(Man(x)\rightarrow Mortal(x))$
Socrates is a man $Man(s)$

Together (as a set of premises / conjunction), you can write: $\big(\forall x\,(Man(x)\rightarrow Mortal(x))\big)\wedge Man(s)$

Important keyword and keyword use is exactly what matches “all.”

Common mistake: using ∃ instead of ∀

Do not translate “All men are mortal” as $\u2203x(Man(x)\u2227Mortal(x))$ . That would mean “there exists a man who is mortal,” which is weaker than “every man is mortal.”

English	FOL form
All men are mortal	$\forall x\,(Man(x)\rightarrow Mortal(x))$
Socrates is a man	$Man(s)$

Translation workflow

Vocabulary

Extract predicates and constants: $Man$ , $Mortal$ , and $s$ ."

Quantifiers

Use $\u2200$ for “all”."

Connectives

Use $\rightarrow$ for “All A are B” (A→B)."

Write final formulas

Output $\forall x(Man(x)\rightarrow Mortal(x))$ and $Man(s)$ ."

Quick checks and edge cases

Knowledge Check

Question 1 of 3

Q1Single choice

Which FOL formula best represents: “All men are mortal.”

$\forall x\,(Man(x)\rightarrow Mortal(x))$

$\exists x\,(Man(x)\rightarrow Mortal(x))$

$\forall x\,(Man(x)\wedge Mortal(x))$

$\exists x\,(Man(x)\wedge Mortal(x))$

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