Question

Consider intuitionistic predicate logic without function symbols. Prove the following extension of the existence property: \(\vdash \exists y \varphi\left(x_{1}, \ldots, x_{n}, y\right) \Leftrightarrow\) \(\vdash \varphi\left(x_{1}, \ldots, x_{n}, t\right)\), where \(t\) is a constant or one of the variables \(x_{1}, \ldots, x_{n} .\) (Hint: replace \(x_{1}, \ldots, x_{n}\) by new constants \(a_{1}, \ldots, a_{n}\) ).

Short answer

The proof shows that a witness \( t \) exists for the provable \( \exists y \varphi \).

Step-by-step solution

Problem Analysis

In intuitionistic predicate logic, the existence property states that if \( \exists y \varphi(x_1, \ldots, x_n, y) \) is provable, then there is a witness \( t \) that satisfies \( \varphi(x_1, \ldots, x_n, t) \). We need to prove that if we have a derivation of \( \exists y \varphi \), we can find a specific term \( t \) such that \( \varphi \) holds with \( y \) replaced by \( t \).

Introduction of Constants

Introduce new constants \( a_1, \ldots, a_n \) to replace the variables \( x_1, \ldots, x_n \). This transformation allows us to work within a framework where the specific variables do not influence the generality of the proof, and they behave like constants.

Reduction to Simple Case

With \( a_1, \ldots, a_n \) as constants, the existential formula becomes \( \exists y \varphi(a_1, \ldots, a_n, y) \). We want to show there exists some constant or variable, \( t \), such that \( \varphi(a_1, \ldots, a_n, t) \) holds.

Application of Existence Property

By the existence property of intuitionistic logic, since \( \exists y \varphi(a_1, \ldots, a_n, y) \) is provable, there must be a particular term \( t \) such that \( \varphi(a_1, \ldots, a_n, t) \) is provable. Thus, \( \varphi(x_1, \ldots, x_n, t) \) holds because \( a_i \) are just instances of \( x_i \).

Conclusion with Generalization

Since we derive \( \varphi(a_1, \ldots, a_n, t) \), and this expression is equivalent to \( \varphi(x_1, \ldots, x_n, t) \) with \( a_i\) replaced back with \( x_i\), we conclude \( \vdash \exists y \varphi(x_1, \ldots, x_n, y) \Leftrightarrow \vdash \varphi(x_1, \ldots, x_n, t) \) using the witness \( t \). This step uses the generalization that constants in the instance do not affect the variable case.

Key concepts

existence property

In the realm of intuitionistic predicate logic, the existence property is a vital concept.
It states that if a statement that claims the existence of an entity is provable, there must be a concrete term (a witness) for the entity.
This witness replaces the existential quantifier to satisfy the rest of the formula.In simpler terms, if a formula like \( \exists y \varphi(x_1, \ldots, x_n, y) \) is provable in a logic system, then you don't just have a theoretical guarantee that such a \( y \) exists.
Instead, you have or can construct a specific value, often denoted by \( t \), that can step into \( y\)'s shoes, making the formula hold true.Some characteristics of the existence property include:

• It's proof-centric. You need a provable assertion to invoke this property.
• It emphasizes constructive proof, aligning well with intuitionistic logic where proofs are not just about showing consistency but about building and identifying specific instances.

To use the existence property effectively, one must constructively show that there exists a particular expression or entity, often demanding substantial logical work to exhibit such a witness.

constants in logic

In logic, constants play a crucial role by serving as specific, non-variable entities in formulas, providing a fixed reference point.
When dealing with intuitionistic predicate logic, introducing constants can help in simplifying and managing logical expressions.In the exercise provided, new constants, such as \( a_1, \ldots, a_n \), are introduced to replace original variables \( x_1, \ldots, x_n \).
When constants replace variables, they allow us to treat them as fixed placeholders, simplifying proofs by removing variable dependencies and focusing on logical equivalence.Some key aspects to remember about constants in logic include:

• They act as specific, non-changing values within a logical expression.
• Using constants can make it easier to identify and apply logical equivalences, especially within proofs.
• They help ensure that the logical structure is maintained while enabling more straightforward substitution and transformation processes.

This method aligns with the notion of generalization in logical proofs, igniting flexible transitions between particular and general scenarios, which is quite useful in constructing proofs in intuitionistic settings.

logical equivalence

Logical equivalence is a powerful tool in logic, allowing us to interchange expressions without changing their truth values.
Two expressions are logically equivalent if, no matter the variable assignments, they yield the same truth values.In the context of this exercise, we assert a logical equivalence between \( \exists y \varphi(x_1, \ldots, x_n, y) \) and \( \varphi(x_1, \ldots, x_n, t) \).
The idea is that proving the existence of \( y \) is equivalent to showing a specific term \( t \) satisfies the original predicate.Here are key elements to understanding logical equivalence:

• It allows converting complex expressions into simpler or more manageable ones while maintaining truth.
• In proofs, it lets us replace proven equivalent expressions freely.
• It often involves systematic transformations, such as substituting variables or constants, maintaining logical consistency.

Recognizing logical equivalences is essential for constructing valid conclusions and building efficient proofs in intuitionistic predicate logic, especially when working within a framework that mandates constructive reasoning.