7.4.1. Validity of Argument

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.

The concept of functional completeness is crucial in logic. We define a set of logical operators as functionally complete if it can express any possible logical statement. In simpler terms, if we have a selection of logical operators (like AND, OR, NOT), we need to be able to form any logical proposition solely using those operators. This is significant because it shows that these operators can generate all logical truths in propositional logic.

Imagine having a basic toolkit with only a hammer and a screwdriver. You could build anything as long as you have nails and screws, similar to how logical operators work. If you have tools that can help construct anything needed, that toolkit (or set of logical operators) is deemed complete.

If I am given a proposition which indeed involves only these three operators, I do not have to do anything. But what about a compound proposition where I have an occurrence of implication? In that case, I can use this logical identity that p → q is logically equivalent to negation of p disjunction q.

In propositional logic, whenever we encounter an implication (p → q), we can convert it into an equivalent expression. This transformation is based on the logical identity that states an implication can be expressed as a combination of negation and disjunction. This means instead of saying 'if p then q,' we can say 'either not p or q.' This allows us to eliminate implications from our logical expressions, utilizing only the logical operators we started with.

Think of it this way: saying 'If I go to the party, then I will have fun' can be seen as 'Either I don’t go to the party, or I have fun.' This transformation helps to simplify decision-making without losing the core meaning of the statement.

What if my expression has bi implication (↔)? I do not have to worry, what I have to do is I can use the identity that the bi implication is nothing but the conjunction of two individual implications.

For biconditional statements (p ↔ q), which mean 'p if and only if q,' we can translate this statement into two separate implications: 'p implies q and q implies p.' This means that if p is true, q is also true and vice versa. Understanding how to manipulate biconditionals helps maintain the functional completeness of the set of logical operations.

Consider a friendship agreement where 'You can be my friend if I can be yours.' It implies mutual agreement: if you agree to be friends, I also agree, which can be broken down into two simpler statements about each party's involvement.

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.

It turns out that we can still achieve functional completeness with fewer operators. If we have just disjunction (OR) and negation (NOT), we can still express all logical statements. This is based on how we can represent conjunctions (AND) using De Morgan's laws, which state that negation of a conjunction is equivalent to a disjunction of negations. Essentially, knowing how to manipulate these basic operators can enable us to reconstruct any logical expression.

Imagine making a sandwich. You can create a sandwich with just bread and any filling by using the bread (disjunction) and the idea of ‘not having bread’ (negation), which would imply you have something else or doing something different without bread (the alternative sandwich). So, just two basic ingredients can still provide adequate variety.

The third part now says that you have to show that only the negation operator and the conjunction operator are functionally complete and here we have to show how we can represent a disjunction in terms of conjunction and negation.

In a similar way, we can demonstrate that with only conjunction (AND) and negation (NOT), we can construct disjunction (OR). Using De Morgan’s laws again, we find that 'p OR q' can be restated with conjunctions and negations as 'NOT (NOT p AND NOT q).' This showcases the extensive power of basic logical constructs in building complex operations.

Think of teamwork: if you say 'You and I cannot work together,' you are negating the idea of collaboration. However, if you manage to redefine 'working together' in terms of other concepts, like collaboration being equal to us not being in opposition, you can deduce how diverse approaches (just like conjunctions) can lead to collaboration, surfacing the interplay of opposites.

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