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Distributive law of logic is a fundamental principle in logical expressions. This law allows us to distribute logical operators over others, similar to how multiplication distributes over addition in arithmetic. For example, in logic, the expression

• \((p \land (q \lor r))\)

can be rewritten as

• \((p \land q) \lor (p \land r)\),

and vice versa. This is particularly useful in simplifying logical expressions, making them easier to work with and understand.
In the original exercise, we used the distributive law to simplify

• \((eg p \lor q) \land (p \land q)\)

to an expression involving only \(\land\) and \(\lor\) individually, helping to ultimately verify the logical equivalence.Using this principle can make complex logical statements easier to handle or solve. This simplification process can help compare expressions to determine logical equivalencies.

Logical duality is the principle where logical operators are interchanged, providing a new expression that maintains truth-table symmetries. This principle often involves swapping the \(\land\) (AND) with \(\lor\) (OR) and vice versa. Additionally, the constants True and False are swapped.
For instance, the logical statement

• \( eg (p \land q) \)

has a dual of

• \((eg p \lor eg q)\).

The principle of logical duality helps derive statements that are structurally similar but opposite in logic, providing alternative perspectives or forms of equations.
In the exercise, we found the dual of the equivalence \(p \land q\) by replacing \(\land\) with \(\lor\), resulting in \(p \lor q\). Understanding this concept aids in flexible reasoning and creating different logical expressions from a given one.

Primitive statements, within logical contexts, are the simplest form of statements adorned with no logical operators. They are the foundational building blocks from which complex logical expressions arise. Think of them as elementary units in logic, most often depicted as variables like \(p\), \(q\), or \(r\).
Since they do not contain any logical operators such as AND \((\land)\) or OR \((\lor)\), primitive statements cannot be broken down further. They represent basic truths or propositions that mean the statement is either true or false.
When you have expressions involving these primitive statements, operators are applied to transform them or combine them into more complex forms. Formulae in logic utilize primitive statements to construct statements and prove more involved theorems or laws, just like what happens in the simplification and proving of logical equivalences as demonstrated in the exercise. Hence, understanding primitive statements is crucial for grasping higher-level logical concepts.