Fixing objects in logic

Question

When using variables one encounters forall and existential quantifiers. My question can is: Is there a formal reason that variables bound by quantifiers are fixed after, or why is this intuition justified?

An example to illustrate what I mean. Consider the example $\forall n \in \mathbf{N} \exists A \forall x [x \in A \iff x \in \mathbf{N} \wedge x+n=2]$. Now, given any $n$, this gives me a set. However, whenever I write "given any $n$" (which should mean the same as "for all $n$, right? I find given more intuitive though), I fix a number, I just don't know which one - it is arbitrary but fixed. What is the formal reason that I can regard this as fixed, if there is any? I think this is connected to the concept of a free variable, but I am not entirely sure.

Another example might be: Let $G$ be a graph and $v$ a vertex of $G$ such that $v$ has degree $d(v).$ Here, I have an arbitrary (but fixed) graph, an arbitrary (but fixed) vertex of that graph and a number $d(v)$ bound by an existential quantifier, again a fixed (but unkown) number. Then I can work with the information that has been assumed about those objects.

All of this is intuitive, however I am looking for a formal reason to back this intuition up.

Answer 1

This is one questions arisen with the advancement of logic and foundational studies. Gottlob Frege notes that (Translations from the Philosophical Writings of Gottlob Frege edited by P. Geach and M. Black, Basil Blackwell, 1960, p. 110):

'n' is not the proper name of any number, definite or indefinite... We write the letter 'n' in order to achieve generality... Of course we may speak of indefiniteness here; but here the word 'indefinite' is not an adjective of 'number', but ['indefinitely'] is an adverb, e.g. of the verb 'to indicate'. We cannot say that 'n' designates an indefinite number, but we can say that it indicates numbers indefinitely. And so it is always when letters are used in arithmetic, except for the few cases $(\pi, e, i)$ where they occur as proper names; but then they designate definite, invariable numbers.

One response to this peculiar role of variables is locate it in a linguistic mastery of distributive (as opposed to collective) reference to a domain of discourse (a plurality) while keeping indefinite to an extent to attain generality.

Another suggestion is to posit arbitrary objects as referents of variables. Notice that we do not make claims about the actual cognitive processes of the human agent; we try to discern the smallest distinctions in our understanding of formal systems and give an account of them by rigorous and coherent views from within, just as done by von Neumann ordinals or Zermelo ordinals.

According to the arbitrary objects view set forth by Kit Fine (Reasoning with Arbitrary Objects, Basil Blackwell, 1985, p. 6):

In addition to individual objects, there are arbitrary objects: in addition to individual numbers, arbitrary numbers; in addition to individual men, arbitrary men. With each arbitrary object is associated an appropriate range of individual objects, its values: with each arbitrary number, the range of individual numbers; with each arbitrary man, the range of individual men. An arbitrary object has those properties common to the individual objects in its range. So an arbitrary number is odd or even, an arbitrary man is mortal, since each individual number is odd or even, each individual man is mortal. On the other hand, an arbitrary number fails to be prime, an arbitrary man fails to be a philosopher, since some individual number is not prime, some individual man is not a philosopher.

Hence, an arbitrary object is dependent on the actual (independent) object It takes values from a a value range the context determines. The properties it inherits from the actual object are only and all those that the objects in its value range share (Fine calls this principle of generic attribution). Thus, an arbitrary object is one abstracted from a class of independent objects and does not exist in physically detectable reality (just as 'an average woman with 1,3 children' does not exist in reality).

Fine's view is quite detailed and has the appeal of using a simple word-world correspondence, a relation which we are inclined to hold. But if it definitely shows something, the role variables play is much more complicated than we would thing while using them smoothly. To give an idea about this complication and the difficulties that such a correspondence view could face, the following problem Fine himself has seen may serve well (A Defence of Arbitrary Objects, p. 70):

In a sentence such as 'Let x and y be two arbitrary reals', we will want to say that the symbols 'x' and 'y' refer to two unrestricted and independent arbitrary reals. But to which? It is natural to suppose that 'x' and 'y' refer to two unrestricted and independent arbitrary reals.

