The Standing Invitation

Posts Tagged ‘Logic

No Nonsense

with 3 comments

Recently a copy of A J Ayer’s Language, Truth and Logic passed through S I Towers, and it caused quite a stir. It’s a short book and very readable – and, I was amazed to learn, was written when the author was younger than I am. It is a beautifully argued manifesto of logical positivism.

Philosophy, for most people, is the asking of Big Questions. Is there a god? What happens after we die? Does the world disappear when we close our eyes? What is ‘truth’? What is ‘good’? And these questions are called Big Questions precisely because thousands of years of arguing have got us no closer to answering them.

Logical positivism was an attempt to tackle these issues from a different angle. Rather than attempting to answer these questions, the project of the positivists was to decide whether or not the questions could be answered. Here, briefly, is how they set about it.

Forget about what you can see. Think instead about what you can say.

The human vocal apparatus make it possible for you to generate all sorts of noises. Most noises are just that – noises – but some are words. Most combinations of words are nonsense: “Mill food only here bushes pardon speak and.” However, some combinations are full sentences, like “I am wearing shoes” or “The sky is green”.

The important point is that almost everything you could possibly say is actually nonsense. The things that actually mean anything – sentences – are a tiny minority. What is it about these particular utterances that makes them important? Well, sentences have a structure. They obey rules. They are not self-contradictory, like the sentence “X is and is not Y”, which is meaningless and indistinguishable from noise.

In fact, there are only two kinds of sentences that are worth talking about: sentences describing the world, and sentences describing other sentences. Any other kind of sentence is uninteresting, because hearing them does not increase one’s knowledge of the world. It’s just noise.

Now, how do we know which sentences describe the world? That’s easy: these are the sentences that can be checked against what we observe around us. “The sky is green” is an attempt to describe the world, and it is well-phrased, logical, and verifiable. It just happens to be false, because it does not match observations that show the sky is blue. The sentence “I am wearing shoes” is true (at the moment).

If you know all the meaningful, true sentences about the world, and all the meaningful, true sentences about other sentences, you will know everything that it is possible to know about the universe. Obviously, in our lifetimes we will never have this perfect knowledge. There are some things that we will never know. However, adopting this stance gives us a tool for cutting away the layers of nonsense that surround us and prevent us from understanding the world.

Does god exist? If you mean, does he exist in the world, does he have an actual location and mass and velocity we could check, then the answer is – maybe. We don’t know, but we could in principle find out. But if you mean, does he exist somehow outside the world, in a place we can never experience, then there is no question here to answer, because in that case sentences containing the word “god” are meaningless. It is impossible for an atheist to disprove the existence of god, but at the same time, anybody religious who talks about god is just making noises. What happens after we die? Again, things that happen outside the “real world” are not subject to verification, anyone who talks about it is taking nonsense. Likewise the question about the world disappearing when we close our eyes: it’s not a question that can be meaningfully answered. What is truth? Good correspondence between a sentence and observation. What is good? Whatever people say is good; people argue about it, but they argue by appeal to emotion, not to logic, unless it is to show that one’s values are inconsistent.

A lot of this is not new. Hume, much earlier, said that a book that didn’t talk about things observed or calculated should be cast onto the flames because there was nothing in it worth reading. But what the logical positivists added was the system of formal logic developed by Russell and Wittgenstein. For lovers of clarity and precision of writing, the appeal is still strong.



As always, I am not a philosopher, and could easily be getting aspects of this wrong. If so, I would be delighted to be set right by someone who knows more about it than me.

A J Ayer’s book was Language, Truth and Logic. The reference to Hume comes from his Enquiry.

Written by The S I

April 1, 2012 at 2:34 pm

Doing Maths Without Numbers

with 3 comments

Define ‘one’. Go on, try.

Well, ‘one’ is what you have when there’s only one of something. You have one duck. There – one. But that’s circular. You still need to define ‘one’ for you to have one duck.

All right, it’s one more than zero. ‘One’ more? Circular again, and what’s this ‘zero’ you’re talking about now?

Half of two? Okay, fine, but what’s two? Well, that’s twice one, isn’t it? Circular again. People trust mathematics with their lives, and yet all it seems to rest on is circular arguments. Can’t we do better than this?

Let’s start at the very beginning, with something really fundamental: “things are what they are.” A duck is a duck. A submarine is a submarine.  “It is what it is” is a property shared by all things. Let’s call this the property of identity, of being identical with itself – the property of being what it is.

Everything is identical to itself. If you grouped together everything that is identical to itself, the set would have everything in it. And the set of everything not identical to itself would be empty.

We have defined an empty set – a group with no members. The set of four-sided triangles is empty, because nothing is both a triangle and four-sided. So is the set of things that are not identical to themselves, because nothing is.

Now we’re used to the idea of sets, consider this: what does the set of three ducks have in common with the set of three triangles? Call it the cardinality of the set ­– a measurement of how much is in it.

Imagine the set of all empty sets. What do all these empty sets have in common? Let’s call it zero ­– the cardinality of an empty set. We have now defined zero. If something is the same as zero, then it has the same cardinality as the set of all things not identical to themselves.

Now that we know what zero is, how many things are identical to zero? There’s just one thing that’s identical to zero: zero. If you filled a set with things identical to zero, there would only be one thing in it. The cardinality of the set of all things that are identical to zero, is one.

And the cardinality of the set of all things that are identical to ‘one’ or ‘zero’ is two. And the cardinality of the set of all things that are identical to ‘zero’ or ‘one’ or ‘two’ is three, and so on and so on and so on.

To break the circularity of mathematics, Bertrand Russell defined ‘I have two hands’ as:

“There is such an a such that there is a b such that a and b are not identical and whatever x may be, ‘x is a hand of mine’ is true when, and only when, x is a or x is b.”

And this is how you do maths without numbers.



The BR quote was from A History of Western Philosophy (1945); also see My Philosophical Development (1959).

The layout of the above argument was from a YouTube video on mathematical logic, but I can’t remember where it is; I’ll update this if I find it.

Written by The S I

August 9, 2011 at 10:00 pm

Posted in Science

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