h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Thanks - a wonderful article and witty too, for a mathematician (I should know..) I always thought the 'implies' rule was nonsense, and now I know why: propositional logic is not enough to describe reality. Thanks for the enlightenment!

P.S. In case anyone is wondering about the above, what I mean is that in my view, the only circumstance under which you can determine the truth or falsehood of 'A implies B' is when A is true and B is false (and then it is false). I think that *all* other conditions make its value *indeterminate*. After all:

1. A false, B false: You can't tell whether A implies B because A hasn't become true yet.
2. A false, B true: Same argument as 1.
3. A true, B false: Aha! A does *not* imply B. Note however that A cannot be said to imply ~B (see 4.)
4. A true, B true: (This is where some people disagree with me) You *still* can't tell. Yes, B is true, but we don't know that its truth is in any way caused by A's truth.

Does that make sense? I suppose I'm really talking about cause and effect, which is what you meant about USSR/communism vs. Clean Room/Cinema scenarios, i.e. the former is cause/effect, the latter is more indirect.

I think I'm confused actually...

I've seen this kind of confusion around, and you're not alone. Here is my favorite example: Let P stand for "The tv is on." and Q stand for "The tv is plugged in." I make the claim that P->Q is a true proposition. Why? Because if P is true and Q is false (i.e. the tv is on and the tv is NOT plugged in), we have an impossibility. The truth of P guarantees the truth of Q. Every other case is possible. the tv can be off and not plugged in. It can be on and plugged in. It can be off and plugged in. To show that P->Q is true, you don't just look at the tv and say "hm, it's plugged in and it's on...therefore P->Q." What if the sucker is battery powered? What you need to do is show that P being true while Q is true is an impossibility, and that everything else is ok.

The other thing to keep in mind is that conditional implication is NOT the same thing as cause and effect. To make the claim that P causes Q is impossible to show using simple logic. Causality is an extremely strange concept, and it amazes me that people can use it so casually (not be confused with "causally"). If I forget to close the window and my cat gets out and gets hit by a car, did my not closing the window cause the death? Did the driver not stopping cause the death? Did the cat's mom giving birth cause the death? Did the butterfly who flapped his wings in Michigan to make rain and slick streets here cause the death? Did I cause the hypothetical death of my cat by coming up with this example? Cause and effect is far too complicated to deal with easily. Conditional implication merely says that if you've got the first one, the second one WILL be true. It doesn't say that the first one CAUSES the second one.

Does this help clear things up?

mmm. wonderful article. thanks for finally writing it; can't wait for the one on quantifier logic.

i feel like i should point out for the sake of example, yet another way to think about this is that "P -> Q" is the same as saying "~P v Q" [read "not P, or Q"--keep in mind this is an entirely different thing from "not P or not Q", that is, the 'not' doesn't carry across the 'or'. anyway "not P, or Q" means either P is false OR Q is true...confused yet?].

this sameness, for the uninitiated, owes to the fact that the two statements "P->Q" and "~P v Q" have identical truth tables. when two seemingly different statements have the same truth table we agree that they're saying the same thing.

so what the @#%@ is a truth table?
we list all the possible true/false combinations for P and Q and decide what that says about the trueness or falseness of the statement as a whole.

P Q P->Q ~P v Q
true true true true
true false false false
false true true true
false false true true

since the two columns on the right are identical, the two statements are equivalent--that is, for all intents and purposes, they're saying the same thing.

a handy trick when dealing with propositions is given to us as a consequence of demorgan's laws, and conveniently enough you can 'distribute' a negation (~) through parentheses by 'inverting' all the statements and operations: P becomes ~P, ~P becomes P, ^ becomes v; ~(~P) becomes ~P, as the ~'s cancel out like negative signs would. implications (-&gt are best handled by converting them to an and/or format as explained above.

so ~((P ^ Q) v ~R) is the same as ((~P v ~Q) ^ R). if anybody doesn't believe you, show them truth table:

