Question

Let \(\phi=(p \rightarrow q) \rightarrow((r \wedge \neg s) \rightarrow q)\). For each of the following interpretations of \(p, q, r,\) and \(s,\) compute \(I(\phi)\) using the truth tables for \(\neg, v, \wedge, \rightarrow,\) and \(\leftrightarrow\) (a) \(I(p)=T, I(q)=T, I(r)=F,\) and \(I(s)=T\) (b) \(I(p)=T, I(q)=F, I(r)=T,\) and \(I(s)=F\) (c) \(I(p)=F, I(q)=T, I(r)=T,\) and \(I(s)=F\) (d) \(I(p)=F, I(q)=F, I(r)=T,\) and \(I(s)=F\)

Short answer

(a) T, (b) T, (c) T, (d) F.

Step-by-step solution

Define the expression and truth table operations

The given expression is \( \phi = (p \rightarrow q) \rightarrow ((r \wedge eg s) \rightarrow q) \). To evaluate \( \phi \), we will use truth tables for the logical operations \( eg \) (not), \( \wedge \) (and), and \( \rightarrow \) (implies). We'll compute each interpretation one by one as given in the problem.

Evaluate Interpretation (a)

For \( I(p) = T \), \( I(q) = T \), \( I(r) = F \), \( I(s) = T \):1. Evaluate \( p \rightarrow q \): Since \( T \rightarrow T = T \).2. Evaluate \( eg s \): Since \( eg T = F \).3. Evaluate \( r \wedge eg s \): Since \( F \wedge F = F \).4. Evaluate \( (r \wedge eg s) \rightarrow q \): Since \( F \rightarrow T = T \).5. Overall \( (p \rightarrow q) \rightarrow ((r \wedge eg s) \rightarrow q) \): Since \( T \rightarrow T = T \).So, \( I(\phi) = T \).

Evaluate Interpretation (b)

For \( I(p) = T \), \( I(q) = F \), \( I(r) = T \), \( I(s) = F \):1. Evaluate \( p \rightarrow q \): Since \( T \rightarrow F = F \).2. Evaluate \( eg s \): Since \( eg F = T \).3. Evaluate \( r \wedge eg s \): Since \( T \wedge T = T \).4. Evaluate \( (r \wedge eg s) \rightarrow q \): Since \( T \rightarrow F = F \).5. Overall \( (p \rightarrow q) \rightarrow ((r \wedge eg s) \rightarrow q) \): Since \( F \rightarrow F = T \).So, \( I(\phi) = T \).

Evaluate Interpretation (c)

For \( I(p) = F \), \( I(q) = T \), \( I(r) = T \), \( I(s) = F \):1. Evaluate \( p \rightarrow q \): Since \( F \rightarrow T = T \).2. Evaluate \( eg s \): Since \( eg F = T \).3. Evaluate \( r \wedge eg s \): Since \( T \wedge T = T \).4. Evaluate \( (r \wedge eg s) \rightarrow q \): Since \( T \rightarrow T = T \).5. Overall \( (p \rightarrow q) \rightarrow ((r \wedge eg s) \rightarrow q) \): Since \( T \rightarrow T = T \).So, \( I(\phi) = T \).

Evaluate Interpretation (d)

For \( I(p) = F \), \( I(q) = F \), \( I(r) = T \), \( I(s) = F \):1. Evaluate \( p \rightarrow q \): Since \( F \rightarrow F = T \).2. Evaluate \( eg s \): Since \( eg F = T \).3. Evaluate \( r \wedge eg s \): Since \( T \wedge T = T \).4. Evaluate \( (r \wedge eg s) \rightarrow q \): Since \( T \rightarrow F = F \).5. Overall \( (p \rightarrow q) \rightarrow ((r \wedge eg s) \rightarrow q) \): Since \( T \rightarrow F = F \).So, \( I(\phi) = F \).

Key concepts

Logical Operations

Logical operations form the backbone of truth tables, helping us evaluate the truth values of complex propositions. The primary operations used in logic include:

• Not (\(eg\) ): This unary operation negates the truth value of a proposition. If a statement is true, applying 'not' makes it false, and vice versa.
• And (\(\wedge\)): This binary operation results in true only if both operands are true. Otherwise, it returns false.
• Or (\(\vee\)): Unlike 'and', this operation is true if at least one of the operands is true. It is false only if both are false.
• Implies (\(\rightarrow\)): This somewhat tricky operation is often counterintuitive. It is false only when a true statement implies a false one. In all other cases, it is true.
• Biconditional (\(\leftrightarrow\)): This is true if both operands have the same truth value, either both true or both false.

Understanding these operations helps in constructing truth tables, which visually display the relationship between their inputs and outputs. By building a truth table, you can systematically break down complex logical expressions and solve them step by step.

Propositional Calculus

Propositional calculus is an area in logic that focuses on propositions and how they relate through logical connectives. It involves building logical expressions using

• individual propositions (like p, q, r)
• logical connectives (like \(\wedge, \vee, eg, \rightarrow,\)and \(\leftrightarrow\))

Propositional variables can either hold a true value, represented by T, or a false value, represented by F. By stringing these variables together with logical operations, we form compound propositions. These are evaluated using truth tables to see under what conditions they hold true or false.A key aspect of propositional calculus is determining the validity of logical statements. Through systematic evaluation, often using truth tables, you can ascertain whether an expression is a tautology (always true), a contradiction (always false), or contingent (sometimes true, sometimes false). This systematic breakdown is critical in fields like computer science and mathematics, where logical consistency is essential.

Truth Value Evaluation

Truth value evaluation refers to determining the truth or falsity of propositions or compound propositions based on given interpretations or assignments. In the context of truth tables, this involves taking assigned truth values to the propositional variables(e.g., \(I(p) = T, I(q) = F\))and evaluating the entire compound expression.For the proposition \(\phi = (p \rightarrow q) \rightarrow ((r \wedge eg s) \rightarrow q)\),we evaluate each part sequentially:

• Begin with the innermost expressions, like \(eg s\),building out to the whole.
• Evaluate each logical operation in the context of its given truth assignments, applying rules such as "\(T \rightarrow F\)is false".
• By constructing a complete truth table, you document each stage of the evaluation, ensuring accuracy in complex logical assessments.

Truth value evaluation is crucial not only in logic but also in reasoning processes and proof validation in broader scopes, where precision in determining logical outcomes dictates solution correctness.