Classical Logic without Bivalance

This paper applies Sandqvist's semantics for classical logic without bivalence to metamathematics, demonstrating how this anti-realist framework resolves $\omega$-incompleteness, establishes induction as meaning-constitutive, and provides an elementary consistency proof for Peano Arithmetic without relying on transfinite ordinals or transcendent truth.

The Gist

Here is an explanation of the paper "Classical Arithmetic Without Bivalence" by Alexander V. Gheorghiu, translated into simple, everyday language using analogies.

The Big Picture: A New Way to Do Math

Imagine you are trying to build a house (Mathematics). For a long time, architects (logicians) have argued about what the house is made of.

• The Formalists say: "It's just a game of rules. We move symbols around like chess pieces. If we follow the rules, the house stands."
• The Realists say: "The house is built on a real, invisible foundation that exists outside our minds. The rules just describe this pre-existing structure."

The author of this paper, Alexander Gheorghiu, is trying a third approach called Inferentialism. He asks: What if the meaning of math doesn't come from a hidden foundation or just a game of symbols, but from how we actually use the words?

Think of it like a game of Telephone. In this view, the meaning of a word (like "number") isn't a picture in your head or a physical object; it's defined entirely by the rules of how you pass that word to your neighbors. If you say "5," the rules of the game dictate exactly what you can say next.

The Problem: The "Missing Link"

In traditional math, there's a weird glitch called $\omega$-incompleteness.

• Imagine you have a rule that works for every single number you can write down: "1 is safe," "2 is safe," "3 is safe," and so on forever.
• But, the system fails to prove the general rule: "All numbers are safe."

It's like a security guard checking every single person entering a building one by one. He checks Person 1, Person 2, Person 3... and finds them all safe. But he refuses to say, "Everyone in the building is safe," because he thinks there might be a "ghost" person hiding in the shadows that he hasn't checked yet.

Traditional math assumes these "ghosts" (unnamed numbers) exist. Gheorghiu says: No, they don't. In his system, if you can check every number you can name, then you have checked everything. There are no ghosts.

The Solution: The "Rulebook" Approach

Gheorghiu uses a framework developed by a philosopher named Sandqvist. Here is how it works in simple terms:

1. The Base (The Rulebook): Imagine a notebook where you write down the basic "material" rules of the game. For example, "If I have a block, I can stack another block." This is your Base.
2. Support (The Check): We ask: "Does this rulebook support the idea that 'All blocks are stable'?"
3. The Twist: In this system, the only things that exist are the things you can build with the blocks you have. You don't need to imagine invisible blocks.

The Magic Trick: Proving Math is Safe

The biggest question in math is: Is our system consistent? (i.e., Can we prove that $1 = 0$? If we can, the whole system collapses).

Usually, to prove math is safe, you need to use "super-math" (like infinite sets or transfinite numbers) that is more powerful than the math you are trying to prove. This feels like cheating. It's like using a sledgehammer to prove a screwdriver is safe.

Gheorghiu's Innovation:
He proves that Peano Arithmetic (the standard rules for counting) is consistent using only the standard rules of counting (induction).

• The Analogy: Imagine you want to prove a bridge is safe. Instead of building a giant crane to test it (which is overkill), you simply walk across it, step by step, showing that every single step holds your weight.
• He defines a "weight" for every number.
  • $0$ has weight 0.
  • $S(0)$ (which is 1) has weight 1.
  • $S(S(0))$ (which is 2) has weight 2.
  • If you add numbers, you add their weights.
• He shows that in his "Rulebook," you can never create a situation where a heavy number equals a light number (like $1 = 0$).
• Because the "Rulebook" works perfectly with simple counting, the whole system is safe. No sledgehammer needed.

Addressing the Critics

The paper anticipates two main objections:

1. The "Too Complicated" Objection: Critics say, "Your definition of 'support' is too complex; it's just as hard as the old math you're trying to replace."

