Question

Find prenex normal forms for (a) \(\neg((\neg \forall x \varphi(x) \vee \forall x \psi(x)) \wedge(\exists x \sigma(x) \rightarrow \forall x \tau(x)),\), (b) \(\forall x \varphi(x) \leftrightarrow \exists x \psi(x)\), (c) \(\neg(\exists x \varphi(x, y) \wedge \forall y(\psi(y)) \rightarrow \varphi(x, x)) \rightarrow \exists x \forall y \sigma(x, y)))\), (d) \(((\forall x \varphi(x) \rightarrow \exists y \psi(x, y)) \rightarrow \psi(x, x)) \rightarrow \exists x \forall y \sigma(x, y)\).

Short answer

Convert logical formulas to sequence of quantifiers followed by a quantifier-free formula, applying logical equivalences and normal forms.

Step-by-step solution

Eliminate Implications and Biconditional in (a)

The formula is \( eg((eg \forall x \varphi(x) \vee \forall x \psi(x)) \wedge(\exists x \sigma(x) \rightarrow \forall x \tau(x)) \). Replace the implication with its equivalent form: \((A \rightarrow B) \equiv (eg A \vee B)\). The formula becomes \( eg((eg \forall x \varphi(x) \vee \forall x \psi(x)) \wedge (eg \exists x \sigma(x) \vee \forall x \tau(x)))\).

Apply Double Negation and De Morgan's Laws in (a)

Use the equivalence \(eg \exists = \forall eg \) and \(eg \forall = \exists eg \) to simplify. First, use De Morgan's Laws to distribute the negation: \( eg((eg \forall x \varphi(x) \vee \forall x \psi(x)) \wedge (eg \exists x \sigma(x) \vee \forall x \tau(x)))\) becomes \((\forall x \varphi(x) \wedge eg \forall x \psi(x)) \vee (\exists x \sigma(x) \wedge eg \forall x \tau(x))\).

Simplify Quantifiers for (a)

Change \(eg \forall x \psi(x)\) to \(\exists x eg \psi(x)\) and \(eg \forall x \tau(x)\) to \(\exists x eg \tau(x)\). The formula becomes \((\forall x \varphi(x) \wedge \exists x eg \psi(x)) \vee (\exists x \sigma(x) \wedge \exists x eg \tau(x))\).

Convert to Prenex Normal Form for (a)

Move all quantifiers to the front. Collect similar quantifiers, in this case, the existential quantifiers: \( \forall x \varphi(x) \vee \exists x (eg \psi(x) \wedge \sigma(x) \wedge \exists x eg \tau(x)) \). Simplify further: \( \forall x \varphi(x) \vee \exists x (eg \psi(x) \wedge \sigma(x) \wedge eg \tau(x)) \).

Eliminate Implications in (b)

The original formula is \(\forall x \varphi(x) \leftrightarrow \exists x \psi(x)\). Replace the biconditional with its logical equivalent: \((A \leftrightarrow B) \equiv ((A \rightarrow B) \wedge (B \rightarrow A))\). The formula becomes \((\forall x \varphi(x) \rightarrow \exists x \psi(x)) \wedge (\exists x \psi(x) \rightarrow \forall x \varphi(x))\).

Convert Implications for (b)

Convert the implications using \((A \rightarrow B) \equiv (eg A \vee B)\). The formula becomes \((eg \forall x \varphi(x) \vee \exists x \psi(x)) \wedge (eg \exists x \psi(x) \vee \forall x \varphi(x))\).

Simplify using Quantifiers for (b)

Change \(eg \forall x \varphi(x)\) to \(\exists x eg \varphi(x)\) and \(eg \exists x \psi(x)\) to \(\forall x eg \psi(x)\). The formula becomes: \((\exists x eg \varphi(x) \vee \exists x \psi(x)) \wedge (\forall x eg \psi(x) \vee \forall x \varphi(x))\).

