The Timpson Lyceum · First-Order Predicate Logic (2024)

Theory of Definition & Kurt Gödel's Proof of God - Lecture 13 (Logic)

Overview

This lecture closes out logic proper by examining the fourth component of an axiomatic system: definitions. We first revisit Aristotle's classical theory of definition (defining a species by its genus plus a differentiating characteristic) and his four rules, then show why those rules — though they suffice for most subjects — are not rigorous enough for formal systems. From this we build the modern theory: two governing criteria (eliminability and non-creativity) and the precise rules for defining predicates, operators, and constants. The lecture then changes gears to preview two ideas beyond first-order logic — modal logic and second-order predicates — and applies them to a famous worked system: Kurt Gödel's ontological argument for the existence of God.

Key Concepts

The four components of an axiomatic system

Axioms — the starting assumptions we prove from.
Primitives — the operators, terms, and relations appearing in the axioms that are not symbols of logic.
Theorems — statements derivable from the axioms.
Definitions — shorthands introducing new symbols, the focus of this lecture.

Aristotle's theory of definition

A definition specifies a species by stating its genus (a broader collection) and a differentiating characteristic (what selects the subcollection). Example: an equilateral triangle (species) is a triangle (genus) with all sides equal in length (differentia).
The genus is relative to the species being defined (like "parent"/"child"). The same term can be a genus in one definition and a species in another.
Repeating the process leads back to categories — top-level genera (e.g. "substance") that are never themselves species and are left undefined.

Aristotle's four rules of definition

States the essence — not merely a property unique to those things. Defining man as a "featherless biped" fails: pluck a chicken and you have a featherless biped that is not a man.
Not circular — "a woman is anyone who identifies as a woman" tells you nothing.
Stated in the positive — defining force as "a non-kinematic notion" says only what it is not.
Not figurative or obscure — "beauty is eternity gazing at itself in a mirror" conveys no usable content.

Why Aristotle's rules are insufficient. A definition can obey all four rules yet still corrupt a system. The bad operator $x \star y = z \;:\!\Longleftrightarrow\; (x < z) \wedge (y < z)$ lets one prove both $1 \star 2 = 3$ and $1 \star 2 = 4$ , hence $3 = 4$ , which collapses to $0 = 1$ — a contradiction. So we need stricter, structural criteria.

The modern starting point. Instead of beginning from undefined categories, formal systems build every definition out of primitive symbols and logical symbols.

Terminology. The definiendum is the side carrying the new symbol; the definiens is the defining expression. The colon in $\;:\!\Longleftrightarrow\;$ (or $:=$ ) marks the definiendum side.

The two criteria every definition must satisfy

Eliminability — a formula $S$ introducing a new symbol is eliminable iff for every formula $S_1$ containing the new symbol there is a formula $S_2$ not containing it with $S_1 \Leftrightarrow S_2$ . Definitions are mere conveniences; you can always rewrite back into primitives.
Non-creativity — let $\mathcal{P}$ be the conjunction of all current axioms, definitions, and theorems. $S$ is non-creative iff there is no formula $T$ (in which the new symbol does not occur) such that $\mathcal{P} \wedge S \vdash T$ while $\mathcal{P} \not\vdash T$ . A definition must never let you prove something new that doesn't even mention the new symbol. The $\star$ example violated this: it proved $3 = 4$ .

Rules for defining a predicate $P(x_1,\dots,x_n) :\!\Longleftrightarrow S$

$x_1,\dots,x_n$ are distinct variables.
$S$ has no free variables except $x_1,\dots,x_n$ (bound variables are fine). Example: $\text{IsLoved}(x) :\!\Longleftrightarrow \exists y\, L(y,x)$ — the $y$ must be bound, or it is meaningless.
$S$ contains only logical, primitive, or previously defined symbols.

Rules for defining an operator $O(x_1,\dots,x_n) = w :\!\Longleftrightarrow S$

Same first three rules as predicates, except $w$ may also appear as a free variable in $S$ .
Uniqueness: there must exist a unique $w$ satisfying $S$ — proved as a theorem before the definition. Division omits this without the restriction $y \neq 0$ : if $b/0 = a$ meant $b = 0 \cdot a$ , then $0/0$ equals every number, so the result is not unique.

Rules for defining a constant $C = w :\!\Longleftrightarrow S$

$w$ is the only free variable in $S$ ( $C$ cannot be free — it is a constant, not a variable).
$S$ uses only logical, primitive, or previously defined symbols.
A unique $w$ must satisfy $S$ . Example: $\pi := \tfrac{\text{circumference}}{\text{diameter}}$ , valid only after proving this ratio is identical for all circles.
For operators and constants you may use $=$ in place of $\Leftrightarrow$ ; e.g. $2 := 1+1$ .

