Nominalistic Logic: NL-to-Format

• NL-to-Format is a formal system that merges higher-order intensional type theory with explicit predicate nominalization to treat predicates as individuals.
• It employs a classical sequent calculus and a flexible comprehension scheme, allowing for set-like constructions without extensionality and infinity axioms.
• Its innovative nominalization axiom supports the derivation of Peano’s postulates, bridging the gap between predicate logic and arithmetic formulations.

Nominalistic Logic (NL) is a formal system presenting higher-order intensional type theory—following Paul Gilmore's ITT—but distinguished by its explicit treatment of predicate nominalization. The system is formulated as a classical sequent calculus and is characterized by a nominalization axiom (N) that enables, under precisely mandated syntactic conditions, the representation of certain predicates as individuals. NL incorporates a flexible comprehension axiom, yet notably omits both the extensionality and infinity axioms. Despite this, it is expressive enough to allow the derivation of Peano's postulates for arithmetic by virtue of its nominalization machinery.

1. Syntax, Types, and Terms

NL is fundamentally typed and based on the lambda calculus with a rich type system reminiscent of higher-order logic. The syntax is specified as follows:

• Terms: Constructed via application and lambda abstraction:

$$t ::= pt \;\mid\; \lambda x. p \;\mid\; x \;\mid\; c$$

where $x$ is a variable, $c$ a constant, $t$ a term, and $p$ is a term of function type.

• Types: Types $\tau, \sigma$ are formed inductively:

$$\begin{align*} \tau & ::= \sigma \mid \imath \ \sigma & ::= \tau\sigma \mid o \end{align*}$$

with $o$ (formulas/truth values), $\imath$ (individuals), and function types as needed.

• Type Formation Rules:

$\begin{align*} x : \tau & \text{ for variable $x$} \ c : \tau & \text{ for constant $c$} \ p : \tau\sigma,\, t:\tau &\implies pt : \sigma \ x:\tau,\, p:\sigma &\implies \lambda x.p:\tau\sigma \ p:\imath &\text{if $p$ is nominalizable} \end{align*}$

Nominalizability: A term $p:\sigma$ is nominalizable iff all free variables of $p$ are of type $\imath$.

• Formulas: Terms of type $o$ are formulas; closed formulas are sentences.
• Variable Binding/Reduction: Alpha, beta, and eta conversion are as in standard simply-typed lambda calculus. Operator priorities are: $$\neg, =, \land, \lor, \rightarrow, \leftrightarrow, \lambda$$ ($\forall,\exists$ equated with $\lambda$ in binding precedence).

2. Sequent Calculus

NL's deductive system is a Gentzen-style sequent calculus with sequents $\Gamma \vdash \Delta$, where $\Gamma$ and $\Delta$ are (possibly empty) lists of formulas. The inference system includes:

• Structural Rules: Similarity/reduction (S), thinning (T), exchange (E), contraction (C).
• Logical Rules:
  • A primitive $c$ is interpreted as "neither ... nor ..." (for negative connectives and quantifiers).
  • Negation: $\neg p \coloneqq c p p$
  • Disjunction: $p \lor q \coloneqq \neg c p q$
  • Conjunction, implication, and biconditional are derived using $\neg$ and $\lor$ as usual.
  • Existential quantification: $\exists x. p \coloneqq \neg c \lambda x. p$
  • Universal quantification: $\forall x. p \coloneqq \neg \exists x. \neg p$
• Identity: Syntactic identity $s = t$, as well as logical equivalence $s \equiv t$ (extensional equality at various types).

NL's inference rules for (disjunction/negation) and quantification use the special constant $c$, exploiting its role as a primitive negative.

3. The Nominalization Axiom (N) and Nominalization Mechanism

The defining feature of NL is the nominalization axiom (N):

$$\vdash~ p = q \;\leftrightarrow\; p \doteq q \tag{N}$$

where $=$ is the nominal identity at type $\imath$, and $\doteq$ is an extensional equality of type $\imath\imath o$. Both $p$ and $q$ must be nominalizable—i.e., be of type $\imath$ and have free variables only of type $\imath$.