Convert to Prenex Normal Form for (b)

Combine the quantifiers: \(\exists x (eg \varphi(x) \vee \psi(x)) \wedge \forall x (eg \psi(x) \vee \varphi(x))\).

Eliminate Implications in (c)

The formula involves \(eg(\exists x \varphi(x, y) \wedge \forall y(\psi(y)) \rightarrow \varphi(x, x)) \rightarrow \exists x \forall y \sigma(x, y)\). Ensure to remove implications using \((A \rightarrow B) \equiv (eg A \vee B)\).

Apply De Morgan's and Simplification in (c)

Apply De Morgan's laws and negate quantified statements appropriately. Rewrite expressions using \(eg \exists = \forall eg \) and \(eg \forall = \exists eg \). Carefully expand \(eg ((\exists x (\varphi(x, y) \wedge \exists y eg \psi(y)) \vee eg \varphi(x, x)) \rightarrow \exists x \forall y \sigma(x, y))\) until full conversion.

Step 11-12: Further Convert and Simplify (Continue As Necessary for Each Exercise)

Follow through similar pathways as in the early steps by handling negations and bringing all quantifiers upfront for prenex conversion, while ensuring no further simplification is possible.

Key concepts

Propositional Logic

Propositional logic is a branch of logic that deals with statements in mathematical form. It looks at how these statements, or propositions, can be combined and manipulated. These propositions are treated like mathematical variables, but instead of numerical values, they hold truth values (either true or false).
At the core of propositional logic are logical operations, such as:- **Conjunction** (\(\land\)), which signifies 'and'.
- **Disjunction** (\(\lor\)), which means 'or'.
- **Negation** (\(eg\)), which flips the truth value.
- **Implication** (\(\rightarrow\)), which represents 'if...then'.
- **Biconditional** (\(\leftrightarrow\)), which is 'if and only if'.

These basic operations allow us to create complex propositions and define rules for transforming them. Understanding these logical operations is key to solving problems in propositional logic efficiently.

Quantifiers

Quantifiers are symbols used in logic to express the idea of 'all' or 'some' within the universe of discourse. They are integral to predicate logic, which extends propositional logic by dealing with more complex logical statements.
- **Universal Quantifier** (\(\forall\)) means 'for all', indicating that a property or statement applies to all elements in a set.
- **Existential Quantifier** (\(\exists\)) signifies 'there exists', showing that there is at least one element in a set satisfying a property.

In logical expressions, quantifiers are used together with variables and predicates. For example, \(\forall x \varphi(x)\) means the proposition \(\varphi(x)\) is true for every \(x\). Handling quantifiers properly is crucial for converting expressions into prenex normal form, where all quantifiers are at the front of the formula.

Logical Equivalence

Logical equivalence refers to when two statements express the same thing in logic, even if they appear different. It's like having two different sentences that mean the same thing. This concept is utilized through transformations that ensure the truth values remain unchanged.
Key laws include:
- **De Morgan’s Laws**, which relate conjunctions and disjunctions through negation, such as \(eg (A \land B) = eg A \lor eg B\).
- **Double Negation**, which states \(eg (eg A) = A\).
- **Implication transformations**, where \(A \rightarrow B\) is equivalent to \(eg A \lor B\).

Applying these equivalences systematically helps simplify formulas and convert them to standard forms like the prenex normal form, which is a canonical form for logical formulas.

Double Negation

Double negation is a principle that states that negating a proposition twice will yield the original proposition. Thus, \(eg(eg A) = A\). It's akin to reversing a reversal, thus leaving the original stance unchanged.
Double negation can be very useful in simplifying logical expressions or in converting them to other forms. When trying to reach prenex normal form, recognizing where double negation can be used allows you to reduce complex expressions neatly.
This technique is handy when working with negations involving quantifiers, as it helps in flipping and organizing them closer to the standardized prenex structure. It's also a fundamental rule that maintains the logical equivalence during transformations.