   • Gheorghiu's Reply: "We aren't trying to replace math with a simpler game. We are trying to explain what math means. It's okay if the explanation is complex, as long as it matches how we actually think and reason."

2. The "Circular" Objection: Critics say, "You used the concept of 'natural numbers' to prove that 'natural numbers' are consistent. That's circular!"

   • Gheorghiu's Reply: "It's not a vicious circle; it's a virtuous one. To understand what a 'number' is, you must understand how to count. You can't understand the concept of a number without using the rules of counting. So, using counting to prove counting is safe is actually the only honest way to do it."

The Conclusion

This paper argues that we don't need to believe in a magical, invisible world of numbers to do math. We don't need to assume there are "ghost numbers" we can't name.

Instead, math is a self-contained game of rules. If we follow the rules of how we use numbers (induction), we can prove the game is safe, consistent, and complete, without needing to look outside the game for a "real" foundation.

In a nutshell: Math isn't about discovering a hidden universe; it's about mastering the rules of the conversation we have with each other about numbers. And as long as we follow the rules of the conversation, the conversation makes perfect sense.

Technical Summary

Here is a detailed technical summary of the paper "Classical Arithmetic Without Bivalence" by Alexander V. Gheorghiu.

1. Problem Statement

The paper addresses foundational issues in metamathematics, specifically the tension between formalism and denotationalism (model-theoretic semantics) regarding the justification of classical logic and arithmetic.

• The Core Issue: Traditional formalism struggles to explain $\omega$-incompleteness (where a theory proves $\phi(n)$ for every numeral $n$ but fails to prove $\forall x \phi(x)$) without treating it as a brute feature of the calculus. Denotationalism explains this via non-standard models but relies on a realist metaphysics (mind-independent structures) and a truth-conditional theory of meaning, which anti-realists (following Dummett) reject.
• The Specific Challenge: How can one provide an anti-realist account of classical reasoning that justifies the consistency of Peano Arithmetic (PA) without appealing to transfinite ordinals, recognition-transcendent truth, or a pre-existing ontology?
• The Gap: While inferentialism (grounding meaning in inferential roles) has been developed, applying it to the essential questions of metamathematics—specifically proving the consistency of PA using only elementary methods—remains an open challenge.

2. Methodology

The paper utilizes Sandqvist's semantics for classical logic without bivalence (2009), an inferentialist framework where meaning is determined by "support" relations rather than truth conditions.

• Framework:

  • Base ($B$): A set of atomic rules representing an agent's material inferential commitments.
  • Support ($\Vdash_B$): A judgment defined inductively (Figure 1) determining when a formula is supported by a base. Crucially, the universal quantifier is defined syntactically: $\Vdash_B \forall x \phi$ iff $\Vdash_B \phi[x \mapsto t]$ for all closed terms $t$.
  • No Bivalence: The system does not assume every sentence is either true or false in a model; validity is defined via derivability within bases.

• Key Definitions:

  • Numerically Definite Theory: A theory $\Gamma$ is numerically definite if for every closed term $t$, $\Gamma \Vdash (t = \bar{n})$ for some numeral $\bar{n}$. This ensures the theory "knows" its ontology is exhausted by the numerals.
  • Arithmetic Base ($A$): A specific construction of atomic rules (Figure 2) that encodes the axioms of PA (equality, successor, addition, multiplication) as inference rules.

3. Key Contributions and Results

A. Resolution of $\omega$-Incompleteness

The paper demonstrates that in Sandqvist's semantics, $\omega$-incompleteness vanishes for numerically definite theories.

• Proposition 5: If a theory $\Gamma$ is numerically definite, it is $\omega$-complete. If $\Gamma \Vdash \phi(\bar{n})$ for all numerals, then $\Gamma \Vdash \forall x \phi(x)$.
• Mechanism: Because the universal quantifier ranges only over the closed terms of the language, and the theory guarantees every term is equal to a numeral, the quantifier effectively ranges over the numerals. There are no "unnamed" entities in the ontology.

