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I wanted to ask about the word "let" and it's usage in more formal language like in mathematics.I understand the implied meaning "let x be a human" is intended to declare x to be a human. I think I've never thought of it before, but now thinking about it deeply I realized that I haven't really a clear idea of, for a lack of better words, why exactly (in terms of the meaning of each word in the sentence) does the sentence imply a declaration. Like, why semantically does this sentence declare x as an human? Or maybe knowing how this particular usage of "let" was derived would help.

I understand in standard English the word "let" to mean "allow" or "permit". So let me guess here but it seems to at least make a little sense to interpreting "let x be an human" as "allow the symbol x to be an human", where "be" here also means "refers" or "represent". It's like you are prompting the reader to "allow" this symbol in their minds to refer to an human, this as a result declares it. Is this what most people think when they read this?

I mean, this makes sense in most contexts, though one thing that seems to be weird with this interpretation is this almost implies an asking for permission from the reader but you kind of just want them to actually allow it and not reject it. But I guess this also is a problem when you say "suppose the symbol x represents an integer" because they can refuse to suppose it I guess.

Now I'm speculating but maybe this is why people often attach "then" after these statements to sort of signify "if you did suppose this then".

Really appreciate the help!

Edwin Ashworth
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Zihao Wang
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    If you look up let in a dictionary, you'll see that it's just a separate sense of that verb (one derived from the "allow, permit" meaning but not identical to it). – alphabet Mar 05 '24 at 03:38
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    Let's be honest. You can't shoehorn all the dozens of ways we use *let* into trivial derivations from the sense *allow.* The full OED has 16 separately-derived entries for *let, and this one (which originally meant to forbear, desist* 8 centuries ago) has no less than 28 sub-definitions. – FumbleFingers Mar 05 '24 at 03:39
  • You must allow that x is a human in order to proceed. – Tinfoil Hat Mar 05 '24 at 03:51
  • @FumbleFingers My bad, I really meant that thinking of it as "allow" seemed to make sense to me at the moment. Maybe I'm looking for the reason why we use it this way or how this usage and this particular sense got derived. Like what was the way they were using "let" at the time and how did this new meaning derive from that? Maybe this would help me really understand why we use "let" this way. – Zihao Wang Mar 05 '24 at 03:52
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    No, in math we don't ask for permission to let x = 1 (as in "suppose the symbol x represents an integer"). We are setting it as true for now. It's called a given: Let [us start with] x = 1. Put another way, it's a mathematical special use of language. – Yosef Baskin Mar 05 '24 at 04:23
  • https://en.wikipedia.org/wiki/Let_expression – HippoSawrUs Mar 05 '24 at 06:33
  • "Let" means "cause" in "let him know"; usually only done with "know" these days but could be used more widely in the past; there's "He lete make a proclamacion þorȝ all his Empire." ("He had a proclamation made through all his Empire") in a 15th century translation of Gesta Romanorum. – Stuart F Mar 05 '24 at 13:05

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In such contexts Let could be glossed "for our purposes here, accept as a given that ..."

TimR
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  • Yeah I agree. It's really just been glossed over in my mind ever since I started using it. Though, I think what I'm really trying to get at is what exactly is the intended action, is it imperative or is it declarative for example. If it's imperative who are we addressing it to? We still use this even if we are alone for example. I want know what exactly are people thinking when they say this. For example if I say "Allow x to be a human", my mind says okay, sure I'll allow/let that be true. I'm accepting (allowing in my mind) that this symbol x is representing a human. – Zihao Wang Mar 05 '24 at 23:27
  • Like, I know it's to declare something or to assume something. But from the some ways we use it for example "we let x be a person" or "I let x represent a integer" which, well, I think means We/I "allow"(in our brains) x to take on the meaning we want. I really think that it should stem from something else other than just taking this usage as a new usage. Because it's not just used in mathematical contexts like "Let this stick figure represent a person" or "Let this rose denote the extent of his love for her". – Zihao Wang Mar 05 '24 at 23:39
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Let is an imperative.

It is the imperative form of the conditional "If you let (allow) X [in this example] be a human."

Here let/allow = raise no objection to X being a human (+ the implied "then we will continue with our argument as if that were so.")

Compare "Oh, let him drive the car if he wants to." which would then have the implication of "then he will stop complaining." or a similar contextual result.

Greybeard
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  • There is also 'ordinary' "let", as in "They let us have out ball back". Contrasts with 1st person imperative "let", as in "Let's get our ball back". – BillJ Mar 05 '24 at 12:32
  • And there is also a kind of quasi-conditional let: "With these new all-weather tires, I'm good to go, let it rain, let it snow". – TimR Mar 05 '24 at 18:37
  • @TimR I'm not sure how relevant that is, since the action (rain, snow) is happening spontaneously, it's not something we cause. – Barmar Mar 05 '24 at 21:01
  • @Barmar: the quasi-conditional use of let is not an imperative involving human agency."let it rain, let it snow" could be paraphrased "come rain or come snow" or "even if it should rain or snow" or "no matter if it should rain or snow". – TimR Mar 05 '24 at 21:54
  • @TimR Yeah, but I don't see how that's comparable to "let x be 3". This means that x is expected to be 3, not just might be 3 and we're OK with it. – Barmar Mar 05 '24 at 22:01
  • @Greybeard I feel that this explanation is more of what I was looking for and I was considering cases without formal contexts, like in the situation when people say "Let this medal represent the hard work of our people etc..." or like when people draw a circle and say "Let this circle represent our knowledge etc...". These contexts feel similar to the formal usage I described as well and I agree it's kind of an imperative in this sense. – Zihao Wang Mar 05 '24 at 23:10
  • @Greybeard Though sometimes we may be considering something by ourselves, why do we still use the imperative there? In this case who are we addressing?

    I realize this is also true for many other words in formal language like I may say "Define x to be an integer" but this is also imperative form or "Suppose x is an integer" is also imperative. Maybe I want to let the symbol x represent some number, but people have to say "Let" in an imperative form to be formal, why is that? Who is the person they are directing the imperative demand to?

     – Zihao Wang Mar 05 '24 at 23:29
  • @Barmar I was adding another usage of let complementing BillJ's "ordinary" let in the first comment above. – TimR Mar 05 '24 at 23:45