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I was thinking about whether there is a difference between the symbol 'x' and the expression/formula containing it.

For example we discuss 'x' the symbol but equally I can talk about 'x' the expression, if a wff is defined as a subset of sequences of symbols, then the formula 'x' is a sequence containing one position with which the symbol 'x' is occurs.

Is there a formal difference between these two? Mathematically a sequence is a function mapping elements to positions, mathematically this makes them different, however if I analyse a sequence 'x+y' I see that there is one occurence of both the symbol 'x' and sequence 'x'(sequence with only the symbol 'x' in it) however both correspond to the same position in the same sequence.

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  • @MauroALLEGRANZA mistake, I think I had the tag there from before.
    – Confused
    Commented Dec 15, 2022 at 12:56
  • Outside mathematics, it makes little sense to speak of a "sequence" of objects... Maybe we can consider the example of a family: "a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship)." It seems that a family has specific properties in a society and it is different from the individual that belongs to it, also when the family has only one member. Commented Dec 15, 2022 at 13:11
  • If you consider the sequence of symbols 'x+y' and 'x' they are different: the first one has three elements while the second one only one. Again, no philosophical issue here... Commented Dec 15, 2022 at 13:12
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    It seems to me like you're asking the difference between a candy and a box of candy when there's only one candy left in the box. A sequence by definition os a collection of objects. A collection of one object isn't really a sequence unless the other objects are implied. Commented Dec 15, 2022 at 19:19
  • One is a containeR, the other one is a containeD. There's a clear conceptual dialectical, complementary, etc. polarization there.
    – RodolfoAP
    Commented Dec 16, 2022 at 6:13

2 Answers 2

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I'm not sure there's enough information in the answer to warrant posting it as a site-appropriate answer proper, but here it is:

In the mathematics of permutations, for a set of n elements, you can put those elements in n!-many sequences. For example, if you have 4 elements, you can put them in 24 sequences. The cases of an empty set and a set with only one element then have relatively trivial sequences to their names, since 0! = 1 (stipulated) and 1! = 1.

Pragmatically/in natural language, we tend not to have a reason to think of such trivialities as sequences, since we often speak of e.g. "a sequence of events," and it would conserve time in written or spoken word usage to refer to "an event" when we are considering only one event. So pedantically, a sequence can contain one stage, but I'm only a pedant some of the time, a peasant the rest, and in this case I'd tend to side with my fellow peasants.

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Yes there's a difference; the symbols and an expression consisting of symbols are two different objects with different functions that you can apply to them, producing different results.

A common example that's widely seen in computer programming is the difference between a list of objects and the objects from which we compose lists, which I will here call atoms. If I declare my set of atoms to be letters and numbers I can make lists such as (x 1 y 2), (x) and (2), and apply functions such as head and tail to these:

head (x 1 y 2)  ⇒ x
tail (x 1 y 2)  ⇒ (1 y 2)
head (3)        ⇒ 3
tail (3)        ⇒ ()

You cannot apply head or tail to an atom; they operate only on lists. It makes no sense to ask, "what is the head of 3" as atoms have no head or tail.

Notice that head takes a list and produces an atom, but tail takes a list and produces another list. You can see from this that (at least some) functions take not any other object as their argument, but only objects of specific types. By distinguishing between the symbol 'x' and the expression consisting of only the symbol 'x' you have created two types in your language, making them two different objects.

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