There are solutions offered to such problems, but it may be pointed that they take on somewhat ad hoc character.

According to some authors, variables over pluralities can be eliminable, and the way it is eliminated is the explanation. They give as examples Quine's technique of elimination variables and combinatory logic among others. Though it is not possible to give an assessment of such techniques and systems briefly, I can say that it is quite dubious whether Quine really succeeded in "explaining away" the idea of variable over plurality, or he just gave another formalism that sweeps it under the rug. Combinatory logic essentially seeks singular reference as an argument to a function and leaves the domain issues to the environment (typically, computational).

As an upshot, it seems that just as we find natural consonance across different usages of one and the same word, we use the symbols with a similar elasticity -a precious faculty. We may observe this with the equality sign more clearly; it signifies different kinds of identity and equality, we pass from one to another smoothly to the point that we sometimes find it too pedantic to mark them out. However, it should be remarked, the task of a principled formal analysis of variables, in particular, over pluralities, still remains not fully addressed.

Answer 2

Actually, "givenness" is different from the stipulated use of quantifiers in first-order logic.

You will find discussion of "givenness" in Markov's "Theory of Algorithms." Markov had preferred a constructive point of view and followed certain traditions in Russian mathematics. To avoid intuitionism, he introduced a notion of "strengthened implication" based upon "givenness."

Another discussion of "givenness" can be found in chapter 19 of Russell's "relative view of quantity." His axioms do not assert the truth of reflexive equality. They do, however, include symmetry and transitivity. So, for any "given" A, reflexiveness can be proven.

You can use something like,

$$ \forall x \forall y ( x = y \leftrightarrow \exists z ( x = z \wedge z = y ) ) $$

$$ \forall x \forall y ( ( x = x \wedge y = y ) \rightarrow ( x = y \leftrightarrow y = x ) ) $$

for transitivity and symmetry. A universal quantifier must be understood as a restricted form using a reflexive equality,

$$ \forall x ( \Phi(x) ) $$

$$ a = a \rightarrow \Phi(a) $$

An existential quantifier would be the expected conjunction,

$$ \exists x ( \Phi(x) ) $$

$$ a = a \wedge \Phi(a) $$

This formalism requires different inference rules for quantification which are essentially constructive. The propositional connectives can be classical. However, the hexagon interpretation needs to be used. Because the hexagon interpretation is not distributive, any relations which are used must be introduced into a proof explicitly with biconditionals.

What occurs with the expansion of a universal quantifier when the antecedent of the quantified formula is, itself, a reflexive equality statement is a formula which is the antecedent in a scheme called "contraction" by Eric Schecter. Contraction implies non-contradiction. So, while the redundancy is strange, it enforces a classical condition on reflexive equality. This is somewhat important as "equality in the presence of apartness" in intuitionism does not have a decidable equality.

The very first paragraph of Section 74 of Kleene's "Introduction to Metamathematics" states that a formalization of ordinary mathematical practice would present as a sequence $S_1$, $S_2$, $S_3$, ... of formal systems. This formalism is specifically designed to respect that.

Answer 3

Tankut Beygu has given you an excellent answer.

The book burners will probably remove it as too philosophical.

The relationship between his answer and mine resides in his remarks about the sign of equality. Recall that recursion theory must introduce an auxiliary form of equality to address "definedness." Definedness is certainly a method used in ordinary mathematical practice. And the section from Kleene's book mentioned in the other answer is precisely about the transition to "formalization" with respect to definedness.

The use of this auxiliary form of equality is comparable to the principle of indiscernibility of non-existents in free logic.

An excellent site for lurking foundational subjects is the FOM mailing list. They have an October 2015 thread on free logic which will certainly validate Mr. Beygu's observation that your question involves issues which have not been worked out.

https://cs.nyu.edu/pipermail/fom/2015-October/thread.html

There is nothing the matter with first-order logic as "a way of studying mathematics." But, it breaks ordinary practice. So, its extra-mathematical stipulations may legitimately be questioned.

Free people know they have to tolerate book burners. It is sad that universities are producing them by the score.