Chapter 2: Problem 6
Show \(\not \models \exists x \varphi \rightarrow \forall x \varphi\).
Short Answer
Expert verified
The entailment is not valid; a counterexample was shown.
Step by step solution
01
Understanding the Symbols
In this exercise, we need to prove that \( \exists x \varphi \rightarrow \forall x \varphi \) is not a valid entailment. Here, \( \exists x \varphi \) means "there exists an \( x \) such that \( \varphi \) is true" and \( \forall x \varphi \) means "for all \( x \), \( \varphi \) is true." The notation \( ot \models \) indicates we have to show it is not a logical consequence.
02
Considering the Implication
The implication \( \exists x \varphi \rightarrow \forall x \varphi \) will be false if \( \exists x \varphi \) is true but \( \forall x \varphi \) is false. Therefore, we need to find a model or an interpretation where this is the case.
03
Creating a Counterexample
Let's choose a simple model with a universe consisting of two distinct elements, say \( a \) and \( b \). Define \( \varphi(x) \) such that \( \varphi(a) \) is true and \( \varphi(b) \) is false. In this universe, \( \exists x \varphi \) is true because \( \varphi(a) \) is true. However, \( \forall x \varphi \) is false because \( \varphi(b) \) is false.
04
Analyzing the Model
In this model, the premise \( \exists x \varphi \) holds, but the conclusion \( \forall x \varphi \) does not. Therefore, the implication \( \exists x \varphi \rightarrow \forall x \varphi \) is false under this interpretation. This demonstrates that the entailment \( \exists x \varphi \rightarrow \forall x \varphi \) cannot be valid in all models.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logical Entailment
In predicate logic, logical entailment is a fundamental concept. It refers to a situation where one statement logically follows from another. This means that if the first statement (premise) is true, then the following statement (conclusion) must also be true.
In formal notation, this is represented as \( A \models B \), meaning statement \( A \) logically entails statement \( B \). However, for the exercise given, we're discussing a situation where logical entailment does not occur.
In our case, \( \exists x \varphi ot\models \forall x \varphi \). This indicates that it is not universally the case that if "there exists" an \( x \) such that \( \varphi \) is true, it logically follows "for all" \( x \), \( \varphi \) is true. Understanding when logical entailment does not take place is as critical as knowing when it does, as it helps identify false assumptions.
In formal notation, this is represented as \( A \models B \), meaning statement \( A \) logically entails statement \( B \). However, for the exercise given, we're discussing a situation where logical entailment does not occur.
In our case, \( \exists x \varphi ot\models \forall x \varphi \). This indicates that it is not universally the case that if "there exists" an \( x \) such that \( \varphi \) is true, it logically follows "for all" \( x \), \( \varphi \) is true. Understanding when logical entailment does not take place is as critical as knowing when it does, as it helps identify false assumptions.
Existential Quantifier
The existential quantifier, denoted as \( \exists \), is an essential component of predicate logic. It is used to express that there is at least one element in the domain for which a particular property holds.
In the formula \( \exists x \varphi \), it states "there exists an \( x \) such that \( \varphi \) is true." This quantifier is particularly useful when discussing properties or statements that do not hold for the entire set, but are true for some members of the set.
In our example, \( \exists x \varphi \) indicates that at least one specific \( x \) makes the statement \( \varphi \) true. It acts as a bridge linking a specific case to a broader logical discussion, highlighting the existence of an instance of truth in the universe of discourse.
In the formula \( \exists x \varphi \), it states "there exists an \( x \) such that \( \varphi \) is true." This quantifier is particularly useful when discussing properties or statements that do not hold for the entire set, but are true for some members of the set.
In our example, \( \exists x \varphi \) indicates that at least one specific \( x \) makes the statement \( \varphi \) true. It acts as a bridge linking a specific case to a broader logical discussion, highlighting the existence of an instance of truth in the universe of discourse.
Universal Quantifier
Predicate logic also features the universal quantifier, denoted as \( \forall \). This quantifier is used to denote that a statement applies to all elements within a certain domain.
The formula \( \forall x \varphi \) reads as "for all \( x \), \( \varphi \) is true," meaning the property \( \varphi \) must hold for every possible value of \( x \). While the existential quantifier speaks to cases where something is true for at least one item, the universal quantifier strengthens a statement, requiring it to hold without exception for all items under consideration.
In the context of our exercise, \( \forall x \varphi \) demands a level of universality that is not automatically guaranteed by a singular instance of truth, which \( \exists x \varphi \) provides. This shows that a statement being true for a specific instance does not automatically imply it is true universally.
The formula \( \forall x \varphi \) reads as "for all \( x \), \( \varphi \) is true," meaning the property \( \varphi \) must hold for every possible value of \( x \). While the existential quantifier speaks to cases where something is true for at least one item, the universal quantifier strengthens a statement, requiring it to hold without exception for all items under consideration.
In the context of our exercise, \( \forall x \varphi \) demands a level of universality that is not automatically guaranteed by a singular instance of truth, which \( \exists x \varphi \) provides. This shows that a statement being true for a specific instance does not automatically imply it is true universally.
Counterexample in Logic
A counterexample in logic is a powerful tool used to demonstrate the falsehood of a proposed generalization or statement. To disprove a statement universally, one needs to find just one example where the statement fails.
In our exercise, a counterexample is provided to challenge the implication \( \exists x \varphi \rightarrow \forall x \varphi \). By outlining a model with two elements \( a \) and \( b \) where \( \varphi(a) \) is true and \( \varphi(b) \) is false, we see that \( \exists x \varphi \) holds while \( \forall x \varphi \) does not.
This counterexample effectively shows a situation where the initial premise is true, yet the consequent fails to hold. Thus, using counterexamples is fundamental in logic for proving the incorrectness of hypothesized entailments and refining our understanding of logical structures.
In our exercise, a counterexample is provided to challenge the implication \( \exists x \varphi \rightarrow \forall x \varphi \). By outlining a model with two elements \( a \) and \( b \) where \( \varphi(a) \) is true and \( \varphi(b) \) is false, we see that \( \exists x \varphi \) holds while \( \forall x \varphi \) does not.
This counterexample effectively shows a situation where the initial premise is true, yet the consequent fails to hold. Thus, using counterexamples is fundamental in logic for proving the incorrectness of hypothesized entailments and refining our understanding of logical structures.
