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I came across the equivalences of conditionals and biconditionals. In the conditionals case, it's $P \implies Q \equiv ( \lnot P) \lor Q$, and the biconditional is $P \iff Q \equiv ( \lnot P \land \lnot Q ) \lor (P \land Q )$.

My question is, if both conditionals and biconditionals can be boiled down to negations, conjunctions and disjunctions, then why are conditionals and biconditionals used in logic?

I imagined at first that it's sort of like programming, where it would simplify control flow, but then I learned there aren't any more "complex" connectives like them, so why are those two special enough to use? Why aren't there more symbolizations of complex propositional forms?

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    $\begingroup$ Only convenience. We can use only $\{ \lnot, \lor \}$ or $\{ \lnot, \land \}$. But $\to$ is very useful for its use in Modus Ponens, that is very natural. See Adequacy of connectives $\endgroup$ Feb 21, 2023 at 15:31
  • $\begingroup$ And see Logical connectives. $\endgroup$ Feb 21, 2023 at 15:33
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    $\begingroup$ This is the kind of question to which user Bram28 (give them a ping) might have a cosy or diverting answer to. $\quad$ The conditional form is probably more intuitive except when its antecedent & consequent are respectively false & true, and except when it is ¬A→B ("A unless B", in which case A∨B communicates the meaning more directly). $\endgroup$
    – ryang
    Feb 21, 2023 at 15:47
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    $\begingroup$ You can even go with just one operation: NAND (also called the Sheffer stroke) alone is functionally complete. But such a system isn't particularly pleasant to use (let alone read!). $\endgroup$ Feb 21, 2023 at 16:44
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    $\begingroup$ @ryang Thanks for the heads-up :) $\endgroup$
    – Bram28
    Feb 21, 2023 at 16:51

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Please note that all truth-functional expressions can be boiled down to NAND's. So given that, you might as well ask: why are we even using conjunctions, disjunctions, and negations?

Also note that I should be able to program all computers by directly writing down long strings of $0$'s and $1$'s. So why do we have languages like C++, Python, or JavaScript?

Why, indeed, talk about planets when ultimately they are nothing but big collections of atoms? Hell, once I know all fundamental forces of physics, why even do chemistry, biology, cognitive science, sociology, etc?

I hope you start to see the answer: it is super useful to have 'higher-level' or 'macro-level' perspectives. Indeed, given the inherent cognitive limitations of the human mind, we really need those. Think about it: why do we do statistics, when all it amounts to is lossy data compression? Ultimately it is because we are of limited intelligence, because if we were God, we would have no need for statistics. In general, an infinitely smart being can look at the raw 'low-level' descriptions, and 'see' everything there is to divine. As humans, however, our explanations, predictions, and general analyses, abilities, and understanding is greatly enriched by high-level concepts and perspectives.

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Wouldn't you rather read: "if $ABC$ is a triangle then the sum of angles $A,B,C$ is 180°" than "$ABC$ is not a triangle or the sum of angles $A,B,C$ is 180°" . Most of theorems in mathematics are of the form "if such and such then you can conclude this and that". Even when there is no antecedent, we usually understand a theorem as meaning "if these axioms hold and if the theorems we have already proved are taken as true, then so on and so forth." You can rewrite this with only "or" and "not", but it will not mean much for most maths students, even mathematicians.

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There is at least one major reason to include implication, namely that it allows us to use intuitionistic logics in a way that is translatable into established mathematical reasoning.

Further, it is well known that there are problems in mathematics outside the scope of its best theories. For example, the Continuum Hypothesis has been proven to be independent of ZFC; so, if one wants to have a worldview that is sensitive to truth, but also sensitive to the limits of proof systems for mathematics, then some sort of non-Classical implication is needed.

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Although it is true that the conditional and biconditional can be boiled down to simpler functions in classical two-valued logic, this may be misleading when attempts are made to extend logic beyond the classical limits.

The biconditional expresses that two propositions are equally true or false, or have the same truth value. In the theory of mathematical structure, this has the properties of an equivalence relation. (Reflexive, Symmetric, Transitive)

The conditional expresses that the conclusion is not less true than the hypothesis. In the theory of mathematical structure, this has the properties of an ordering relation. (Reflexive, Antisymmetric, Transitive)

These properties make extended chains of reasoning possible.

In variants of logic, for instance multivalued logics, modal logic, or intuitionism, the neat correspondence between the conditional and the biconditional and the other logical operations (negation, conjunction and disjunction) breaks down. The conditional and biconditional have to be otherwise defined or their properties somehow asserted independently in order for them to be useful.

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