Would a phrase like 'x is an unspecified object' be part of my meta-language? As x is a variable, such an expression is not meaningful in relation to any object in my interpretation, however we specifically refer to x as an object, but not in a way that we can create a predicate for my object x.

It is strange because we refer to the object, but it's the object's relation to its reference.

  • From context, "x is an unspecified object" is a shorthand for "let x denote an unspecified object" (x is not literally an object, or even object template), so yes, it is a meta-statement about use of a symbol.
    – Conifold
    Sep 24, 2022 at 13:35
  • What do you mean by "unspecified"? Does it include class-member variables as in "Let x be any even number"? Does it include variables that refer to an unknown individual as in in "Let x be whoever stole my tuna sandwich out of the fridge"? Sep 24, 2022 at 15:47
  • @DavidGudeman More like 'class members'
    – Confused
    Sep 24, 2022 at 16:21
  • @Conifold "From context, "x is an unspecified object" is a shorthand for "let x denote an unspecified object" (x is not literally an object, or even object template)" No. It would be either let 'x' denote an unspecified object or indeed x is an unspecified object. Sep 24, 2022 at 16:42
  • 2
    Have you heard of variables such as your x ranging over the domain of the usual one-sort FOL are not non-logical symbols? Sep 25, 2022 at 1:31

1 Answer 1


Would a phrase like 'x is an unspecified object' be part of my meta-language?

To state "x is an unspecified object" is a regular but prior assertion that any subsequent reasoning will have to comply with and respect as an assumption. There is no meta level which cannot be straightforwardly described as an assumption.

There is no logical difference between "x is an unspecified object" and "let 'x' denotes some object".

Surely, we would also all agree that there are plentiful objects in the real world, or depending on context in some fictional or conceptual world, and so we would have to agree that, given our specification, x is one of them. We don't say which it is but humans constantly use indefinite descriptions which have the same linguistic function.

So "x is an unspecified object" is perfectly meaningful, including in the sense that we can subsequently always further specify what sort of object it is, for example, some object in the real world like a milk shake, or some mathematical concept etc.

This is also why there would be no difficulty using a predicate to further specify that x is for example an even number or what the Unicorn is now looking at.

The notion of meta-language only becomes useful for theories. A theory is the meta-level of its object. When the object is itself a theory, it may become necessary to make a formal distinction between the object and the theory about the object.

This distinction may be achieved using various means but an ordinary meta-level distinction occurs when we use quotation marks to separate what we are saying from the language we are talking about. For example: The statement "Trump has lost the election" is true. This may also be achieved using a new lexical scheme specific to the meta-level.

So there is no meta-language in "x is an unspecified object", where x is just identified as an unspecified object. The sentence has the same logical structure as "Joe is a man", where "a man" just means "an unspecified man".

There is meta-language in "let 'x' denotes some object", where "x" is clearly isolated by single quotation marks. In the sentence, "x" does not yet refer to anything. However, the sentence effectively identifies "x" as the name for something, namely, some object, so that in the subsequent discussion, the name "x" will be taken to denote (or refer to) some object, and x will be understood to be some object.


The statement "John is an unspecified object" doesn't make sense but only because we all read "John" as a name so that it is always understood as referring to a specific object, i.e., the only object in the context named "John". Asserting that John is an unspecified object contradicts our linguistic assumption that the name "John" by definition refers to a specific object.

  • so it's in our logical language, but I cannot make a predicate x is an unspecified object', 'x is an object' works under the interpretation 'x=john' but ' john is an unspecified object' makes no sense. Is it that the statement only makes sense outside of a particular interpretation?
    – Confused
    Sep 24, 2022 at 19:13
  • @user1007028 Good point. See my edited answer. Sep 25, 2022 at 10:20
  • I guess in the general context of the language we can say 'x is an unspecified object' and this would be true, but in an interpretation where we assign 'x' the statement no longer applies, just as generally, the assertion 'x+1=2' has no meaning until we interpret x=1 then it becomes true? Just like 'he is an unspecified person' kind of makes sense in some cases but not if 'he' is actually a specific person.
    – Confused
    Sep 25, 2022 at 10:44
  • @user1007028 "'x+1=2' has no meaning until we interpret x=1 then it becomes true?" It has to mean something to us if we are to interpret it in any way. 2. "'he is an unspecified person'" Pronouns always either refer back to some object previously discussed or to something which is readily identified in the context, so like "John", "he" is always used to refer to a specified object. Sep 25, 2022 at 16:28
  • it depends entirely on context then, I also guess the issue is also the word 'is' can be used somewhat differently, thank you for your detailed answers.
    – Confused
    Sep 25, 2022 at 17:20

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