Part 1: ARGUMENTS
The entire purpose of the logic we're looking at is assessing arguments. So it'll be helpful to understand exactly what an argument is. Here's a working definition:
ARGUMENT: A series of propositions consisting of premises which are purported to support a conclusion.
Of course, to understand this definition, we need to know what a proposition, premise, and conclusion are.
A proposition is a special kind of statement - it's one that can be true or false. Obviously, questions like 'What time is it?' can't be true or false. And neither can commands like 'Shut the door.'
A proposition says something about the world. Here are some examples of propositions:
1) It is sunny outside.
2) Mercury is the closest planet to the sun.
3) All mammals lay eggs.
Notice that 3 is false. But that's okay, it's still a proposition. Remember, these are statements that can be true OR false. As it turns out, there are some philosophers who have some strong arguments about what is and isn't a meaningful proposition. But that discussion is for an Analytic Philosophy class. We don't really care about these things in logic. If it's something to make sense to say it's true or false, then it's a proposition.
So an argument consists of premises and a conclusion. The premises are propositions that give you a reason to accept the conclusion, which is also a proposition. The conclusion is what you're supposed to, well, conclude! Here's as example:
1) All men are mortal.
2) Socrates is a man.
3) Therefore, Socrates is mortal.
In this argument, 1 and 2 are the premises which support 3, the conclusion. You can usually tell the conclusion by keywords like 'therefore' 'so' and 'thus.'
Look back at the definition of an argument - notice it says that the premises are PURPORTED to support the conclusion. That just means that they are intended to give support - but they may fail miserably. The result would be a bad argument, but it's still an argument. Here's an example:
1) Monkeys like bananas.
2) I like bananas.
3) Therefore, I'm a monkey.
In this argument, the premises lend very little support to the conclusion. You may even have an argument where the premises have nothing at all to do with the conclusion. But these are still arguments - just really bad ones!
So now you know what an argument is. Up next, we'll go over the basics of how to assess an argument. This, remember, is the central goal of logic (at least, the logic we're talking about).
It's worth noting here that the kind of logic we'll be talking about is called PROPOSITIONAL LOGIC. This logic deals with, you guessed it, propositions. Overall, it's very weak - there are many arguments it can't assess.
More powerful logical systems like predicate logic and modal logic can handle more arguments. But you have to walk before you can run, and this kind of logic is a very good place to start. If you can understand this, you'll have a much easier time learning more powerful logical systems.
These are the basics, so if there are any questions, please post them. It's vital that you understand these definitions so that the next part will make sense.
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Your reductio makes perfect sense to me. But then again, one could use a reductio against a variety of arguments that incorporate claims about infinity. But this doesn't mean that there is an actual contradiction in the premises.
I suppose we could try to formalise the argument, but I'm thinking that this will be beyond my limited abilities as a logician. Trying to formalise the notion of an infinite universe, for example, is going to be very problematic. I'm not even sure how it would be done. We would need something more than a mere property statement (it would seem), but I'm just not sure what that 'something more' would amount to. At any rate, it's something fun to think about!
But this doesn't mean that there is an actual contradiction in the premises.
But, in this case, there is! Let's look at my first outline, going from steps three to four:
3) Then, the density (mass per volume) of living things in the universe is 0, since the volume of the universe is infinite.
4) Then, from fact 1, there are no life forms in the universe as the total mass of all life forms is zero.
Basically, what I am saying in (4) is that since the denisity of life throughout the entire universe is zero, the mass must be zero as well. In step three, I defined density as mass per volume.
So, equivalently, what I am saying in (4) is that if mass/volume = zero, then mass must equal 0. Or, mass/volume = 0 only if mass = 0.
In step three, I also state that mass/volume = 0 if the volume is infinite. Therefore, the statement: mass/volume = 0 only if mass = 0 is actually false! It is possible to have zero density but positive mass in an infinite area. This means that step four does not logically follow from step 3.
I said this proof was sneaky, and it is. Most people know from basic algebra that if x/y = 0, then x must be 0, and therefore don't catch onto the logical contradiction. But, within my proof I explicitly state that there is another way for a quotient to be zero. And then in the very next step I just ignore that fact.
For those who are interested:
Mathematically, you could see that this proof contained a critical error more easily than if you were just following the logic (though I claim it is still possible to find the error with little knowledge of math). This is because the division involving infinity actually uses a different operator than regular division. So, in my proof, I used the infinite kind of division (which involves limits and such) in step three, but pretended like I had used the regular kind of division in step four.
