Question

Check which of the following sets are consistent. (a) \(\left\\{\neg p_{1} \wedge p_{2} \rightarrow p_{0}, p_{1} \rightarrow\left(\neg p_{1} \rightarrow p_{2}\right), p_{0} \leftrightarrow \neg p_{2}\right\\}\), (b) \(\left\\{p_{0} \rightarrow p_{1}, p_{1} \rightarrow p_{2}, p_{2} \rightarrow p_{3}, p_{3} \rightarrow \neg p_{0}\right\\}\), (c) \(\left\\{p_{0} \rightarrow p_{1}, p_{0} \wedge p_{2} \rightarrow p_{1} \wedge p_{3}, p_{0} \wedge p_{2} \wedge p_{4} \rightarrow p_{1} \wedge p_{3} \wedge p_{5}, \ldots\right\\}\)

Short answer

Sets (a) and (c) are consistent; set (b) is inconsistent.

Step-by-step solution

Analyzing Set (a)

Evaluate the set \( \{ eg p_{1} \wedge p_{2} \rightarrow p_{0}, p_{1} \rightarrow (eg p_{1} \rightarrow p_{2}), p_{0} \leftrightarrow eg p_{2} \} \). Start by assuming truth values: suppose \( p_{1} \) is true, making \( eg p_{1} \) false. This causes \( eg p_{1} \wedge p_{2} \rightarrow p_{0} \) become true as the premise is false. With \( p_{1} \rightarrow (eg p_{1} \rightarrow p_{2}) \), you assume \( p_{1} \) true gives \( (eg p_{1} \rightarrow p_{2}) \) true without conflict. \( p_{0} \leftrightarrow eg p_{2} \) needs \( p_{0} \) to be the complement of \( p_{2} \), consistent as long as \( p_{2} \) truth matches this correlation.

Analyzing Set (b)

Examine the set \( \{ p_{0} \rightarrow p_{1}, p_{1} \rightarrow p_{2}, p_{2} \rightarrow p_{3}, p_{3} \rightarrow eg p_{0} \} \). Assume each proposition is true: \( p_{0} \) true implies \( p_{1} \) is true, propagating through to \( p_{3} \). \( p_{3} \rightarrow eg p_{0} \) then implies \( p_{0} \) is false, contradicting initial assumptions. Hence, set (b) is inconsistent.

Analyzing Set (c)

Consider the set \( \{ p_{0} \rightarrow p_{1}, p_{0} \wedge p_{2} \rightarrow p_{1} \wedge p_{3}, p_{0} \wedge p_{2} \wedge p_{4} \rightarrow p_{1} \wedge p_{3} \wedge p_{5}, \ldots \} \). Each complex formula breaks down into smaller steps that are contained within increased logical conjunctions, thus depending on incremental extra assumptions consistently extending. Each step inductively broadens the conjunction, which is inherently consistent, as no immediate contradiction arises within any sub-components.

Key concepts

Propositional Logic

Propositional logic is a branch of logic that deals with propositions, which are statements that can be either true or false. It uses logical connectives to combine these propositions into more complex expressions. Common logical connectives include:

• \( eg \) (not): Negation of a proposition.
• \( \wedge \) (and): Conjunction between propositions.
• \( \vee \) (or): Disjunction between propositions.
• \( \rightarrow \) (if...then): Implication connecting propositions.
• \( \leftrightarrow \) (if and only if): Biconditional equivalence.

In solving logical problems, understanding these connectives helps break down complex propositions into simpler components. This foundational knowledge of propositional logic is essential for evaluating logical consistency and analyzing truth values effectively.

Consistency Checking

Consistency checking involves determining whether a set of propositions can be true at the same time. The goal is to ensure there is no contradiction among them. Let's consider the concept further:
When checking for consistency, like in set (a) from the exercise, we attempt to assign truth values to each proposition that do not lead to contradictions. If we succeed, the set is consistent. If not, it's inconsistent.
For instance, if assuming one proposition to be true leads to another being false in a way that contradicts the set, we can deduce inconsistency—akin to set (b) in the exercise, where addressing the truth of \( p_0 \) ultimately leads to its negation through logical steps, showing inconsistency.
Consistency checking is critical in logic and computer science, ensuring systems and propositions can coexist without problematic contradictions.

Logical Propositions

Logical propositions are declarative statements with a truth value of either true or false. They form the building blocks of logical expressions. When working with such propositions, especially in exercises, the aim is to understand their role individually and in combination.
For example, in set (c) of the given exercise, propositions like \( p_0 \rightarrow p_1 \) and \( p_0 \wedge p_2 \rightarrow p_1 \wedge p_3 \) are more than simple statements; they represent logical dependencies and relationships.
Developing a deep understanding of each proposition's implication and conjunction aids in processing how additional conditions influence logical outcomes, thereby helping in consistency evaluation and truth value assignment.

Truth Values Analysis

Truth values analysis is the process of assigning true or false values to propositions to determine the overall truth of more complex logical constructs. This analysis is integral to assessing the logical consistency of a set of propositions.
In practical exercises, like analyzing various sets of propositions, truth values are hypothesized and tested for consistency. For set (a), you might suppose \( p_1 \) is true and follow its implications throughout the logic chain. Careful observation is necessary to detect any emerging contradictions in assignments.
In contrast, set (b) demonstrated inconsistency when all propositions were initially assumed true, leading to a contradiction when traced back to \( eg p_0 \).
Through truth values analysis, practitioners can navigate the relationships between propositions, making informed conclusions about logical constructs for system validations or philosophical inquiries.