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# Is concatenation a logical AND?

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As a person with no significant background in mathematics and computer science I thought that concatenation and AND are logically identical if not very similar because both add something to something yet not by numerical addition (natural addition) so I found it sensical to say "x AND y" is weight to "x . y". But of course I might be wrong, so I ask:

Is concatenation (for example, as with . in PHP) a logical AND?

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tl;dr: No.

From an engineering perspective, you might be asking if a concatenation operator can be used in place of a logical ‘and’ operator. This is obviously specific to a particular language, but I'm not aware of any language in which this is the case, and I've seen a few languages. For most languages with static types, using a concatenator when a logic operator is expected won't even type check; for the others, concat often ends up working more like an ‘or’ than an ‘and’, as detailed in 8063's answer.

From a mathematical perspective, you might be asking if the concept of logical ‘and’ generalizes in any interesting ways such that it might include concatenation under some conditions. ‘Interesting’ here is somewhat subjective, so I can't say absolutely not, but I will say that the typical perspective that a mathematician would take on the generalization of logical ‘and’ and ‘or’ is that they are the two operations of a lattice, specifically a Boolean lattice (a.k.a. a Boolean algebra). One of the requirements for lattice operations is that they must be commutative—that is, the order of their arguments mustn't matter. Another is that they must be idempotent—that is, any value combined with itself via the operation must be unchanged. Concatenation violates both of these in general. One can conceive of a generalization of a lattice (or semilattice, if you only care about ‘and’ and not ‘or’) that doesn't carry these restrictions, but one would be giving up very much of what it means to be a lattice in that case, and consequently very very much of what it means to be the logical ‘and’.

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You're right that you can make some sort of connection between addition, concatenation, and logical ‘and’—these are all operators that can be considered as monoids. A monoid is a very general concept: it comprises a particular type of things that can be combined, a unique value of that type which is ‘empty’, and two laws for combining values:

• The associative law: (a + b) + c = a + (b + c)
• The identity law: a + empty = empty + a = a

Mathematicians usually define monoids by their operator and their empty value as a pair, so we could call the additive monoid (+, 0), the string monoid (., ""), and the ‘and’ monoid (&&, true).

There are lots of other monoids too, even on the same types! For example, the multiplicative monoid (*, 1) and the ‘or’ monoid (||, false) are just as valid as (+, 0) and (&&, true). (set union, empty set) is another easy example of a monoid. But subtraction can't be made into a monoid, because it isn't associative. And set intersection isn't a monoid because you can't have an identity (unless you're working in a set theory where there is a ‘universe’ set, in which case that set becomes the ‘empty’ set of the intersection monoid because it's the only set that can satisfy the identity law, even though it is as far from literally empty as a set can be).

In a very nice sense, the string monoid (or its more general version, the list monoid) is the prototypical monoid. This is because concatenation throws away exactly what the monoid laws say doesn't matter (the groupings by which values are combined, and any empty values), and nothing else. You can always represent a monoidal computation by grouping everything into lists first, concatenating them, and then running the monoid operation on the elements of the resulting list; but you can't say the same for any monoid that isn't isomorphic to the list monoid.

So, while I don't think that you can reasonably call concatenation a sort of logical ‘and’, it's more legitimate to say that logical ‘and’, as a monoidal operation, is a sort of concatenation (with some extra work at the end)! The same could equally be said for logical ‘or’, addition, multiplication, and other monoidal operations like set union, so don't get too hung up on the word ‘and’, even though it is often a very natural word for monoids.

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It is OR.

Assimilate FALSE to the empty string, and TRUE to any other string:

ε CONCATENATE ε = ε
ε CONCATENATE something = something
something CONCATENATE ε = something


What about both sides at TRUE:

something CONCATENATE something = somethingsomething


But because any non empty string assimilates as TRUE:

something CONCATENATE something = something

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