Question

Let the ternary connective \(\$$ be defined by \)\llbracket \$\left(\varphi_{1}, \varphi_{2}, \varphi_{3}\right) \rrbracket=1 \Leftrightarrow\( \)\left.\llbracket \varphi_{1} \rrbracket+\llbracket \varphi_{2}\right]+\left[\varphi_{3}\right] \geq 2\( (the majority connective). Express \)\$$ in terms of \(\vee\) and \(\neg\).

Short answer

\( \$\left(\varphi_1, \varphi_2, \varphi_3\right) \equiv \neg(\neg \varphi_1 \vee \neg \varphi_2) \vee \neg(\neg \varphi_1 \vee \neg \varphi_3) \vee \neg(\neg \varphi_2 \vee \neg \varphi_3) \)

Step-by-step solution

Understand the Problem Statement

We need to express the ternary connective \(\\(\) using the logical operators OR (\(\vee\)) and NOT (\(eg\)). The connective \(\\)\) is defined such that it returns true (1) if at least two out of the three inputs are true.

Identify Scenarios for the Connective

For three logical variables \(\varphi_1, \varphi_2, \varphi_3\), the expression \(\llbracket \$\left(\varphi_1, \varphi_2, \varphi_3\right) \rrbracket = 1\) when at least two inputs are true.

Construct Logical Expressions for 'At Least Two True'

To achieve a result of 1 for at least two true inputs, consider the following scenarios: \(\varphi_1 \land \varphi_2\), \(\varphi_1 \land \varphi_3\), and \(\varphi_2 \land \varphi_3\). This can be expressed in logic using ANDs (\(\land\)) and ORs (\(\vee\)).

Translate 'AND' to 'OR' and 'NOT'

Convert each AND operation into OR and NOT operations: \(\varphi_1 \land \varphi_2\) can be written as \(eg(eg \varphi_1 \vee eg \varphi_2)\). Apply similar transformations to the other ANDs.

Combine Translated Expressions

Combine the expressions obtained in Step 4 using OR to cover all scenarios where at least two inputs are true: \[ eg(eg \varphi_1 \vee eg \varphi_2) \vee eg(eg \varphi_1 \vee eg \varphi_3) \vee eg(eg \varphi_2 \vee eg \varphi_3) \].

Key concepts

Ternary Connective

In propositional logic, a ternary connective is a logical operation that involves three inputs. It is used to determine the truth value of a compound proposition. This specific ternary connective is defined as a 'majority connective.' It assesses the inputs

• \(\varphi_1\)
• \(\varphi_2\)
• \(\varphi_3\)

to see which are true. The result is true if at least two inputs are true. Understanding ternary connectives is key to simplifying complex logical expressions by reducing them into simpler operations, like the popular AND, OR, NOT operators, making them easier to handle in logical proofs.

Logical Operators

Logical operators are building blocks for constructing logical expressions in propositional logic. They help in developing advanced logical functions through simple operations, such as:

• NOT (\(eg\)): This operator negates, or inverts, the truth value of a proposition.
• AND (\(\land\)): This is true only if both propositions it connects are true.
• OR (\(\vee\)): This operator returns true if at least one of the propositions is true.

The exercise uses OR and NOT operators to express the specific ternary connective by restructuring the AND operations into a combination of OR and NOT. This conversion is crucial for scenarios where certain operations are restricted or specific transformations are needed, providing flexibility in designing logical circuits and functions.

Truth Tables

Truth tables are vital tools for visualizing the outcome of logical operations. They map out the possible truth values of given propositions and their resulting outputs under different logical operations. In the context of the ternary connective problem, a truth table can help illustrate:

• When each input alone is true or false.
• The combinations of inputs that result in the majority connective becoming true.

A clear truth table will list all eight possible input combinations (since each of the three propositions has two possible truth values), showing how many inputs must be true for the ternary connective to return true. This visualization is a fundamental step in understanding how complex logical expressions behave under various conditions.

Propositional Logic

Propositional logic is the segment of logic dealing with propositions, which are statements that can be either true or false. It uses logical operators to form complex statements from simpler ones. The aim in our exercise is to express a ternary connective (a compound statement with a specific behavior) using basic operations. Consider these components in propositional logic:

• Atomic propositions, which are simple declarative statements.
• Compound propositions, formed by combining atomic ones using logical operators.

In practice, propositional logic is leveraged extensively in computer science for algorithm design and verification, automated theorem proving, and in building logic circuits. Understanding how to manipulate expressions in propositional logic, as in this exercise, is fundamental for effectively analyzing and constructing logical statements in any logical assessment or computational solution.