The logic of not merely connectives (and, implies, not, inclusive or, xor, etc) but also quantifiers

[Boojatta]

In neglecting to credit Frege with what he was the first person to formulate, has the world copied not only his insights into distinguishing between valid and invalid reasoning, but also his focus on what is a very restricted subset of all possible statements that are either definitely true or definitely false?

After all, if it is actually true that, for all real numbers q and r, if r isn't equal to zero, then (q/r) times r = q ...

... then it follows that if ((q/0) times 0) = q then r = 0. However, faced with a claim of that nature, most teachers of mathematics in grades 1 to 12 would without hesitation mark the statement as false and/or prohibited and/or meaningless and/or an indication of student inability to comprehend.

Evidently, formation rules are taken to be very important. Two people cannot discuss a formula unless both agree that it is at least potentially meaningful and that it doesn't violate the rules for formation of well-formed formulas (wffs). So, before we even get to the question of validity of reasoning, we need to deal with the question of formation of sentences. If Frege focused attention on an unnecessarily narrow and rigidly circumscribed collection of possible sentences, then the quantificational logic that he introduced could be misleading and of only limited utility.

Here's an example of an alternative to Frege's approach to the logic of quantifiers:
a blog entry about independence-friendly quantification

[bluerum]

1. "if r isn't equal to zero" is pretty clear

[Boojatta]

2. The quantifier "for every" comes before the if-then statement

Do you agree that, for all real numbers q and r, if r isn't equal to zero, then ((q/r) times r) = q?

The sub-formula "if r isn't equal to zero, then ((q/r) times r) = q" can be replaced with its contrapositive: "If (q/r)*r doesn't equal q, then r = 0."

After replacement, we obtain:
For all real numbers q and r, if (q/r)*r doesn't equal q, then r = 0.

Now, zero is a real number. If it's true for all real numbers q and r, then it's true in particular for r=0 and for any real number q.

In other words, we have derived this conclusion:
For every real number q, if (q/0)*0 doesn't equal q, then 0 = 0.

Assuming that a statement that begins with the so-called "universal" quantifier "for all" (aka "for every") is true, it follows that the statement obtained by substituting any specific real-number value for that variable is a true statement.

Rules for well-formed formulas ordinarily involve basic questions of syntactic form, and don't depend on whether or not a given statement is true or false. The special role of the number zero involves semantics and not merely mathematical syntax. Of course, in the syntax of ordinary English, things are different and we have to be careful about one versus many and so on. Hence we see neologisms such as "file(s)" to avoid the need to vary the English language label depending on the value of a constant, and the impossibility of choosing a single English language label for the entities being counted if the count isn't a constant but is a variable ranging from zero to three (for example).

[Boojatta]

3. Link to an article about what is called Independence Friendly Logic

http://plato.stanford.edu/entries/logic-if/