B. Induction as Meaning-Constitutive

• Proposition 6: In a numerically definite theory, the principle of mathematical induction is a consequence of the definition of the universal quantifier and the equality axioms.
• Significance: Induction is not an external axiom added to the system but is constitutive of the understanding of natural numbers within this framework. It follows from the fact that surveying the numerals suffices for the whole space.

C. The Collapse of the Q–PA Gap

• Result: The distinction between Robinson Arithmetic ($Q$) and Peano Arithmetic ($PA$) becomes semantic rather than proof-theoretic.
• Explanation: In model theory, $Q$ and $PA$ differ because $PA$ has induction over a domain that may contain non-standard elements. In this inferentialist semantics, the theory fixes the ontology. If the theory is numerically definite, the domain is the numerals. Thus, the "gap" where induction adds power over $Q$ disappears; the gap is an artifact of model-theoretic assumptions, not the inferentialist framework.

D. Elementary Consistency Proof for PA

The paper provides a direct proof of the consistency of Peano Arithmetic ($PA \not\Vdash \bot$).

• Construction: The author defines a specific Arithmetic Base ($A$) (Figure 2) containing rules for equality and arithmetic operations.
• Weight Function: A weight function $w(t)$ is defined on closed terms via primitive recursion:
  • $w(0) = 0$
  • $w(S(t)) = w(t) + 1$
  • $w(t_1 + t_2) = w(t_1) + w(t_2)$
  • $w(t_1 \cdot t_2) = w(t_1) \cdot w(t_2)$
• Proof Logic:
  1. By induction on the structure of derivations in base $A$, it is shown that if $t_1 = t_2$ is derivable, then $w(t_1) = w(t_2)$.
  2. Since $w(1) = 1$ and $w(0) = 0$, the equation $1 = 0$ cannot be derived.
  3. Therefore, $A$ does not support $\bot$ (contradiction).
  4. Since $A$ supports all axioms of $PA$, $PA$ is consistent.
• Constraint: The proof uses only ordinary induction on natural numbers. It explicitly avoids transfinite ordinals and does not rely on a model-theoretic interpretation.

E. Soundness and Completeness

• The paper establishes that if $PA$ is consistent in the inferential semantics ($PA \not\Vdash \bot$), it is consistent in classical logic ($PA \nvdash \bot$) because the semantics is sound with respect to classical derivability.
• It addresses the case where the language is extended with infinite constants (to achieve completeness). The consistency proof holds even in this enriched setting, as the new constants can be treated as alternative names for zero without altering the weight function's properties.

4. Significance and Philosophical Implications

• Anti-Realist Justification of Classical Logic: The paper successfully demonstrates that classical arithmetic can be justified without committing to a realist metaphysics (mind-independent structures). The "truth" of arithmetic statements is grounded in the inferential roles of terms.
• Pragmatic Circularity: The proof relies on induction to prove the consistency of induction. The author argues this is a "pragmatic" (virtuous) circularity, not a vicious one. In an inferentialist framework, induction is constitutive of the concept of natural numbers; therefore, using induction to justify the system is necessary and legitimate.
• Rejection of Hilbert's Programme Constraints: The result is not a contribution to Hilbert's Programme (which sought a finitist proof of consistency). Instead, it shows that consistency can be established within a framework that accepts general inductive reasoning but rejects the need for transfinite methods or realist ontology.
• Ontological Economy: The framework resolves the problem of $\omega$-incompleteness by strictly limiting the ontology to what is inferentially determined by the theory (the numerals), eliminating the need for "hidden" non-standard models.

Conclusion

Gheorghiu's paper provides a robust metamathematical framework where Peano Arithmetic is proven consistent using only elementary induction, grounded in an inferentialist semantics that resolves $\omega$-incompleteness by defining the ontology of arithmetic strictly through the theory's inferential commitments. This offers a viable anti-realist foundation for classical reasoning, bypassing the need for model-theoretic realism or transfinite ordinals.