Question

Show that HA \(\vdash \varphi \vee \psi \leftrightarrow \exists x(x=0 \rightarrow \varphi) \wedge(x \neq 0 \rightarrow \psi))\),

Short answer

The equivalence holds because each side of the expression defines the same truth conditions.

Step-by-step solution

Understand the Logical Connectives

The problem involves showing that the disjunction (logical OR) is equivalent to a particular existential statement with implications. The symbol \(\vee\) stands for OR, \(\exists\) is the existential quantifier ("there exists"), and \(\rightarrow\) is the implication (logical IF-THEN). The equivalence \(\leftrightarrow\) means both sides have the same truth value.

Translate the Logical Statement

The logical statement \(\varphi \vee \psi \leftrightarrow \exists x(x=0 \rightarrow \varphi) \wedge (x eq 0 \rightarrow \psi)\) means we need to show these two expressions are equivalent. This equivalence suggests that one side supports the truth conditions of the other side and vice versa.

Analyze the Disjunction

The expression \(\varphi \vee \psi\) represents that at least one of \(\varphi\) or \(\psi\) is true. If \(\varphi\) is true, \(\varphi \vee \psi\) is true. Similarly, if \(\psi\) is true, then \(\varphi \vee \psi\) is also true.

Break Down the Existential Statement

The existential statement \(\exists x(x=0 \rightarrow \varphi) \wedge (x eq 0 \rightarrow \psi)\) means there exists some \(x\) such that both implications are true: if \(x=0\), then \(\varphi\) is true, and if \(x eq 0\), then \(\psi\) is true. This setup divides the possibilities of \(\varphi 's\) and \(\psi's\) truth values across different cases.

Showing Equivalence from Left to Right

Assume \(\varphi \vee \psi\) is true. If \(\varphi\) is true, then \(x=0\) satisfies \(x=0 \rightarrow \varphi\). If \(\psi\) is true, then \(x eq 0\) satisfies \(x eq 0 \rightarrow \psi\). Thus, the existential statement is held true by suitable choice of \(x\), proving one direction of the implication.

Showing Equivalence from Right to Left

Assume \(\exists x(x=0 \rightarrow \varphi) \wedge (x eq 0 \rightarrow \psi)\) is true. Depending on the value of \(x\), if \(x=0\), then \(\varphi\) must be true. If \(x eq 0\), then \(\psi\) must be true. In either case, at least one of \(\varphi\) or \(\psi\) is true, hence \(\varphi \vee \psi\) is true

Key concepts

Disjunction

Disjunction, represented by the logical symbol \( \vee \), signifies the logical "or" in intuitionistic logic. It states that at least one of the involved propositions must hold true for the entire expression to be true. For instance, in the expression \( \varphi \vee \psi \), the disjunction denotes that either \( \varphi \) or \( \psi \) (or both) are true.
Disjunction plays a key role in logical reasoning, allowing for flexibility where multiple potential truths are acknowledged.
Key points to remember about disjunction:

• It only requires at least one true component to hold the overall truth.
• If both propositions in a disjunction are false, then the disjunction is false.
• Disjunction is commutative, meaning \( \varphi \vee \psi \) is logically equivalent to \( \psi \vee \varphi \).
• When proving equivalence in logic, understanding disjunction helps recognize interdependencies between statements.

In intuitionistic logic, disjunction doesn't assume anything about excluded possibilities, contrasting with classical logic's law of excluded middle, emphasizing how intuitionistic logic constructs truth without assuming predefined negations.

Existential Quantifier

The existential quantifier \( \exists \) is used in logic to express that there exists at least one instance in a domain for which a property or condition holds true. In the statement \( \exists x(x=0 \rightarrow \varphi) \wedge (x eq 0 \rightarrow \psi) \), the quantifier indicates that there exists a value for \( x \) making both implications valid simultaneously. The existential quantifier is fundamentally about showing existence rather than certitude for all instances.
A few integral aspects of the existential quantifier include:

• It asserts the existence of at least one instance where the condition is met.
• It does not provide information about how many such instances exist.
• Understanding existential quantifiers is crucial in forming expressions that need to prove the possibility rather than certainty.

This concept plays into confirming diverse truth conditions, thus connecting to disjunction where multiple outcomes are valid, enhancing comprehension of formulating and validating existential claims in intuitionistic logic.

Logical Equivalence

Logical equivalence \( \leftrightarrow \) indicates that two statements or expressions share the same truth value in all conceivable scenarios. When we say \( \varphi \vee \psi \leftrightarrow \exists x(x=0 \rightarrow \varphi) \wedge (x eq 0 \rightarrow \psi) \), both sides are true under the same conditions.
Logical equivalence confirms that one statement can reliably substitute another without altering truth assessments.

• This involves verifying truth tables where both expressions yield identical truth values across all variable interpretations.
• Understanding equivalence helps in simplifying logical expressions and arguments.
• Equivalence needs demonstration through analysis, showing mutual truth support.

Logical equivalence is prominent in crafting elegant proofs, streamlining argument complexity by substituting complex expression clusters with simpler or alternate phrases that maintain identical truth conditions.

Implication

Implication, represented by \( \rightarrow \), constitutes the logical "if-then" relationship between two statements. In our context, implications such as \( x=0 \rightarrow \varphi \) express that whenever \( x=0 \) is true, \( \varphi \) must be true as well. It's a directional truth claim, meaning if the antecedent (first part) holds, then the consequent (second part) follows.
Important considerations for implications include:

• An implication is always true if the antecedent is false, regardless of the consequent's truth value.
• It's vital in assertions about dependent truths, aiding in constructing conditional scenarios in proofs.
• Understanding implication is crucial for evaluating complex statements, especially under intuitive logic where the existence of examples plays a more critical role than absolute universal truths.
• In logical expressions, ensuring both sides adhere evenly to conditions requires analyzing implications thoroughly.

Building an understanding of implications offers insights into structured reasoning, where conclusions are reliably derived from assumed premises leading into broader logical constructs.