Σ⊢θ Implies Σ⊢(∃x)(θ) In First-Order Logic Exploring Provability And Quantifiers

In the realm of mathematical logic, particularly within first-order logic, a fundamental question arises: If a formula θ is provable from a set of premises Σ (denoted as Σ⊢θ), does it necessarily follow that the formula (∃x)(θ) is also provable from Σ (denoted as Σ⊢(∃x)(θ))? This question delves into the heart of quantifiers and their interaction with provability, touching upon crucial concepts such as variable binding, free variables, and the rules of inference in first-order logic. This article aims to provide a comprehensive exploration of this question, examining the conditions under which the implication holds, the potential pitfalls, and the deeper logical principles at play.

Unpacking the Notation and Core Concepts

Before diving into the intricacies of the question, it's crucial to establish a clear understanding of the notation and underlying concepts. Let's break down the key components:

• Σ (Sigma): Represents a set of formulas within a given language L. These formulas serve as the premises or axioms from which we aim to derive other formulas.
• θ (Theta): Denotes a formula in the language L. This is the formula we are trying to prove or derive from the premises in Σ.
• ⊢ (Turnstile): This symbol signifies provability. Σ⊢θ means that the formula θ can be proven from the set of formulas Σ using the rules of inference of the logical system.
• (∃x) (Exists x): This is the existential quantifier. (∃x)(θ) asserts that there exists at least one value for the variable x that makes the formula θ true.

With these definitions in place, we can rephrase the central question: If we can prove θ from Σ, can we automatically conclude that there exists some x for which θ holds, and can we prove this existential statement from Σ as well?

The Importance of Free Variables

A crucial aspect to consider when dealing with quantifiers is the concept of free variables. A variable in a formula is considered free if it is not bound by any quantifier (either existential or universal). For example, in the formula P(x), the variable x is free. However, in the formula (∃x)P(x), the variable x is bound by the existential quantifier. The distinction between free and bound variables is essential because it affects how we can manipulate and infer new formulas.

The question of whether Σ⊢θ implies Σ⊢(∃x)(θ) hinges on whether the variable x appears as a free variable in θ. If x is not free in θ, then the implication might not hold, and we need to be cautious about introducing the existential quantifier.

Exploring the Implication: When Does Σ⊢θ Imply Σ⊢(∃x)(θ)?

The implication Σ⊢θ implies Σ⊢(∃x)(θ) does not universally hold in first-order logic. It holds true under specific conditions, primarily when the variable x is free in the formula θ and under certain restrictions within the proof system. To understand this better, let's delve into the reasoning behind it.

The core idea is that if θ is provable from Σ, then θ must be true in all models that satisfy Σ. If x is free in θ, then θ essentially makes a statement about the object denoted by x. If θ is true in all models of Σ, then there must exist at least one object (the one denoted by x) that makes θ true. This is the intuition behind why the existential quantifier might be introduced.

However, this intuition must be formalized within the proof system. Most proof systems for first-order logic include an existential generalization rule (or a similar rule) that allows us to infer (∃x)(θ) from θ, provided that x is free in θ. This rule captures the idea that if we have proven θ for a specific object (denoted by x), then we can generalize to the existence of an object satisfying θ.

The Existential Generalization Rule

The existential generalization rule can be formally stated as follows:

If we have a derivation of θ from Σ, and x is a free variable in θ, then we can infer (∃x)(θ) and add it to our derivation. This can be represented as:

Σ ⊢ θ
------ (Existential Generalization)
Σ ⊢ (∃x)(θ)

The condition that x must be free in θ is crucial. If x is already bound in θ, then introducing another existential quantifier might change the meaning of the formula and lead to incorrect inferences. For instance, consider the formula (∀x)P(x). If we were to apply existential generalization without the free variable condition, we might incorrectly infer (∃y)(∀x)P(x), which is a different and potentially stronger statement.

Counterexamples: When the Implication Fails

To further illustrate the importance of the free variable condition, let's examine scenarios where the implication Σ⊢θ implies Σ⊢(∃x)(θ) fails to hold. These counterexamples highlight the subtleties of quantifier manipulation and the need for careful application of inference rules.

Consider the case where θ is a closed formula, meaning it contains no free variables. For example, let θ be the formula P(a), where 'a' is a constant symbol. Suppose Σ⊢P(a). Can we conclude Σ⊢(∃x)P(x)? In this specific scenario, if our language contains at least one constant symbol, then the inference holds because P(a) directly asserts the existence of an object satisfying P. However, if θ were a more complex closed formula where the existential quantifier's introduction isn't directly implied, the implication may falter.

Another important situation to consider is when the language L is empty, meaning it contains no constant symbols. In this case, even if Σ⊢θ, we cannot necessarily infer Σ⊢(∃x)θ. The existential quantifier asserts the existence of an object, but if our language lacks the means to refer to objects (through constant symbols), we cannot establish this existence. This is a more nuanced example that emphasizes the relationship between the language, the formulas, and the interpretations we can construct.

The Role of Proof Systems and Completeness

The validity of the implication Σ⊢θ implies Σ⊢(∃x)(θ) also depends on the specific proof system we are using. A proof system is a set of axioms and inference rules that define what constitutes a valid proof. Different proof systems may have different rules for handling quantifiers, and some systems might not have an explicit existential generalization rule.

In a sound and complete proof system for first-order logic, the implication Σ⊢θ implies Σ⊢(∃x)(θ) (when x is free in θ) will hold. Soundness means that every provable formula is also logically valid (true in all models), and completeness means that every logically valid formula is provable. Gödel's Completeness Theorem guarantees the existence of such proof systems for first-order logic.

However, it's important to note that even in a complete proof system, the existential generalization rule (or an equivalent rule) is necessary to bridge the gap between θ and (∃x)(θ). Without such a rule, we cannot simply assume that the existence of an object satisfying θ follows from the provability of θ itself.

Completeness and Semantic Implications

The completeness theorem ties the syntactic notion of provability (⊢) to the semantic notion of logical consequence (⊨). Σ⊢θ means θ is provable from Σ, while Σ⊨θ means θ is true in all models that satisfy Σ. Completeness tells us that if Σ⊨θ, then Σ⊢θ. This connection is crucial for understanding why the existential generalization rule is valid.

If Σ⊢θ, then by soundness, Σ⊨θ. If x is free in θ, and θ is true in all models of Σ, then it must be the case that there exists an object in each of those models that satisfies θ. This semantic argument provides further justification for the existential generalization rule, as it demonstrates that the inference from θ to (∃x)(θ) preserves truth in all models.

Practical Implications and Examples

To solidify our understanding, let's consider some practical implications and examples.

Suppose we are working in a first-order logic formalizing arithmetic. Let θ be the formula x + 0 = x, where x is a free variable. This formula states that adding 0 to any number x results in x itself. If we can prove θ from a set of axioms for arithmetic (Σ), then we can use existential generalization to infer (∃x)(x + 0 = x). This means we can prove that there exists at least one number x such that x + 0 = x, which is a fundamental property of arithmetic.

However, if we were to consider a formula like (∀y)(y = y), which is a tautology (always true), we cannot directly apply existential generalization to introduce another quantifier. The formula already asserts a universal property, and adding an existential quantifier might lead to confusion or misinterpretations.

Exercise 2.7.1.1: A Deeper Dive

Referencing the original context, Exercise 2.7.1.1 from