P Q R ~((P ^ Q) v ~R) ((~P v ~Q) ^ R)
T T T F F
T T F F F
T F T T T
T F F F F
F T T T T
F T F F F
F F T T T
F F F F F

however i recommend not using the words 'distribute' and 'invert' in this context within hearing distance of a mathematician; they're liable to give you a stern lecture. it's simply helpful to think about things in this way, since you're used to using the ideas with numbers.

a small but tangential point is that there are two different kinds of 'or' -- the inclusive 'or' and the exclusive 'or'.
the inclusive 'P or Q' means " P is true, or Q is true, or both P and Q are true". the exclusive indicates "P is true, or Q is true, but not both."
the romans had two different words for 'or' depending on which context they were talking about. i was once told that the latin for the inclusive or (the one we use) started with a 'V', which is why we use the v-shaped upside-down carat for the 'or' symbol. but he might have been lying, any latin students could tell you better than ego.

it's also worthwhile to point out that the biconditional, usually represented <=>, is simply an 'if-then' that goes both ways.
for example, P <=> Q says P -> Q and Q -> P. it is important to realize that biconditional statements are reversible, whereas one-way conditionals are not. it's sort of like how 2 squared is always 4, but the square root of 4 is not necessarily 2.

in discussing propositions, keep in mind that a conditional statement can have 3 pesky relatives: its converse, its inverse, and its contrapositive.
if P->Q, the contrapositive ~Q->~P will always hold; their truth tables are identical. think of a contrapositive as a twin brother.
if i'm asleep, i'm happy; therefore, if i'm not happy, i'm definitely not asleep.

the converse Q->P is tricky, as it tells you nothing about the original statement. if i'm happy, that doesn't necessarily mean i'm sleeping.
the inverse is the contrapositive of the converse, ~P->~Q. but it doesn't come up very much in practice.

this kind of stuff is the basis of proofs in mathematics. it turns out to be a very nice system, even though many an aspiring computer scientist has changed his major after taking a course in logic. following a stream of hieroglyphic letters and arrows across a page is a little confusing and intimidating at first. buy with practice it becomes an efficient and convenient shorthand for explaining things.

if you're really interested in blowing your mind, try to conceive of what would happen to the whole of mathematics if P and ~P meant the same thing, true were false [and we all got killed on the next zebra crossing?]. i have a strong unfounded personal conviction that we could unravel chaos theory if we dropped our primary assumption that things ought to make sense in the first place, but then i have a very vivid imagination. for instance it would seem that an infinite, random list of digits contains all possible patterns, and hence no patterns. ouch.

if anyone would like to discuss russell or cantor i would be enthralled to read about their work.

oops. apparently the guide doesn't like extraneous spaces. so the truth tables are probably impossible to read...sorry.

Yes, bravo, great entry!!!!! Three or four weeks of lectures condensed into a few paragraphs. (still, you didn't repeat yourself as much as my lecturer did, so that cuts the time down ;P ) But if I could make one tiny suggestion.... truth tables (and thanks mockturtle for the good attempt - bummer about the spaces) - it just makes it so much easier to understand cos it all makes sense!!! That and maybe even truth trees - although these take a little getting used to.

Thanks and keep up the good work - maybe I can afford to skip more lectures with this service ;P

OK. Conditional implication is not the same thing as cause and effect - fine. However, my problem is this; what the hell is the use of conditional implication if it doesn't actually mean 'implies'?

In your TV example, I understand what P and Q stand for, but I still think that, in making the leap to 'P->Q' being a true proposition, you have made reference to certain facts in reality, i.e. that you know that it is impossible for the TV to be on when it is not plugged in. Surely this kind of philosophical level-changing is exactly what is prohibited in propositional logic?

Can you explain in exactly what way it is useful to have conditional implication in propositional logic if it does not actually mean what it appears to mean?

Sorry, still confused!!

Implication is important as it's the fundamental basis of the argument, which is sort of the whole point of logic. An argument is built up of a series of propositions called premises and one proposition called the conclusion. In order for an argument to be valid, the truth of the propositions must impy the truth of the conclusion, not by virtue of outside knowledge, but by virtue of the rules of logic themselves. Here's a simple, if inane example using two basic propositions which I'm not even going to define in real-world terms:
There is one premise: P&Q (remember that this means that both P & Q must be true for this to be true)
The conclusion is: P

This makes sense, if both P and Q are true, then certainly (whatever P and Q are, P must be true)

You express this relationship via the conditional implication:
(P&Q)->P

You'd have to be thinking on another plane to reject the above statement, certainly the truth of both P&Q guarantees the truth of just P.

