Talk:Free logic

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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?

I have modified the formulas accordingly. The formula A was not meant to be atomic where Ax is a monadic predicate. Nortexoid 02:02, 13 July 2005 (UTC)

[edit] Unicorns

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)

That is not true, since we can infer existential generalizations from unquantified sentences by the valid inference schema $P(a_1,...,a_n) \supset \exists x_1,\dots,\exists x_nP(x_1,...,x_n)$ . Nortexoid 01:38, 6 January 2006 (UTC)

Let U be the predicate "there is a unicorn": it is indeed valid that $U(a) \rightarrow (\exists x.U(x))$ , but one cannot deduce $(\exists x.U(x))$ from this without the additional premise

U (a)

(you cannot show that there is a unicorn without postulating a specific unicorn, even in classical, non-free, formulations of FOL).

You're right--it will be removed. Nortexoid 02:39, 6 January 2006 (UTC)

The point of my remarks is that the assertion

From (1) in the unrestricted predicate calculus, if A is 'is a unicorn' it can be inferred that a unicorn exists.

in the article is wrong. Please change it.

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)

That is true but nowhere in the article have I said that the substitution is permissible in classical FOL. In any case, feel free to rewrite the article if you wish. Nortexoid 02:39, 6 January 2006 (UTC)

[edit] Rewrites

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 $(\forall x.Fx \rightarrow Gx) \rightarrow (\exists x.Fx \wedge Gx)$ 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)