The Implicative Function is a propositional function with two arguments p and q, and is the proposition that either not-p or q is true, that is, it is the proposition ~ p v q. Thus if p is true, ~ p is false, and accordingly the only alternative left by the proposition ~ p v q is that q is true. In other words ii p and ~p v q are both true, then q is true. In this sense the proposition ~ p v q will be quoted as stating that p implies q. The idea contained in this propositional function is so important that it requires a symbolism which with direct simplicity represents the proposition as connecting p and q without the intervention of ~ p . But “implies” as used here expresses nothing else than the connection between p and q also expressed by the disjunction “not-p or q” The symbol employed for “p implies q}” i.e. for ” ~pvq ” is “p \supset q.” This symbol may also be read “if p, then q.” The association of implication with the use of an apparent variable produces an extension called ” formal implication.” This is explained later: it is an idea derivative from ” implication ” as here defined. When it is necessary explicitly to discriminate ” implication ” from ” formal implication,” it is called “material implication.” Thus ” material implication” is simply “implication” as here defined. The process of inference, which in common usage is often confused with implication, is explained immediately.

I dont understand this when i compared to implication function read in modern logic books which just give a truth table for if p then q.

I am confused…

But you write in the first part:

“Here are a few metamathematical statements.

-1- “There are infinitely many prime numbers” is true.

-2- The continuum hypothesis cannot be proved using the standard axioms of set theory.

-3- “There are infinitely many prime numbers” implies “There are infinitely many odd numbers”.

-4- The least upper bound axiom implies that every Cauchy sequence converges.

In each of these four sentences I didn’t make mathematical statements. Rather, I referred to mathematical statements.”

I beg to differ slightly.

First, many metamathematical statements are mathematical statements in a larger theory and can often be treated as mathematical objects (for example Model theory).

Second, I would have drawn important distinctions between those four (but it was not exactly the subject of your post which is already very detailed).

Further, each of your four sentences implies a specific mathematical universe with minimal logical and set-theoretic axioms for it to be meaningful and unambiguous. I think this is important to point out to young mathematicians.

In these sentences we have most of the time silent implications together with an explicit “implies” (see below).

There is a topological analogy: most properties of, say a knot, depend on the space it is embedded in.

Not that these sentences do not have the same implied strength or the same immediate relevance for the mathematician, undergraduate or not. So I prefer to rephrase them with parameters and implicit hypothesis.

-1- Theorem A is true (in implied theory T)

-2- Axiom C is independent from Axiom-System S

-3- Theorem A has Corollary B (in implied theory T common to A and B)

-4- Axiom L (added to implied Theory R) gives it the strength to prove Theorem V.

The first sentence is of the most common kind for a mathematician.

The third sentence is very common as well and is a very small step from -1-.

Both -1- and -3- are used so frequently that the distinctions between mathematics and metamathematics is blurred as in common metalinguistic sentences people use every day : “Please, can you finish your sentence?” or “Do not answer this question!”

The fourth one is of strong metamathematical character and of interest to most mathematicians, because Theorem V is useful and a common way to express continuity. It could be paraphrased/expanded : one of the solutions to create a mathematical universe where you can have a notion of continuity for your analysis theorems is to have a Theory R consistent with Axiom L and add this axiom L to R, creating Theory R2 and go on with finding limits.

But the second one is the strongest of all, the most “meta” and the only one to be explicit about its metamathematical context. It is part of a family of statements of about “relationships between logical contexts in which you can do mathematics”. You can call that meta-trans-peri-mathematics or meta-meta-metamathematics.

It would be very difficult to find an equivalent to -2- in a non mathematical situation. It would be considered at best very subjective or dogmatic such as “You cannot speak about the “Gestalt” philosophical concept in english without using the german word “Gestalt” or another philosophical german word of equivalent depth and power. You will always fail if you try.”

