Basic logic — connectives — IMPLIES | Gowers's Weblog

]]>Basic logic — connectives — IMPLIES | Gowers's Weblog

]]>“$P\implies Q$ unless $P$ and $\neg Q$.”

to emphasize the use of quotation marks; however, should this not be

“$P\implies Q$” unless “$P$ and $\neg Q$.”?

with a quotation mark after Q in the first implication?

]]>When you say to bad, I am going to is use the double-line arrow anyways, why not just use a single line arrow for if..then?

]]>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.

]]>I must have been talking about Terry.

]]>The interpretations of “implies” happen in the context of propositional, or zero-order, logic. So, I would scratch “Godel completeness (meta) theorem” and write “completeness (meta) theorem” (for propositional logic), at least since that came as known before Godel’s result. Second, unless I’ve misunderstood something, the completeness (meta) theorem says if “p|=q”, then “p|-q”. For these two to come as equivalent, if that’s what you meant (and I don’t mean to assert that you did mean this), you also need soundness… if “p|-q”, then “p|=q”.

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.

]]>Oops, it’s supposed to be “I think Gowers’s post…”

]]>You’ve talked about the appendix basic mathematical “logic” in your Analysis I to me before, especially about this “implies” issue:-)(http://terrytao.wordpress.com/books/analysis-i/#comment-49187). I Gowers’s post is a very nice complementary material for that appendix.

]]>Hello Doug, I agree one can make things more complicated, and then a case of the type that I put forward falls apart. But I was not offering an argument that should compel those who already know about these things. My objective was the more modest one of getting beginners to stop worrying and get on with practising the connectives, until what might have seemed unnatural comes to be natural.

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.

]]>I like your approach here. But, one can still have a commitment to truth-functionality here and not have “T” for the (F, F), (F, T) lines of the truth table, if say one wants to give up {T, F} as the truth set and have a 3-valued, many-valued (with “many” meaning at least 3), or infinite-valued truth set instead. In other words, you’ll need an a priori commitment to *two-valued truth functionality* for your approach to work. I think the same applies to Gowers’s presentation here. That said, if you postulate that ->(F, F)=U, and ->(F, T)=U where U represents a third truth value not equal to T or F and “->” indicates material implication, then statement forms like, in Polish notation, CpCCpqq no longer hold as theorems or tautologies.

]]>The phrase, “I don’t mean to be rude, but” is not without philosophical interest …

]]>I interpreted your Thatcher-tsunami example as: Let

“Margaret Thatcher was Prime Minister of the UK at some time ” and “there was a tsunami in Japan shortly before time t”. With this interpretation, is not true, since there was tsunamis before Thatcher became Prime Minister. On second reading I see that you wrote that it was "two fixed statements", so it was probably my own fault. ]]>

“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”.

]]>Actually, upon reflection I would probably withdraw my second comment: the completeness theorem equates “logical deduction” with “material implication in all possible worlds”, rather than “material implication with parameters”, which is a slightly different concept. (For instance, one normally doesn’t consider the prime ministership of Thatcher to be a variable parameter.)

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