Question

Show \(\models \forall x(\varphi(x) \leftrightarrow \exists y(x=y \wedge \varphi(y)))\) and \(\vDash \forall x(\varphi(x) \leftrightarrow \forall y(x=y \rightarrow \varphi(y)))\), where \(y\) does not occur in \(\varphi(x)\)

Short answer

Both logical equivalences are universally valid in any domain.

Step-by-step solution

Understanding the Problem Statement

The exercise requires us to demonstrate the validity (\(\models\)) of two logical equivalences. These equivalences involve quantified statements, where \(\forall\) represents 'for all' and \(\exists\) represents 'there exists'. We need to show that each statement is universally valid.

Analyzing the First Equation

We need to show that \(\models \forall x(\varphi(x) \leftrightarrow \exists y(x=y \wedge \varphi(y)))\). This means that if \(\varphi(x)\) holds for all \(x\), then it is equivalent to there existing some \(y\) such that \(y\) is equal to \(x\) and \(\varphi(y)\) holds.

Proving the First Equation's Validity

For any element \(a\) in the domain, by the nature of equality, \((a=a)\) is true. Therefore, \(\exists y(a=y \wedge \varphi(y))\) is true if \(\varphi(a)\) is true, demonstrating equivalence. Similarly, if \(\exists y(a=y \wedge \varphi(y))\) is true, then \(\varphi(a)\) must be true. Hence, this equivalence holds universally.

Analyzing the Second Equation

We need to show that \(\vDash \forall x(\varphi(x) \leftrightarrow \forall y(x=y \rightarrow \varphi(y)))\). Here, our goal is to prove that for any \(x\), \(\varphi(x)\) is equivalent to for all \(y\), if \(x=y\) then \(\varphi(y)\) must be true.

Proving the Second Equation's Validity

If \(\varphi(a)\) is true for some \(a\), then for any \(y\), \(a = y\) implies \(\varphi(y)\) is true. Conversely, if \(\forall y(a = y \rightarrow \varphi(y))\) is true, then specifically for \(y = a\), \(\varphi(a)\) is true. This proves the equivalence is always valid.

Key concepts

Quantifiers

Quantifiers are fundamental in logic, particularly in expressing statements about whole sets of objects. In logical expressions, we often encounter two main types of quantifiers: universal quantifiers and existential quantifiers. The universal quantifier is denoted by the symbol \(\forall\) and translates to "for all," indicating that a statement applies to all elements within a specific domain. For example, \(\forall x \varphi(x)\) means that the predicate \(\varphi(x)\) is true for every \(x\).
On the other hand, the existential quantifier is represented by \(\exists\), meaning "there exists." It specifies that there is at least one element in the domain for which the statement holds true. For instance, \(\exists x \varphi(x)\) conveys that \(\varphi(x)\) is true for at least one \(x\).
In the exercise, understanding these quantifiers is key to grasping why the logical expressions are valid. We are to show that a universally quantified expression is logically equivalent to another expression involving either the existential or universal quantifier in specific forms.

Validity

Validity in logic refers to a scenario where, if the premises are true, the conclusion must necessarily be true. In the context of our exercise, validity involves demonstrating that the logical equivalences we are analyzing are true under all interpretations or structures. This means showing that no matter what specific elements we choose from the domain, the equivalences hold.
For the given logical expressions, the task is to verify this unconditionally. When we say a logical statement is valid, we mean that it's impossible for the premises to be true and the conclusion false across any possible interpretation.

• For the statement \(\forall x(\varphi(x) \leftrightarrow \exists y(x=y \wedge \varphi(y)))\), it means that for any instance where \(\varphi(x)\) holds, there is equivalently a \(y\) such that \(x=y\) and \(\varphi(y)\) holds.
• Similarly, \(\forall x(\varphi(x) \leftrightarrow \forall y(x=y \rightarrow \varphi(y)))\) signifies that if \(\varphi(x)\) is true for all \(x\), it correlates with the criterion that for every \(y\), if \(x = y\), then \(\varphi(y)\) is also true.

Each equivalence ensures that the truth is preserved despite any changes in variable assignments, thereby proving their validity.

Equality

Equality is a crucial concept in logic and mathematics, providing a basis for comparing objects or elements. In logical proofs, we often use equality to establish connections between different variables or terms within an argument or equation.
The equality symbol \(=\) indicates that the terms on either side of it are identical in value or simply refer to the same object.

• In the exercise, one of the key points under scrutiny is understanding how equality impacts the logical equivalences. For instance, \(\exists y(x=y \wedge \varphi(y))\) uses equality (\(x = y\)) to create a binding, showing that if \(\varphi\) applies to any \(y\) that is equal to \(x\), then it seamlessly aligns with \(\varphi(x)\).
• The construct \(\forall y(x=y \rightarrow \varphi(y))\), further emphasizes the role of equality by suggesting that if \(x\) equals any \(y\), then \(\varphi(y)\) must hold true, reinforcing the universal truth of \(\varphi\).

Through equality, these expressions convey a consistent truth, maintaining logical coherence in different contexts and interpretations.

Logical Proofs

Logical proofs are structured arguments that validate the truth of a statement through a series of logically connected premises leading to a conclusion. In proving logical equivalences, like those in our exercise, the process involves showing that both statements imply each other.
Essentially, a logical proof needs to confirm that if one side of the equivalence is true, the other side must be true as well, and vice versa. This bilateral truth establishes equivalence.
To break down the given exercise:

• For proving \(\forall x(\varphi(x) \leftrightarrow \exists y(x=y \wedge \varphi(y)))\), the proof involves showing that for any instance \(a\) of \(x\), if \(\varphi(a)\) is true, there exists a corresponding \(y\) such that \(x = y\) and \(\varphi(y)\) holds, and conversely.
• In proving \(\forall x(\varphi(x) \leftrightarrow \forall y(x=y \rightarrow \varphi(y)))\), the argument constructs that if \(\varphi(a)\) holds, then for all \(y\), \(a = y\) leads to \(\varphi(y)\), and vice versa. Thus, confirming the mutual validity of each condition.

These proofs are fundamental in establishing logical equivalences, ensuring reasoning is sound and arguments are robust.