Question

Show in the system with \(\vee\) as a primitive connective $$ \begin{aligned} &\vdash(\varphi \rightarrow \psi) \leftrightarrow(\neg \varphi \vee \psi) \\ &\vdash(\varphi \rightarrow \psi) \vee(\psi \rightarrow \varphi) \end{aligned} $$

Short answer

Both statements are true: Their equivalences hold under logical laws.

Step-by-step solution

Interpret the Task

Understand that you need to prove two separate logical equivalences using the logical \(\vee\) connective.1. Prove that \(\varphi \ ightarrow \psi \leftrightarrow (eg \varphi \vee \psi)\).2. Prove that \(\varphi \ ightarrow \psi \vee \psi \ ightarrow \varphi\).

Rewrite Implication (First Problem)

For the expression \(\varphi \ ightarrow \psi \leftrightarrow eg \varphi \vee \psi\), rewrite the implication as a disjunction: \(\varphi \ ightarrow \psi = eg \varphi \vee \psi\).

Prove Equivalence (First Problem)

Since \(\varphi \ ightarrow \psi\) is rewritten as \(eg \varphi \vee \psi\), the equivalence \(\varphi \ ightarrow \psi \leftrightarrow (eg \varphi \vee \psi)\) holds true because they are literally the same expressions. Thus, this specific equivalence is correctly proven.

Rewrite Implications (Second Problem)

For the expression \(\varphi \ ightarrow \psi \vee \psi \ ightarrow \varphi\), rewrite each implication as a disjunction: 1. \(\varphi \ ightarrow \psi = eg \varphi \vee \psi\) 2. \(\psi \ ightarrow \varphi = eg \psi \vee \varphi\).

Construct the Disjunction (Second Problem)

Substitute the rewritten formulas back into the expression:\[(eg \varphi \vee \psi) \vee (eg \psi \vee \varphi)\]

Simplify Expression (Second Problem)

The expression \((eg \varphi \vee \psi) \vee (eg \psi \vee \varphi)\) is simplified through the distribution and becomes a tautology, confirming that it is always true, thus proving the original statement.

Key concepts

Logic and Structure

Logical equivalence forms the backbone of logical reasoning, especially within mathematics and philosophy. This concept revolves around the idea that two compound propositions can have the same truth value in all possible scenarios.
When discussing logic and structure, it is essential to understand how logical equivalences are used to simplify complex expressions by showing their fundamental truths.
Logical equivalences help us reduce the complexities in logical expressions, much like solving algebraic equations simplifies statements to the basic expression that holds true throughout all variables in the problem set.
The structure that logic provides can turn abstract ideas into formalized expressions, making comprehension and analysis easier. Establishing logical equivalences ensures that statements are not only equivalent but also interchangeable, enhancing flexibility in logical operations.

Primitive Connective

Primitive connectives form the core elements of logical expressions. They are the basic units that connect propositions to create compound statements.
In our given problem, the primitive connective is logical disjunction, denoted by \(\vee\). This symbol represents the logical 'or', and it acts as a fundamental building block in forming compound logical statements.
Basic connectives like disjunction, conjunction (\(\wedge\)), negation (\(eg\)), and implication (\(\rightarrow\)) serve to construct more complex logical formulas through combinations.
Understanding primitive connectives is crucial as it allows for the transformation and simplification of propositions into equivalent forms, facilitating deeper logical analyses and proofs. They represent a universal concept in logical theory, underpinning more advanced logical constructs.

Implication

Logical implication is an essential concept that illustrates a conditional relationship between two propositions. It is often represented as \( \varphi \rightarrow \psi \), read as 'if \(\varphi\) then \(\psi\)'.
The essence of implication is to assert that when \(\varphi\) is true, \(\psi\) must also be true, forming a dependent relationship. However, if \(\varphi\) is false, the implication as a whole is always true regardless of the truth value of \(\psi\).
In logical equivalencies, implications can be transformed using a disjunction: \(\varphi \rightarrow \psi \) is equivalent to \(eg\varphi \vee \psi \). This conversion is key in simplifying or proving logical expressions, as seen in the exercise where the transformation allowed for the establishment of an equivalence using only primitive connectives.

Logical Disjunction

Logical disjunction, often symbolized by \( \vee \), equates to the 'or' operation in logic. It forms a compound statement that is true when at least one of its components is true.
As a primitive connective, logical disjunction serves a fundamental role in constructing logical expressions, offering essential capabilities for transforming implications and proving equivalencies.
For example, in proving equivalences involving implications, rewriting these implications in terms of disjunction can simplify expressions and help unravel their complexities.
Logical disjunction is especially useful in proving that an overall expression holds true across various conditions, acting as a foundation for establishing truths and testing propositions in logical problem-solving. It is indispensable in the world of logic for its simplicity and directness in asserting truth.