I don't understand why there are 5 propositional connectives (conjuction, disjunction, implication, negation and equivalence) that are regularly given a priori in propositional calculus? If the goal was to minimize the number of them why was equivalence given a priori even though it can be defined as a conjucture of an implication and its reverse (A->B)&(B->A) = (A<->B) ? Likewise, if the goal was to have all possible resulting truth table results be available from these connectives, why isn't the negated implication -(A->B) given as one of them? In general the question is why these specific 5 propositional connectives are used as such, instead of others.
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2$\begingroup$ There are more... see Logical connectives. $\endgroup$– Mauro ALLEGRANZAMar 23 at 8:02
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$\begingroup$ Interesting, didn't know that, thank you @MauroALLEGRANZA $\endgroup$– InquisitorMar 23 at 8:05
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2$\begingroup$ If the goal was to minimize the number, then we only need one. See "Sheffer stroke". $\endgroup$– Gerry MyersonMar 23 at 8:12
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It is because they relate closely to words, phrases, and other linguistics constructions we often use in English: 'and', 'or', 'not', 'if ... then ..', and 'if and only if'. And I am sure most other natural languages have counterparts to these as well, because they represent important ideas and concepts. It helps us think and understand logical relationships.
And sure, we don't need all these connectives for the purposes of doing propositional logic. We can, for example, just use the NAND or just the NOR connective. However, having the connectives that we typically use is certainly very useful.
Consider: we should be able to program by writing long string of $1$'s and $0$'s. But it would be crazy to insist that we do exactly that. Having higher-level programming languages makes programming a hell of a lot easier. Same for logic.
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These are the most useful operations. Negation is a special case, because it is a unary operation. Of four possibilities, the other three are not practically useful.
For the binary operations, the conjunction and disjunction are the most useful. Since these are analogous to multiplication and addition as well as having connections to concepts occurring in natural language, they are obviously useful. Most of the other connectives can be expressed as combinations of those two and negation, and are both less often required and more easily understood as these combination.
The conditional and biconditional are special cases that need a little more attention.
In most mathematics, operations and relations are considered distinct: Addition and subtraction yield other numbers, but comparison yields truth values. Since logic deals with truth values, and because the 16 possible truth tables for binary operations or relations can all be expressed with combinations of one or two other operations, the distinction between operations and relations is obscured.
The conditional is best understood as a relation. In the case of two-valued logic, this can be defined in terms of negation and disjunction, but in a way, this is an overly simplistic and misleading definition. It is better understood as a mathematical ordering relation which expresses the idea that the consequent is not less true that the precedent, or the conclusion is at least as true as the hypothesis. This is the idea behind the concepts of implication and entailment. The simple definition in terms of negation and disjunction cannot be successfully extended to multi-valued logics. In other variants of logic, the ordering properties of the conditional are simply assumed to hold. Whether the simple material conditional is adequate for logic in general is a much-debated question.
The biconditional expresses the idea of logical equivalence, or that two expressions or propositions are equally true or false, and is likewise best understood as a relation. In two-valued logic, it is a mathematical equivalence relation which expresses the idea that two expressions being compared have the same truth value. This can also be defined in terms of negation and conjunction, but once again, this only works in two values. The definition as a two-way conditional is more conceptually sound and can be employed in multi-valued systems or systems when truth values are not defined at all. It is only a mathematical equivalence if the conditional (however it is defined) is a mathematical ordering relation.