The remarquable thing about mathematics is that we can reach a so strong level of implication in our discourse about it.

http://xkcd.com/704/

That said, I wouldn’t interpret either of these as having anything to do with material implication, or perhaps better the material conditional, at least not so easily. I do agree that “p|-q” can get interpreted as “p implies q” with “implies” meant in the sense of “logical deduction”. But, if you read “p implies q” in the sense of “if p, then q”, then you’ve said (p->q). If interpreted semantically, which might seem more fitting than syntactically “|-(p->q)”, then “p implies q” means |=(p->q). By completeness one can infer |-(p->q). Now, the sense of “p implies q” in terms of “logical deduction” p|-q, comes as related to that of “material implication” |-(p->q) by the deduction (meta) theorem and its converse. In other words, if p|-q, then |-(p->q) (the deduction metatheorem), and if |-(p->q), then p|-q (the converse of the deduction metatheorem). If you mean iff p|=q, then and only then |-(p->q), then you’ll need both completeness and the deduction metatheorem and its converse.

So completeness plays a role here, sure. But, completeness doesn’t equate things here… completeness along with the deduction theorem and its converse “equates” “material implication” with “logical deduction”, at least if “material implicaton” means |=(p->q) and “logical deduction” means p|-q.

The context may be relevant here. I teach logic for the sake of logic, to people who may not be mathematically inclined. One probably has less trouble teaching logic for the sake of (and as a part of) mathematics, to mathematicians.

“If is both even and odd, then .”
I would say n had to be the zero-function

You write two different places, but it is shown as on my computer (strange, since I didn’t have the problem with AO instead of B in your “and and or”-post).

Your proof of “If is rational then there is an integer that is both even and odd” reminds me the joke/anecdote where Russell claims that he can prove anything, assuming the 1+1=1. Someone challenged him: “you can’t use 1+1=1 to prove that you are the pope”, to which he answer “I am one and the pope is one thus the pope and I are one”.

May I suggest an alternative way, which is really a way of selling the seemingly odd truth-table for “if p then q”, but which also sells the parallel apparent oddity for implication, because they really ought to keep in step. It feels satisfying to have that much connection between object language and meta-language.

People are happy to accept the first two lines of the truth table: T and T give you T, and T and F give you F. Worries are all about the two lines where p is false: F T giving T, and F F giving T.

But consider the alternatives, given that one is committed to having something truth-functional. T and F (for the value of the whole conditional) would make “If p then q” equivalent to q, which doesn’t seem right. F and T would make it equivalent to “p if and only if q”, which really ought to be different from “if p then q”. Finally, F and F would make it equivalent to “p and q”, which again ought to be something different.

At that point, someone will ask why we should be so fixated on truth-functionality. Well, it kind of helps to hold the whole system together.

The wikipedia page on the “use-mention distinction” has some nice discussion and examples of the distinction between a statement, and a reference to that statement.

The two interpretations of “implies” as “material implication with parameters” and “logical deduction” are connected by the Godel completeness theorem, which roughly speaking says that the two senses are equivalent (assuming one’s logic is consistent and at most first-order, of course).

I like to interpret “If A, then B” as “B is at least as true as A”, as I discussed in this Buzz.

A related issue came up recently for me. Several times in the past I’ve seen students use the word “suggests” in proofs, as in “P suggests that Q is true”. The most recent time I saw this, having recently discussed the “implies” connective in detail, it occurred to me that in colloquial English, “suggest” is a (not quite exact) synonym for the most common usage of “imply”, namely, to hint that something is true without directly saying so. This meaning of “imply” is much weaker than its mathematical meaning. I don’t know whether the students who write “suggest” in place of “imply” are misunderstanding what “imply” means in mathematical English, or if they’re simply overgeneralizing the synonym relationship between those words.

By contrast, nobody says ‘if X, then Y’ when Y is known to be true because it would be pointless to say it – it doesn’t convey any new information. The difficult one is ‘if X, then Y’ where X is known to be false – is it a vacuous statement, or is it an assertion about some hypothetical alternative reality, and if so, is it an assertion about causation? The interpretation of ‘counterfactual conditional’ sentences can be a tricky business. There is also a limit to how far people will go with such counterfactuals – it’s OK if the premise is merely false, but it’s not allowed to be ‘nonsensical’ (which does not mean ‘syntactically invalid’).