This is all really interesting. But in the interest of full disclosure - I'm definitely out of my league at this point. I'm happy to talk about lots of different logical systems and the theory behind logic. But I can't meaningfully contribute to this, although I am learning a lot!
Any equation dealing with infinity causes any constant to approximately equal zero.
I have to admit that I am not happy with how the word "logic" is thrown around in the politics forum like a cheap stone age weapon. Logic is a way to reason and think, it is not a scare word.
/// Mastaplaya ///
I'm curious about the Rule of Contraction and why exactly people thought it was a good idea.
It should be obvious that
A implies that (A implies B) ≠ A implies B
I've never actually heard of the rule of contraction, though I can at least try to explain what's going on here. So you have two sentences in propositional logic:
(1) P -> Q and (2) P -> (P -> Q)
You note that these aren't identical. But they are logically equivalent. In fact, the following sentence is a theorem of prop logic:
(P -> (P ->Q)) <-> (P -> Q)
Basically that sentence just says that (1) and (2) (from above) imply one another. In other words, they have the same truth value in all cases. And that's all that logical equivalence is.
There are basically 2 ways to determine whether a set of sentences are logically equivalent. The easiest way is to just construct a truth table for each one. A truth table looks at every logical possibility and determines whether the sentence is true or false. The sentence P -> Q is false only when P is true and Q is false and is true otherwise. And it turns out that (2) behaves the same way.
Now, semantically, these are saying two different things. That much does seem obvious. But it's important to keep in mind what logical equivalence is saying. Logic isn't concerned with semantic content (well, it is, but not at this level) - all that matters is the truth values of a given sentence. So with this in mind, it turns out that all of the theorems of a given system are logically equivalent. They're all saying very different things, but from a purely logical point of view, they are all equivalent.
The sentence P -> Q is false only when P is true and Q is false and is true otherwise. And it turns out that (2) behaves the same way.
But if we equate (2) to (1), we can put anything in place of Q and still have (2) turn out a "true", as in Curry's paradox.
If P = "P -> Q", we get: P -> (P -> Q) and (P -> Q) -> Q, so P implies that P implies Q and P implying Q also implies Q. If Q is false, "P -> Q" tells us that P is also false, meaning the statement "P -> Q" is false. We can still say it's true that P -> (P -> Q), because P -> P, but this doesn't relate to Q being true.
As I see it, having both P and Q false, we haven't actually said anything about the truth of the statement "P -> Q", because P doesn't need to have any relation to Q.
Ah, I see what you're saying now. I have two thoughts on what's going on.
First thought - problems with material implication
The logical connective (represented by an arrow -> or a horseshoe) that is material implication is a tricky beast. It's really unintuitive and many of my students struggle to understand what's going on. Note that I'm not suggesting you're struggling to understand, I'm just pointing out that it's really, really unintuitive.
Typically when we use conditionals in ordinary language, we are expressing a specific relationship between two things. Sometimes it's a causal relationship - 'If you fall from that height, you're going to get hurt'. Other times it's more of a biconditional - 'If you drive me home, I'll pay you £5'. But the arrow just doesn't work that way. The statement could still be true if you don't fall from that height and you still get hurt. Or if you don't drive me home and I still give you £5. But this is just a feature of how conditionals work in formal logic. There are other logical systems that try to handle the conditional differently, but these are far beyond the scope of this intro to logic thread. Plus, I'm not really familiar with these systems, so I can't give much insight here. My earlier comments about logical equivalence and implication rest upon this reading of conditionals. They're unintuitive, but they're also fundamental to building a sound and complete logical system (even though more powerful logical systems are incomplete, but that's a different matter).
Second thought - Solving the problem
If we take a step back to look at what's going on, we can solve the problem with predicate logic. You mention Curry's paradox, which hinges on what's going on here. A central claim in the paradox is that, whatever P we choose, we have a Q such that P -> (P->Q).
With predicate logic, we can represent this claim using quantifiers. I can't write them here because the quantifiers are an upside-down A and a backwards E. The basic idea, though, is once we properly quantify Curry's paradox, we can avoid it. In short, the claim that 'For every P there is a Q such that P -> (P->Q)' just isn't saying the same thing anymore. So the equivalence relation that was in place in propositional logic no longer holds. Interestingly, this is also true if we reverse the quantifiers -- 'There is a P such that for every Q...'
Moegreche is teaching logic very nicely for free! This is a fun read. I took a philosophy 201 class since it was necessary for computer scientists and it helps with discrete mathematics. Good on you Moe!
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