Question

Show \(\models \forall x(\varphi \rightarrow \psi) \rightarrow(\forall x \varphi \rightarrow \forall x \psi) ; \models(\exists x \varphi \rightarrow \exists x \psi) \rightarrow \exists x(\varphi \rightarrow \psi)\) \(\models \forall x(\varphi \mapsto \psi) \rightarrow(\forall x \varphi \leftrightarrow \forall x \psi) ; \models(\forall x \varphi \rightarrow \exists x \psi) \leftrightarrow \exists x(\varphi \rightarrow \psi) ;\) \(\models(\exists x \varphi \rightarrow \forall x \psi) \rightarrow \forall x(\varphi \rightarrow \psi)\)

Short answer

This statement uses universal and existential quantifiers to show various logical implications and equivalences.

Step-by-step solution

Understanding Implications: Universal Quantifiers

We begin by analyzing the statement \(\models \forall x(\varphi \rightarrow \psi) \rightarrow(\forall x \varphi \rightarrow \forall x \psi)\). This statement asks us to show that if for every \(x, \varphi \rightarrow \psi\) holds, then if \(\forall x \varphi\) holds, \(\forall x \psi\) must also hold. This is a logical implication involving universal quantification.

Analyzing Existential Quantifiers

The second component \(\models (\exists x \varphi \rightarrow \exists x \psi) \rightarrow \exists x(\varphi \rightarrow \psi)\) denotes if there exists an \(x\) such that \(\varphi\) implies \(\psi\) for some \(x\), then there exists an \(x\) such that \(\varphi\) implies \(\psi\) for that \(x\). This applies existential rules through logical implication.

Using Logical Equivalence

The expression \(\models\forall x(\varphi \mapsto \psi) \rightarrow(\forall x \varphi \leftrightarrow \forall x \psi)\) denotes that if each \(x\) yields \(\varphi \mapsto \psi\), then \(\forall x \varphi\) bidirectionally relates with \(\forall x \psi\). This showcases the duality of universal implications directly reinforcing logical equivalence.

Combining Universal and Existential Quantifiers

The expression \(\models(\forall x \varphi \rightarrow \exists x \psi) \leftrightarrow \exists x(\varphi \rightarrow \psi)\) involves both universal and existential quantifiers. It states that an implication with a universal antecedent and existential consequent mirrors that of an existential quantifier applied as an implication.

Implications among Mixed Quantifiers

The final expression \(\models(\exists x \varphi \rightarrow \forall x \psi) \rightarrow \forall x(\varphi \rightarrow \psi)\) underscores if there exists some \(x\) such that \(\varphi\) leads to a universal \(\psi\), then universally for every \(x\), \(\varphi\) results in \(\psi\). It strengthens the necessity of universal results from specific existential instances.

Key concepts

Universal Quantification

Universal quantification is a fundamental concept in first-order logic that allows us to express statements that apply to all elements in a given domain. When we see a statement of the form \( \forall x \varphi \), it means "for every x, \( \varphi \) is true." This concept is immensely useful when establishing properties or truths that must hold universally across all instances.

In the original exercise, statements involving universal quantifiers like \( \models \forall x(\varphi \rightarrow \psi) \rightarrow(\forall x \varphi \rightarrow \forall x \psi) \) are prevalent. This particular statement suggests that if a condition \( \varphi \rightarrow \psi \) holds for all \( x \), then if \( \varphi \) is universally true, \( \psi \) must also be universally true.

Consider universal quantification as laying the groundwork where every possible case must abide by the established rule. It's the logic’s way of enforcing consistency across the spectrum of possibilities.

Existential Quantification

Existential quantification allows us to express that there exists at least one element in the domain for which a particular condition holds true. This is denoted by the symbol \( \exists x \varphi \), which reads as "there exists an \( x \) such that \( \varphi \) is true." Existential quantifiers are important for highlighting specific instances that satisfy a condition without requiring all to meet it.

In the exercise, an expression such as \( \models (\exists x \varphi \rightarrow \exists x \psi) \rightarrow \exists x(\varphi \rightarrow \psi) \) illustrates the use of existential quantification in logical implications. It posits that if some \( x \) satisfies \( \exists x \varphi \), leading to \( \exists x \psi \), then there is an \( x \) for which \( \varphi \rightarrow \psi \).

This can be visualized as finding at least one pathway from condition \( \varphi \) to \( \psi \), relying on the strength of specific instances to denote existence.

Logical Implication

Logical implication is a crucial building block in first-order logic. It expresses a relationship where one statement (the antecedent) being true guarantees that another statement (the consequent) is also true. This is typically represented as \( \varphi \rightarrow \psi \), meaning if \( \varphi \), then \( \psi \). This directionality is vital for constructing logical arguments and proofs.

For example, \( \models (\exists x \varphi \rightarrow \forall x \psi) \rightarrow \forall x(\varphi \rightarrow \psi) \) suggests that if some \( x \) leads to a universal conclusion \( \psi \), then universally, \( \varphi \) implies \( \psi \) for each \( x \). Logical implications help affirm the cause-effect chain in logical reasoning and validate propositions soundly.

Logical implications embed completeness in logical frameworks by ensuring that truth in one aspect mandates truth in another, facilitating coherent logic systems.

Logical Equivalence

Logical equivalence extends logical implication by establishing that two statements are interchangeable in truth value. Represented by \( \leftrightarrow \), logical equivalence states that both directions, \( \varphi \rightarrow \psi \) and \( \psi \rightarrow \varphi \), must be true. This signifies a bi-directional truth flow, cementing that the statements are effectively the same in meaning.

Within the given problem, expressions such as \( \models \forall x(\varphi \mapsto \psi) \rightarrow(\forall x \varphi \leftrightarrow \forall x \psi) \) demonstrate logical equivalence. This asserts that peering through the universal lens, the correspondence between \( \varphi \) and \( \psi \) is consistent and mutual, showing equivalence.

Logical equivalence thus supports showing that different formulations of logic convey the same truth, allowing for fluid transitions between expressions and robust mathematical reasoning.