7.1.1. Lecture - 07

So we will start with question 8, in question 8 we are defining a functionally complete set of logical operators, if you are given a set of logical operators, we say it is functionally complete, if every compound proposition can be converted into a logically equivalent proposition involving only the logical operators, that is given in your collection. So the first part of question 8 asks you the following.

In this section, we explore what it means for a set of logical operators to be functionally complete. A set of logical operators is considered functionally complete if any logical expression can be expressed using only these specific operators. For example, if you are given a set containing conjunction (AND), disjunction (OR), and negation (NOT), you can represent any logical statement with just those three. This is important in logic and computer science as it forms the basis for building circuits and algorithms. The key point is that any complexity in logic can be broken down to just using these three operations.

Imagine you have a toolbox that only has three tools: a hammer, a screwdriver, and a wrench. No matter the construction project you are working on, as long as you have these three tools, you can build anything - from a simple shelf to a complex home. Much like that toolbox, these logical operators can be thought of as basic building blocks for constructing any logical argument.

What about a compound proposition where I have an occurrence of implication? In that case what I can do is I can use this logical identity that p → q is logically equivalent to negation of p disjunction q and I can substitute p → q by this RHS expression.

This part keenly addresses how to handle implications in logical propositions. An implication is a statement of the form 'if p then q', which can be rewritten using disjunction and negation. The identity states that 'p implies q' can be expressed as 'not p or q'. This transformation allows us to express more complex propositions solely using the basic operators of AND, OR, and NOT. By substituting any implications with this identity, we can simplify and ensure our statements are only composed of functionally complete operators.

Think of a traffic rule: 'If it’s raining, then I will carry an umbrella'. This can be transformed into 'Either it’s not raining, or I will carry an umbrella'. The implication is broken down into a more direct expression that highlights the conditions affecting the decision, making it easier to assess the situation.

And by applying this rule repeatedly wherever I have an occurrence of implication I can remove those implications and I will now have an equivalent formula where everything is represented only in terms of conjunction, disjunction and negation.

De Morgan's laws provide a way to negate conjunctions and disjunctions. They state that the negation of a conjunction is the disjunction of the negations, and the negation of a disjunction is the conjunction of the negations. In practical terms, this means that once we transform all implications into disjunctions, we can further simplify our compound propositions by applying these laws, ensuring we maintain only our basic operators.

Imagine a classroom rule: 'No talking or eating is allowed'. If you want to express the opposite of this rule (i.e., people doing one or the other), you would say ‘It is not the case that no one is talking and no one is eating’, which simplifies to ‘Somebody is either talking, or eating, or doing both’. This simplification helps clarify the conditions under which the rules are applied.

So that shows that if you have these three operators namely a conjunction, disjunction and negation, you can represent any statement, any compound proposition and hence this is a functionally complete set of logical operators.

Here we conclude that the set containing conjunction, disjunction, and negation is indeed functionally complete. This means any logical statement can be created by using combinations of these three operators, creating a strong foundation for logical deductions and reasoning. It ensures that no matter how complex the logical structure is, it can always be expressed through these fundamental operators.

Consider programming: every function or algorithm can be broken down to basic operations like assigning values, making choices (if-then statements), and repeating actions (loops). Similarly, logic operates on a foundation where complex logical expressions can derive from these simple components.

Now the second part of the question says that I do not need both conjunction and disjunction to be there. It is sufficient if I just have a disjunction and negation operator and I can represent every statement.

This segment reveals that it's not necessary to have both conjunction and disjunction to achieve functional completeness; having just disjunction and negation is adequate. By substituting conjunctions using the identities we learned earlier, we can express all logical propositions using only these two operators. This allows for a reduced set of operators while still retaining the power to express complex logical relationships.

Imagine you want to communicate with friends using just emojis instead of full sentences. Using a combination of a “happy” emoji (✌️) and a “not” emoji (🚫), you can convey a range of emotions and ideas without needing the full grammar of sentences. Just like that, negation and disjunction can represent all the essentials of logic.

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