7.7.1. Unsatisfiability Proof via Resolution

The set of logical operators is functionally complete if every compound proposition can be converted into a logically equivalent proposition using only these operators. Proving this means demonstrating that we can express any logical statement using just a specific collection of operators.

A functionally complete set of logical operators means you can construct any logical statement using only those operators. In this proof, we show that if you only have conjunction (AND), disjunction (OR), and negation (NOT), then you can represent all possible logical statements. If you come across an implication (p → q), you can replace it with ¬p ∨ q, allowing you to continue using just conjunction, disjunction, and negation for any statement.

Think of these logical operators like tools in a toolbox. If your toolbox has all the necessary tools (AND, OR, NOT), you can build anything (any logical proposition). If you're missing certain tools (like AND), but can recreate their functions using the others (like manipulating OR and NOT), your toolbox is still functional and complete.

Implications (p → q) can be transformed into disjunctions (¬p ∨ q) using logical identities. For bi-implications (p ↔ q), we can express them as the conjunction of two implications (p → q) AND (q → p), and then further replace each implication accordingly.

To handle implications in logical expressions, we can substitute them with equivalent expressions made of conjunctions, disjunctions, and negations. For a bi-implication, which indicates that both propositions imply each other, you can break it down into two implications. Each of those implications can then be transformed into disjunctions, ultimately using only your available operators.

Imagine planning a dinner party. If you say, 'If I cook, then we eat,' (the implication), you could also say, 'If I don't cook, we won't eat,' (the contrapositive). Breaking down implications like this is similar to exploring all possibilities to ensure everyone is satisfied without leaving anything out.

It's sufficient to have just disjunction and negation operators to represent every logical statement by converting conjunctions into equivalent expressions. For example, the expression p AND q can be rewritten as ¬(¬p OR ¬q) using De Morgan's laws.

In logical expressions, you can effectively create all necessary combinations of true and false values, even using just disjunction (OR) and negation (NOT). By transforming conjunctions into expressions involving these two operators, you ensure that no logical operations are lost. This flexibility confirms that disjunction and negation are also functionally complete by showing how you can represent conjunction.

Think of this as a recipe where you use just two ingredients (disjunction and negation) to create a dish that could traditionally require three (including conjunction). By creatively combining those two ingredients (using negation to invert flavors), you realize that you can still create a satisfying meal, representing any logic needed.

Similarly, one can represent all logical operations using just conjunction and negation. The disjunction can be converted to an expression containing only these two operators, confirming this pair's functional completeness.

By employing conjunction (AND) along with negation, we can express disjunctions by representing expressions using NOT and AND. This shows that multiple combinations of two operators can fulfill the requirements of a complete logical system, emphasizing their ability to cover all logical operations.

Similar to cooking, where you might think you need flour, sugar, and eggs to bake a cake, you can still make a great dessert using just flour and eggs, if you fold and twist the ingredients creatively (using negation) to get similar outcomes in different forms.

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