Now there are a whole lot of possibilities for the truth of P and or Q. We don't know which is true, and frankly, we don't care. But in every possible case, things still make sense. If both P and Q are true, then we have a true statement implying a true statement, which is cool. If just P is true, then we get false implies true, which doesn't have a problem. If just Q is true, we get false implies false, another truth value that's okay. And if both are false, we get false implies false again, which is just dandy. The only thing that can give us trouble in an implication is if we were to get true implies false, and that's simply not possible due to the way we set things up.

Now an argument can be valid or invalid regardless of the actual truth values of the propositions in question. It merely depends on the relationships between the possible truth values. For instance (PvQ)->P is an invalid argument. Sure, we may know that P (which stands for "The sky is blue.") is true and Q (which stands for "The sky is orange.") is false, but the actuality of the concepts represented by P and Q has no bearing on the logical, conditional implication. We have to deal with all the possibilities, and one of those possibilities (P being false and Q being true) leads to the truth configuration true implies false, which is our big nono when it comes to conditional implication.

Logic is meant to aid rational thinking, not be a slave to the finicky concepts of whether things actually are true or not. If you're trying to convince someone that something is true, it really doesn't help you just to know yourself that what you're trying to prove is indeed true. What you really need is a good chain of reasoning that leads the other person to believe what you believe (given some premises that you both already believe). Logic doesn't provide you with the truth, but the valid chain of reasoning.

I hope this helps.

By the way, I urge you not to underestimate the complexity of the cause-effect concept, it really doesn't hold together when you try to nail everything down. It's like Jello in that fashion.

OK, I think I get it now.

To be honest, I think my whole confusion derived from the actions of extremely irresponsible textbook authors, writing stuff like "If the sky is orange then I am the Pope", or some such nonsense.

Thanks for your patience, Hippiechick!

Wow, excellent entry and associated conversations! I'm glad someone has made a distinction between implies and causes.

The Latin word you are looking for is vel. Aut is the exclusive or. A subtle distinction, and NOT one that most Latin students are taught--you need one of the big Latin dictionaries, like Lewis & Short or Oxford Latin Dictionary.

However, you did err in one respect: "than ego" is not correct. Ego is declined ego, mei, mihi, me, me. The concept which is expressed in English by that sense of than is expressed in Latin by the ablative of comparison: therefore, me.

The phenomenon you exhibited there is known as slippage linguistically--slipping into a second language, which is usually an unconscious phenomenon. Linguists postulate that "slippage" must always follow the grammatical rules of one language, because grammatical structure is a higher-level morpheme, which requires conscious thought--so what you did was unsurprising. Morphemes and the 4M model will have to be another guide entry . . . but I don't want to publish anything on that until my friend who is working on it finishes her book/journal articles.

Your friendly local amateur linguist,
Kathy

Wow, excellent entry and associated conversations! I'm glad someone has made a distinction between implies and causes.

The Latin word you are looking for is vel. Aut is the exclusive or. A subtle distinction, and NOT one that most Latin students are taught--you need one of the big Latin dictionaries, like Lewis & Short or Oxford Latin Dictionary.

However, you did err in one respect: "than ego" is not correct. Ego is declined ego, mei, mihi, me, me. The concept which is expressed in English by that sense of than is expressed in Latin by the ablative of comparison: therefore, me.

The phenomenon you exhibited there is known as slippage linguistically--slipping into a second language, which is usually an unconscious phenomenon. Linguists postulate that "slippage" must always follow the grammatical rules of one language, because grammatical structure is a higher-level morpheme, which requires conscious thought--so what you did was unsurprising. Morphemes and the 4M model will have to be another guide entry . . . but I don't want to publish anything on that until my friend who is working on it finishes her book/journal articles.

Your friendly local amateur linguist,
Kathy