Meaning and Utility

• Nominalization: Certain predicates (e.g., $\lambda x.P(x)$) with all free variables of individual type can themselves be regarded as individuals—allowing, for instance, sets (or properties) to be named and manipulated as objects within the logic.
• Axiom N: Asserts that two nominalized terms are equal as individuals iff their predicate extensions coincide, essentially enforcing intensional rather than extensionality at the individual level, but only for nominalizable entities.
• — Example: If $P(x)$ and $Q(x)$ are unary predicates (for $x:\imath$), then $N_P = \lambda x.P(x):\imath$ and $N_Q = \lambda x.Q(x):\imath$ are individuals. Then $N_P = N_Q \leftrightarrow N_P \doteq N_Q$ asserts that these individuals are identical iff the predicates coincide over all individuals. This enables quantification and construction over such "named sets" without requiring a comprehension axiom for all predicates.

4. Comprehension Scheme

NL includes a flexible comprehension axiom, permitting the construction of "names" for classes defined by predicates, as long as the predicates are nominalizable. The formalization grants a restricted but powerful abstraction, supporting the definition of sets (or classes) without the hazards of full naive set theory (no Russell paradox). The exact schema is not reproduced here, but the key constraint is nominalizability: only predicates with all free variables of type $\imath$ can be "named" as individuals.

5. Absence of Extensionality and Infinity Axioms

NL is marked by two significant omissions:

• No extensionality axiom: Extensionality, as in set theory (identification of objects with identical memberships), is not adopted. Therefore, predicates (even if coextensive) may yield distinct individuals when nominalized. This distinguishes NL from extensional higher-order or set-theoretic frameworks and underlines its intensional type-theoretic nature.
• No infinity axiom: NL does not assume the existence of an infinite collection (as in ZF or the Peano axioms). Instead, it supports definitions and derivations that construct the natural numbers from within the system by way of nominalization and comprehension.

6. Derivation of Peano’s Postulates via Nominalization

Despite lacking an explicit infinity axiom, NL encodes the natural numbers and Peano’s postulates through its nominalization machinery and comprehension. Key definitions include:

• Zero: $0 \coloneqq \lambda x. x \not\doteq x$
• Successor: $t' \coloneqq (\lambda x y. x \doteq y)\, t$
• General $n$: Built iteratively, $n+1 = n'$

The set-theoretic structure (e.g., the empty set, successor, induction) can thus be realized within NL. The nominalization axiom ensures that successor operations and number equality behave as required, and the comprehension scheme provides for inductive definitions over the domain of individuals.

7. Summary Table: Key Technical Features

Feature	NL Specification
Term formation	Typed lambda calculus (variables, constants, applications)
Predicate nominalization	Only for predicates with free variables of type $\imath$
Axiom (N)	$p = q \leftrightarrow p \doteq q$ for nominalizable $p, q$
Comprehension	For classes defined by nominalizable predicates
Extensionality	Not included; logic is intensional
Infinity	Not postulated; constructed via nominalization
Derivable Peano postulates	Yes, via nominalization and comprehension
Inference system	Classical sequent calculus, structural and logical rules

8. Significance and Logical-Philosophical Position

Nominalistic Logic exemplifies an intensional higher-order logic where certain predicates—specifically, those whose free variables are of type individual—can be treated as objects in their own right. The nominalization axiom precisely manages identity conditions for such objects, bridging predicate logic and class abstraction without recourse to unrestricted set comprehension or extensionality.

NL's design, omitting extensionality and an infinity axiom, supports fine-grained distinctions crucial for philosophical debates (e.g., nominalism vs. platonism) and foundational mathematical formalization, while remaining logically robust enough to capture arithmetic via derived means. This makes NL a distinctive alternative to classical set theory or extensional second-order logic, aligning with the aims of intensional type theory but embedding the act of "naming" predicates into the formal language itself.

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References: Jørgen Villadsen (2008), Nominalistic Logic: From Naive Set Theory to Intensional Type Theory (0812.4814); Paul Gilmore (2001), An Intensional Type Theory: Motivation and Cut-Elimination.