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Is 1b really a substitution instance of 1? I suspect the author wanted to say that 1b is a substitution instance of 1a (and, in the next paragraph, that 1a (instead of 1) implies from "everything identical with Pegasus is Pegasus" to "something is identical to Pegasus"). But then the question becomes: is 1b really a substitution instance of 1?
You can only infer that "there is a unicorn" using your (1) if you have the hypothesis "everything is a unicorn". [this is changed from the original to address the following objection] This is really not a serious problem as the hypothesis seems very unlikely. A better example is the observation that the axiom of the empty set, for which many of us feel a need in Zermelo set theory (or ZFC), is redundant, because logic tells us that there is a set (since the class of sets is the domain of the quantifiers in ZFC) and then Separation gives us the empty set from any set. Randall Holmes 06:34, 5 January 2006 (UTC)
There are other serious problems with what you say here. For example, in standard FOL, there simply can't be a term denoting Pegasus, so you cannot make the substitution you describe. I am reluctant to edit the article myself, because if I do I shall simply rewrite it from first principles.
Randall Holmes 01:57, 6 January 2006 (UTC)
I have rewritten the text so that it doesn't contain any obvious errors. Your 1a was not a substitution instance of (1) in any obvious sense, so I relabelled it as (4); in (1a) and related sentences the scopes of the quantifiers were not what I would expect, so I adjusted them. The scheme is not valid, so I omitted it. The existence predicate is usually a primitive in free logic; the definition you give may actually always be valid, but I need to check that (there are at least two essentially different systems of free logic). Randall Holmes 14:56, 6 January 2006 (UTC)
I added a corresponding clarification request with more detail, but basically some of the equations in this arrow use a while others use a , and so I'm not certain whether both arrows are meant to denote/connote the same thing, and whether that same thing is material conditional or logical consequence (i.e. semantic consequence). 69.143.122.185 ( talk) 16:25, 28 January 2023 (UTC)
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||
|
Is 1b really a substitution instance of 1? I suspect the author wanted to say that 1b is a substitution instance of 1a (and, in the next paragraph, that 1a (instead of 1) implies from "everything identical with Pegasus is Pegasus" to "something is identical to Pegasus"). But then the question becomes: is 1b really a substitution instance of 1?
You can only infer that "there is a unicorn" using your (1) if you have the hypothesis "everything is a unicorn". [this is changed from the original to address the following objection] This is really not a serious problem as the hypothesis seems very unlikely. A better example is the observation that the axiom of the empty set, for which many of us feel a need in Zermelo set theory (or ZFC), is redundant, because logic tells us that there is a set (since the class of sets is the domain of the quantifiers in ZFC) and then Separation gives us the empty set from any set. Randall Holmes 06:34, 5 January 2006 (UTC)
There are other serious problems with what you say here. For example, in standard FOL, there simply can't be a term denoting Pegasus, so you cannot make the substitution you describe. I am reluctant to edit the article myself, because if I do I shall simply rewrite it from first principles.
Randall Holmes 01:57, 6 January 2006 (UTC)
I have rewritten the text so that it doesn't contain any obvious errors. Your 1a was not a substitution instance of (1) in any obvious sense, so I relabelled it as (4); in (1a) and related sentences the scopes of the quantifiers were not what I would expect, so I adjusted them. The scheme is not valid, so I omitted it. The existence predicate is usually a primitive in free logic; the definition you give may actually always be valid, but I need to check that (there are at least two essentially different systems of free logic). Randall Holmes 14:56, 6 January 2006 (UTC)
I added a corresponding clarification request with more detail, but basically some of the equations in this arrow use a while others use a , and so I'm not certain whether both arrows are meant to denote/connote the same thing, and whether that same thing is material conditional or logical consequence (i.e. semantic consequence). 69.143.122.185 ( talk) 16:25, 28 January 2023 (UTC)