Question

Write a formula "saying" that at least four distinct objects satisfy predicate \(P\).

Short answer

\(\exists x_1, x_2, x_3, x_4 (P(x_1) \wedge P(x_2) \wedge P(x_3) \wedge P(x_4) \wedge x_1 \neq x_2 \wedge x_1 \neq x_3 \wedge x_1 \neq x_4 \wedge x_2 \neq x_3 \wedge x_2 \neq x_4 \wedge x_3 \neq x_4)\)

Step-by-step solution

Understanding the Predicate

In logic, a predicate is an expression that can be true or false depending on the values of its variables. Here, we have a predicate \(P(x)\), where \(x\) is an object.

English Interpretation

The requirement of the problem is to express the idea that there are at least four distinct objects for which the predicate \(P\) is true.

Use of Quantifiers

To express 'at least four', we need to use existential quantifiers. The statement "There exist at least four distinct objects" means we need to express four existential quantifications with distinct variable names.

Constructing the Formula

We need to find four distinct objects \(x_1, x_2, x_3, x_4\) such that \(P(x_1)\), \(P(x_2)\), \(P(x_3)\), and \(P(x_4)\) are true, and each object is different from the others.

Building Distinctness

We need to ensure that \(x_1, x_2, x_3, x_4\) are distinct. This can be done by stating inequalities like \(x_1 eq x_2\), \(x_1 eq x_3\), etc.

Final Formula

Combining all pieces, the formula is: \[\exists x_1, x_2, x_3, x_4 \, (P(x_1) \wedge P(x_2) \wedge P(x_3) \wedge P(x_4) \wedge (x_1 eq x_2) \wedge (x_1 eq x_3) \wedge (x_1 eq x_4) \wedge (x_2 eq x_3) \wedge (x_2 eq x_4) \wedge (x_3 eq x_4))\]

Key concepts

Existential Quantifiers

Existential quantifiers are a fundamental concept in predicate logic. They allow us to express the existence of certain objects that satisfy particular properties. In our exercise, an existential quantifier is represented by the symbol \(\exists\). This symbol can be read as "there exists" or "there are." It helps indicate that at least one or more objects meet the condition specified by a formula.

When dealing with more complex conditions, like "at least four distinct objects satisfy a predicate," we stack these quantifiers with different variable names for each object. For example, the statement requires us to use not just one but four existential quantifiers, each representing a different object with the predicate \(P\) being true for each.

The significance of the existential quantifier goes beyond mere existence; it gives us a way to formally express statements about some objects' characteristics within a domain, which is crucial in areas like mathematics, computer science, and philosophy.

Logical Formulas

Logical formulas are expressions that combine predicates, logical operators, and quantifiers to create a statement that can be true or false. The use of logical formulas allows us to create complex statements about objects, even though each variable can only take one value at a time. In our case, the exercise requires a logical formula to express that at least four distinct objects satisfy a specific predicate.

Constructing a logical formula involves several steps:

• Identify Variables: Determine the objects in question. In our task, we labeled them as \(x_1, x_2, x_3,\) and \(x_4\).
• Apply Predicates: Use predicates like \(P(x)\) to specify the properties we are interested in.
• Add Quantifiers: Use existential quantifiers to denote the existence of these objects.
• Ensure Distinctness: Add constraints ensuring that these objects are different.

The logical formula constructed in our solution encapsulates all these elements to create an accurate and meaningful expression.

Distinct Objects

In logical expressions, especially those involving quantifiers, ensuring the distinctness of objects is often necessary. The requirement that objects must be distinct implies that they are different from each other, and in our hypothesis "at least four distinct objects satisfy predicate \(P\)," this is an essential component.

To ensure distinctness, the logical formula includes inequalities between all variable pairs. For example, for the variables \(x_1, x_2, x_3,\) and \(x_4\), we assert that each is not equal to the others, such as \(x_1 eq x_2\), \(x_1 eq x_3\), and so forth, using conjunctions (\(\wedge\)) to link these inequalities together.

This process is crucial because simply asserting existence without considering distinctness could let different variables refer to the same object, misrepresenting the task's requirement. By clearly listing the inequalities, we make sure that each variable indeed points to a separate, unique object.