Question

Show \(\quad(a) \quad \varphi \vdash \neg(\neg \varphi \wedge \psi)\), (b) \(\neg(\varphi \wedge \neg \psi), \varphi \vdash \psi\) (c) \(\neg \varphi \vdash(\varphi \rightarrow \psi) \leftrightarrow \neg \varphi\), (d) \(\vdash \varphi \Rightarrow \vdash \psi \rightarrow \varphi\), (e) \(\neg \varphi \vdash \varphi \rightarrow \psi\)

Short answer

(a) True, (b) True, (c) True, (d) True, (e) True.

Step-by-step solution

Analyze Part (a)

To prove \( \varphi \vdash eg(eg \varphi \wedge \psi) \), we start by considering that \( \varphi \) derives nothing false. Therefore, \( eg \varphi \) must be false. Since \( eg \varphi \) is false, \( eg \varphi \wedge \psi \) is also false because the conjunction of any statement with a false statement is false. Hence, \( eg(eg \varphi \wedge \psi) \) is true.

Solve Part (b)

For \( eg(\varphi \wedge eg \psi), \varphi \vdash \psi \), assume \( eg(\varphi \wedge eg \psi) \) and \( \varphi \) are true. Then \( \varphi \wedge eg \psi \) must be false. The only way for the conjunction to be false, while \( \varphi \) is true, is if \( eg \psi \) is false, meaning \( \psi \) must be true.

Explore Part (c)

To demonstrate \( eg \varphi \vdash (\varphi \rightarrow \psi) \leftrightarrow eg \varphi \), consider that if \( eg \varphi \) is true, then \( \varphi \) is false, making \( \varphi \rightarrow \psi \) true regardless of \( \psi \) (vacuous truth). Therefore, \( (\varphi \rightarrow \psi) \) is equivalent to \( eg \varphi \), as both are true given \( eg \varphi \).

Demonstrate Part (d)

For \( \vdash \varphi \Rightarrow \vdash \psi \rightarrow \varphi \), if \( \varphi \) is a theorem \( (\vdash \varphi) \), then \( \varphi \) is true in all cases. Consequently, \( \psi \rightarrow \varphi \) is also true for any \( \psi \), since the truth value of \( \psi \) does not affect the truth of \( \varphi \).

Confirm Part (e)

To verify \( eg \varphi \vdash \varphi \rightarrow \psi \), if \( eg \varphi \) is true, then \( \varphi \) is false, making \( \varphi \rightarrow \psi \) true by vacuous truth (a false antecedent makes the implication true, regardless of the consequent).

Key concepts

Propositional Logic

Propositional logic is a fundamental part of formal logic that deals with propositions and their relationships. It focuses on operations such as conjunction, disjunction, implication, and negation.

Propositions themselves are simple statements that can either be true or false. In propositional logic, these propositions are represented by symbols like \( \varphi \), \( \psi \), and so on. This symbolic representation makes it easier to analyze and form logical expressions and proofs.

One of the essential aspects of propositional logic is understanding how complex logical statements, known as formulas, are constructed. These statements are built using logical connectives:

• Conjunction \((\wedge)\), which works like the logical “and.”
• Disjunction \((\vee)\), which works like the logical "or."
• Implication \((\rightarrow)\), representing "if...then..." statements.
• Negation \((eg)\), which inverts the truth value of a statement.

By manipulating these structures, we can explore the potential relationships and outcomes of different propositions.

Logical Proofs

Logical proofs are the backbone of formal reasoning, providing the method through which theorems or logical statements are shown to be true. A logical proof involves deriving a conclusion from specific premises using logical operations justified by a set of rules.

To construct a proof, one usually follows a sequence of steps, each justified by a rule of inference or a logical principle. Key rules in propositional logic include the Modus Ponens, where from \( p \rightarrow q \) and \( p \), one can infer \( q \); and Modus Tollens, which allows inferring \( eg p \) from \( p \rightarrow q \) and \( eg q \).

In the context of the original exercise, different logical proofs were applied to demonstrate how statements like \( eg \varphi \vdash (\varphi \rightarrow \psi) \leftrightarrow eg \varphi \) hold true under given conditions. These proofs not only deepen our understanding of logical relationships but also refine critical thinking and problem-solving skills.

Implication in Logic

Implication in logic represents conditional statements or "if...then..." scenarios, symbolized as \(\rightarrow\). It holds a special place in logical reasoning because it captures the dependency relationship between two propositions.

Understanding implications requires recognizing that an implication \( p \rightarrow q \) is logically equivalent to saying that not \( q \) when \( p \) is true would be contradictory. In simple terms, for the implication to fail, you must end up with \( p \) being true and \( q \) being false, which highlights how the truth of \( p \) enforces the truth of \( q \).

In the exercise, implication is explored in statements like \( \varphi \rightarrow \psi \), where understanding the dynamics of truth values helps establish why certain logical conclusions such as vacuous truth come into play, particularly when the antecedent (the "if" part) is false.

Negation in Logic

Negation is a crucial logical operation that reverses the truth value of a proposition. If a proposition \( \varphi \) is true, its negation \( eg \varphi \) is false, and vice versa.

Negation is often used in proofs and logical expressions to express "not" or to imply contradiction. It serves as a pivotal tool in constructing logical arguments, particularly in demonstrating the validity of statements through proof techniques like refutation.

For example, in part (c) of the exercise \( eg \varphi \vdash (\varphi \rightarrow \psi) \leftrightarrow eg \varphi \), understanding negation allows us to see why the implication is true when \( \varphi \) is false. Here, we use negation's capacity to illustrate that even if \( \psi \) varies, the implication holds due to the falsity of \( \varphi \).

Logical Equivalence

Logical equivalence in logic refers to scenarios where different expressions have the same truth value across all possible situations. When two expressions are logically equivalent, denoted \( \leftrightarrow \), they are interchangeable in logical arguments without altering the truth outcome.

For instance, \( eg(eg \varphi \wedge \psi) \) is logically equivalent to \( eg \varphi \vee eg \psi \). This equivalence follows from De Morgan's laws, which provide rules for distributing negation across conjunctions and disjunctions.

In the exercise, part (c) illustrates logical equivalence by demonstrating how \( (\varphi \rightarrow \psi) \leftrightarrow eg \varphi \) holds true given certain conditions. To determine equivalence, transform and simplify logical expressions to show they can represent the same truth conditions. This concept enriches our ability to relate complex logical statements succinctly and accurately.