Question

Show $\models \mathrm{\exists }x(\phi (x)\to \mathrm{\forall }y\phi (y))$. (It is instructive to think of $\phi (x)$ as ' $x$ drinks').

Short answer

The statement is valid because in every interpretation, either everyone drinks or the implication holds vacuously.

Step-by-step solution

Understanding the Statement

We need to show that the statement $\mathrm{\exists }x(\phi (x)\to \mathrm{\forall }y\phi (y))$ is valid, meaning it is true in every interpretation. Here, $\phi (x)$ can be interpreted as a property like ' $x$ drinks'. This implies that there exists some object $x$ such that if $x$ has the property of drinking, then every object $y$ must also drink.

Rephrasing the Statement

The statement $\mathrm{\exists }x(\phi (x)\to \mathrm{\forall }y\phi (y))$ can be rewritten to better understand its logic: 'There exists some $x$ such that if $x$ drinks, then everything drinks'. This essentially indicates a special case where if any one entity has a drinking property, all entities have it.

Assume a Scenario for Clarity

Assume that within our universe, $\mathrm{\forall }x\phi (x)$ is true, meaning every element drinks. In this instance, for any chosen individual $x_0$, it is true that $\phi (x_0)\to \mathrm{\forall }y\phi (y)$ holds because $\phi (x_0)$ and $\mathrm{\forall }y\phi (y)$ are both true. Therefore, there exists such an $x_0$.

Understand Logical Validity

To prove logical validity, consider any interpretation: If $\mathrm{\forall }x\phi (x)$ is false, meaning there exists some $x$ such that $eg\phi (x)$ is true, then $\mathrm{\exists }x\phi (x)$ is false, which trivially makes $\phi (x)\to \mathrm{\forall }y\phi (y)$ true because the implication is true when its antecedent is false. Thus, $\mathrm{\exists }x(\phi (x)\to \mathrm{\forall }y\phi (y))$ holds by vacuous truth.

Summarize the Proof

For every model where $\mathrm{\forall }x\phi (x)$ is true, our statement holds as explained. For any model where $\mathrm{\forall }x\phi (x)$ is false, $\phi (x)$ is false for some $x$, making the implication $\phi (x)\to \mathrm{\forall }y\phi (y)$ trivially true. Hence, in all interpretations, there is some $x$ for which the statement holds.

Key concepts

Logical Validity

Logical validity refers to a situation where a statement or formula is true under all possible interpretations. In other words, if something is logically valid, it doesn't matter how you change the context or the specifics; the statement remains true. This concept is fundamental in mathematics and philosophy, especially in predicate logic. Predicate logic allows for reasoning with statements about objects and their properties.

In logical validity, we want to ensure that the argument holds for every possible scenario. This is often done by considering all interpretations of the statement's variables and showing the statement holds in each case. For example, when dealing with a logically valid statement like $\mathrm{\exists }x(\phi (x)\to \mathrm{\forall }y\phi (y))$, we demonstrate that no matter what circumstances or values $x$ and $y$ take, the statement remains true.

• Logical arguments that are valid carry a strong assurance; if the premises are true, the conclusion cannot be false.
• Understanding logical validity helps ensure clarity and correctness in reasoning.

Quantifiers in Logic

Quantifiers are symbols used in logic to specify the number of instances or the scope of application for a given predicate. The two main types of quantifiers are the existential quantifier ($\mathrm{\exists }$) and the universal quantifier ($\mathrm{\forall }$). These quantifiers play a crucial role in constructing logical statements in predicate logic.

1. **Existential Quantifier ($\mathrm{\exists }$)**: This symbol expresses that there exists at least one object in the domain for which the predicate holds true. For example, $\mathrm{\exists }x\phi (x)$ means there is some object $x$ such that $\phi (x)$ is true, or in our earlier context, "there exists someone who drinks."

2. **Universal Quantifier ($\mathrm{\forall }$)**: This quantifier claims that for all objects in the domain, the predicate expression is true. For instance, $\mathrm{\forall }y\phi (y)$ implies every object $y$ has the property $\phi$, or "everyone drinks."

• Quantifiers help us convey complex ideas compactly and with precision.
• Understanding how to manipulate and interpret quantifiers is critical in formulating and comprehending logical expressions effectively.

Logical Implication

Logical implication is represented by the symbol $\to$. It defines a relationship between two propositions, where the implication $A\to B$ suggests that if proposition $A$ is true, then proposition $B$ must also be true. This concept is pivotal in understanding logical arguments and their structures.

The truth of a logical implication hinges on its antecedent and consequent:

• *Antecedent* ($A$): The condition or premise of the implication.
• *Consequent* ($B$): The result or conclusion that follows the antecedent.

A critical aspect of logical implication is that if the antecedent is false, the implication is considered true "vacuously". This can seem counterintuitive but ensures consistency within logical frameworks. For example, in the statement $\phi (x)\to \mathrm{\forall }y\phi (y)$, if $\phi (x)$ is false, the whole statement is true regardless of $\mathrm{\forall }y\phi (y)$.

Understanding logical implication helps in forming accurate logical and mathematical proofs. It allows for deducing conclusions from given premises and analyzing whether certain propositions necessarily follow from others, thus sharpening critical thinking skills.