Modal logic (preview). Two new operators: $\Box$ ("necessarily") and $\Diamond$ ("possibly"). Key rules:

If $x$ is a theorem, then $\Box x$ (whatever is always true is necessarily true).
$\Box(x \to y) \to (\Box x \to \Box y)$ (distribution).
$\Box x \to x$ (necessity is stronger than truth).
Stacked identical operators collapse to one; in a mixed string, only the last operator matters.
The operators are interdefinable: $\Diamond x \;\equiv\; \neg \Box \neg x$ . Intuitively $\Diamond$ behaves like $\exists$ and $\Box$ like $\forall$ .

Second-order predicates (preview). A first-order predicate (e.g. $\varphi, \psi$ ) takes objects and returns true/false — these are properties. A second-order predicate takes first-order predicates. In Gödel's system, $P(\varphi)$ (" $\varphi$ is a positive property") is the lone primitive; "positive" is left undefined (intuition: courageous = positive, cowardly = negative).

Gödel's ontological argument.

Axiom 1: $P(\varphi) \wedge \Box\,\forall x\,(\varphi(x) \to \psi(x)) \;\to\; P(\psi)$ — a positive property can only entail positive properties.
Axiom 2: $P(\neg\varphi) \Leftrightarrow \neg P(\varphi)$ — every property is positive or negative; lacking a positive property is the same as having a negative one.
Theorem 1: $P(\varphi) \to \Diamond\,\exists x\,\varphi(x)$ — a positive property is possibly instantiated.
Definition 1 (godlike): $G(x) :\!\Longleftrightarrow \forall \varphi\,(P(\varphi) \to \varphi(x))$ — having every positive property.
Axiom 3: $P(G)$ — being godlike is positive.
Theorem 2: $\Diamond\,\exists x\,G(x)$ — possibly, a godlike being exists.
Definition 2 (essence): $\varphi$ is essential to $x$ iff $\varphi(x)$ and every property of $x$ is necessarily entailed by $\varphi$ .
Axiom 4: $P(\varphi) \to \Box\,P(\varphi)$ — positive properties are necessarily positive.
Theorem 3: $G(x) \to (G \text{ is essential to } x)$ .
Definition 3 (necessary existence): $x$ necessarily exists iff each essential property of $x$ is necessarily instantiated.
Axiom 5: necessary existence is positive.
Theorem 4: $\Box\,\exists x\,G(x)$ — necessarily, a godlike being exists.

Worked Example

Theorem 1: if $\varphi$ is positive, then $\varphi$ is possibly instantiated. Symbolic claim, then the proof:

\begin{aligned} &\text{Claim: } P(\varphi) \;\to\; \Diamond\,\exists x\, \varphi(x). \\ &\text{Assume } P(\varphi). \text{ Rewrite the goal: } \Diamond\,\exists x\,\varphi(x) \;\equiv\; \neg\Box\,\forall x\, \neg\varphi(x). \\ &\text{For contradiction assume } \Box\,\forall x\, \neg\varphi(x) \quad(\text{nothing has } \varphi). \\ &\text{Then } \Box\,\forall x\,(\varphi(x) \to \neg\varphi(x)) \text{ holds vacuously } (\varphi(x) \text{ is always false}). \\ &\text{Apply Axiom 1 with } \psi := \neg\varphi: \quad P(\varphi) \wedge \Box\,\forall x\,(\varphi(x)\to\neg\varphi(x)) \;\to\; P(\neg\varphi). \\ &\text{So } P(\neg\varphi). \text{ By Axiom 2, } P(\neg\varphi) \Leftrightarrow \neg P(\varphi), \text{ giving } \neg P(\varphi). \\ &\text{But } P(\varphi) \wedge \neg P(\varphi) \text{ is a contradiction.} \\ &\text{Therefore } \neg\Box\,\forall x\,\neg\varphi(x), \text{ i.e. } \Diamond\,\exists x\,\varphi(x). \qquad \blacksquare \end{aligned}

Summary

An axiomatic system rests on four pillars — axioms, primitives, theorems, and definitions — and this lecture makes definitions rigorous.
Aristotle defines a species by its genus and a differentiating characteristic; his four rules (essence, non-circular, positive, non-figurative) are adequate for everyday subjects but not for formal logic.
A definition can satisfy all four Aristotelian rules yet still wreck a system (the $\star$ operator forcing $3 = 4$ ), so modern logic demands two structural criteria: eliminability and non-creativity.
Only three kinds of symbol can be defined — predicates, operators, constants — each by a short rule set; operators and constants additionally require a uniqueness proof (illustrated by why division by zero is forbidden).
Modal logic adds $\Box$ (necessarily) and $\Diamond$ (possibly), interdefinable via $\Diamond x \equiv \neg\Box\neg x$ ; second-order predicates take properties as arguments.
Gödel's ontological argument uses one primitive ( $P$ , "positive"), five axioms, three definitions, and four theorems to derive $\Box\,\exists x\,G(x)$ — that a being possessing every positive property necessarily exists — a compact showcase of modal and second-order reasoning.