# How can sequences/expressions occur in other sequences/expressions?

I know I specifically wrote a question about Wetzel, however I do not want to invalidate previous answers.

In Quine's 'Mathematical Logic' he discusses occurences of 'expressions' in other 'expressions'.

He suggests that one can 'expression' can occur within another.

Later, Wetzel wrote an article named 'What are occurences of Expressions' which expands on this idea and also the idea of Pierce's Tokens.

Wetzel defines any expression as a finite sequence of symbols, she defines the idea of an occurence based on this.

Wetzel then give the description of occurences in expressions as the following:

For any 'expression' or finite sequence of symbols.

In particular she uses the following sentence:

My issue with both Wetsel and Quine's suggestions are caused by the way they talk about occuring, Wetzel uses the idea that each 'bird' is an occurence of the type 'bird' and this acceptable, however the occurence of bird in any group of animals is acceptable because 'bird' is of a further abstraction that the group it occurs in. The group of birds that 'bird' occurs (by having an occurence in the set) are of very different kinds, one is concrete the other abstract.

If we look at *, using Wetzel's definition. She describes how sequences should have 'occurences' the expression 'Macavity' consists of a sequence of letters, therefore consisting of different occurences of each letter.

In the expression 'Macavity, Macavity there is no one like Macavity' each 'Macavity' has it's own occurence of each letter, corresponding to a position in this much longer sequence that we deifne as the expression of this sentence, yet the type 'Macavity' supposedly consists of occurences of letters itself. Therefore it seems there is no unique type 'Macavity' as each occurence in the sentence consists of it's own occurences of each letter, the occurences being qualitatively identical with the type, something which seems impossible.

If we take the same definition of 'expression' and ignore the the idea of one 'occuring' in another, then there is no issue to define a 'concatenation' operation such that for any expression x and y, x+y contains occurences of the same letters in the same order.

However, if we look at the definition of sequence, it seems having the same letters in the same order should be the same sequence occuring again, however if we discuss the 'occurences' or positions, each place it occurs in another sequence it will have the same elements in the same order but at different occurences or positions in the containing sequence. Perhaps it can be solved by stating that a sequence consists of two parts, which elements are in the sequence, and where they occur relative to each other, I am not entirely convinced on this, but I am currently actively searching for more sources in this direction of thinking.

Given this, why is it that Quine and Wetzel believe that expressions can occur in expressions, or sequence occur in sequences?

I think this can be solved by exploring Sequences and how they can occur in each other.

In computer science they have 'strings' and sequences as data types, in this context they define 'concatenation' and 'substring' but at no point is there any explicit mention of 'occurences'. We can ask, what is in common between the sequence '321' and '321' in the sequence '654321'? This is puzzling because it suggests that it is two things: elements, and position, surely this is the essential definition of 'sequence'? However how do we overcome the fact that we can see occurences in the sequence first sequence and yet the place it 'occurs' has it's own occurences in the containing sequence?

Perhaps this whole connundrum can be solved by one sentence:

'An occurence of a sequence is a sequence of occurences'. The second 'occurences' referring to 'occurences' of letters/symbols elements of the sequence that is 'occuring'.

I'm not sure to what extent this solves it, but it matches with the idea of a token of a sequence being a sequence of tokens.

I would however grealty appreciate any users thoughts on this.

• The issue is answered quite simply for formal language with the so-called inductive definition of terms (and formulas). See e.g. Heinz-Dieter Ebbinghaus & Jörg Flum & Wolfgang Thomas, Mathematical Logic (Springer, 3rd ed, 2021): Def.3.1, page 15. The procedure described in the example show the derivation of a term: every step in the derivation produces a sub-term of the final term. That sub-term is an expression (a finite string of symbols) that occurs in the final expression. Dec 6, 2022 at 12:14