In first-order logic, is this argument valid?

Hi

If I've misunderstood, I'm sorry, but I think that some of the confusion might be due to your use of the phrase 'the actual rules of inference'. I'd begin by distinguishing between different systems of logic. Among the most popular systems are classical logic, intuitionistic logic, relevance logic, fuzzy logic, and paraconsistent logic.

A system of logic will have both a semantic theory and at least one proof system. The semantic theory tells you the truth-conditions of expressions formed out of the logical constants that the system uses. A proof system is basically a set of rules of inference or axioms by which certain formulas can be derived from others. As I see it, the relation between systems of logic and semantic theories is one-one, and the relation between systems of logic and proof systems is one-many.

A proof system S is sound with respect to a semantic theory T just in case (i.e., if and only if) every theorem of S is a tautology with respect to T.

A proof system S is complete with respect to a semantic theory T just in case every formula that is a tautology with respect to T is a theorem of S.

You can see from these definitions of soundness and completeness why the relation between systems of logic and proof systems is one-many: for any semantic theory T, there are potentially many sets of rules of inference or axioms that are sound and complete with respect to T.

Thus, even relative to a particular system of logic, we can't properly speak of 'the (actual) rules of inference': there are potentially many sets of rules of inference that are sound and complete with respect to the system's semantics.

Whether one formula can be derived from another depends on which proof system you're using. For example, P can be inferred from ~~P within any proof system that is sound and complete with respect to classical logic but cannot be so inferred within any proof system that is sound with respect to intuitionistic logic.

With respect to classical semantics, ∀x Px entails ∃x Px, so within any proof system that is complete with respect to that semantics, you can derive the latter from the former. There are, however, semantics with respect to which ∀x Px does not entail ∃x Px. If you want to insist that it is possible that ∀x Px should be true while ∃x Px is false, you'll want to go for an extreme version of free logic (I think you'll want something like what is called 'inclusive logic', which is sensitive to the supposed possibility that the domain of discourse should be empty). I think that a logic is 'free' just in case it allows for the possibility that its constants should fail to refer, thereby invalidating such inferences as that from Fa to ∃x Fx. As far as I'm aware, proof systems that are sound with respect to inclusive logic tend to use the standard rules for the quantifiers, but they place restrictions on these rules. For example, existential introduction works differently, with the result that such formulas as ∃x (Fx v ~Fx) and ∃x (x = x) are no longer derivable as theorems.

I think that one of the motivations for developing free logic was the phenomenon of bearerless names: names, like 'Zeus' and 'Vulcan', that fail to refer. As I understand it, at least some versions of free logic are not inclusive, in the sense that they do not allow for the possibility that the domain of discourse should be empty, but any inclusive logic must be free, in the sense that it will have to allow for the possibility that names (constants) should lacks referents.

I feel that it is certainly an unfortunate consequence of a proof system's being complete with respect to classical semantics that it should permit the derivation of such formulas as ∃x (x = x) as theorems, since intuitively it is (logically) possible that there should be nothing at all (and, thus, that there should exist nothing that is identical with itself). I'm not sure, however, that I'd want to say that your argument is invalid. I think that within inclusive logic, your premise would be read '∀x (∃y (x = y) > Px)': for any x, if x exists, then x is P. Clearly, ∀x (∃y (x = y) > Px) is consistent with ~∃x Px on the supposition that it is possible that the domain of discourse should be empty, so the inference from ∀x Px to ∃x Px is invalid.

I would be interested to know why you'd like this argument to fail. Are you concerned about the ontological status of numbers?

You require that there exists an x.

Here is a valid argument:

Given: A is a set.

Given: For all x in A: x is an integer.

Given: A is non-empty.

Therefore there exists x in A: x is an integer.

Regarding additional details: If we are not given that A is non-empty, then the argument becomes invalid.

Regarding second additional details: What rule(s) of inference were you applying? If you were doing universal elimination/instantiation followed by existential introduction, then we have the same issue: Universal instantiation requires the set to be non-empty. Perhaps the wording isn't clear in many references. Here is a statement of universal instantiation

http://en.wikipedia.org/wiki/Axiom#Examples

Note this wording: "...a term t that is substitutable for x in phi...".

In other words, using universal instantiation pre-supposes the existence of such a term t.

Here's a PDF on universal instantiation that doesn't directly address the empty set, but explicitly specifies that we have an element of the set as part of its steps:

http://www.csd.uwo.ca/~lila/LectureNotes/logic18.p...

You might also be interested in the manipulations in this PDF on page 3 (pages 10 and 11 if you look at the green numbers, also green page 8 which is used in green page 10)

http://cs.anu.edu.au/student/comp2600/lectures/05-...

Sorry I don't know any good books. The best I can offer is that I hang around mathhelpforum.com a lot, and my guess is Ackbeet, emakarov, or Plato (or other regulars) on that forum could give you some good suggestions.

a million: if premise a million and 2 are standard as valid and the underlying assumption that immoral issues could be rejected is standard as authentic the tip is valid in the boundary situations set by way of the premises - even nevertheless, you could desire to nevertheless reject it for different motives. If the "consequently" is study as "for those motives" then confident, if study as "as a universal case" then no. #2 Premise a million states a particular condition for books being written for atheism yet would not point out different motives those books would be written do no longer exist. consequently, you could no longer argue the tip on the inspiration of premise 2. #3: works for me, if premise a million and 2 are standard as valid. i assume technically there's a 0.33 non-theist purpose industry who nevertheless have faith in miracles - which i might strongly argue against as miracles tend to be defined as "interventions by way of a god". So for me it rather works, till people want to argue for a various definition of "miracle".

Every white hole is a purple dragon

Therefore there exists a white hole that is a dragon.

if that is over your head how about

Every transcendental integer is divisible by π.

Therefore there is some transcendental integer.

First order logic usually does not include the "some" or "all" quantifiers.

I would think it would be

if it an integer, it is an x

I agree with you that the statement is not a first-order logic argument

maybe a second-order?

Yes, it is valid, as the conclusion follows from the premise. But if the premise is true or not is another story.

Of course it is valid. If every x is something, then there can be no x which isn't that something.