Logic and truth values

[robvalue]

I've been very interested in logic recently, but it's not a subject I have studied much. I did a bit of basic Boolean algebra, and I've read that there are a lot of different logic systems that can be used.

I started from the idea that a particular statement can have a truth value of 0 (false) or 1 (true).

(a) Is a sentence by default true or false? In computer programming (what I've done anyway), it is the case that any statement is true until it includes a false element. The point I'm getting at is what would be the truth value of an empty statement? One that says nothing at all. I would assume it would be true.

(b) What about a statement that is nonsense? One that makes no sense, either at all, or in the syntax of accepted statements. Would it be treated as actually saying nothing, so would revert to (a)? Or is it just not allowed?

I find also the idea of paradoxical statements interesting, generally ones that reference themselves in some way. For example, "this statement is false." I have read that there are several ways of approaching this problem, I think my favourite is to add in the implication that if a statement is being made, it is asserting that it is true in what it is saying. So it becomes, "This statement is true. This statement is false." And then it can be seen to be false.

The problem with the liar paradox is that the opposite of false is true. But I wonder if you could define truth value to be 1 if it is consistent, and 0 otherwise. So the liar paradox cannot have truth value 1, since this does not check out. So it has value 0. I may have not thought that through properly though.

If anyone has any links to websites which address these kinds of thing particularly well (I already checked out wikipedia hehe) then feel free! Thanks

[robvalue]

By the way, I love maths I get the same kind of looks when I say that as you get when you say you're an atheist. I was a maths teacher/tutor for many years, and it was only the demonic way that children behave these days that put an end to my career. I still did some tutoring afterwards though, I am best when teaching one on one.

[naimless]

"This statement is true. This statement is false." And then it can be seen to be false.

False.

[Diablo]

By the way, I love maths I get the same kind of looks when I say that as you get when you say you're an atheist. I was a maths teacher/tutor for many years, and it was only the demonic way that children behave these days that put an end to my career. I still did some tutoring afterwards though, I am best when teaching one on one.

I'm a maths graduate too and I know the feeling. I was so bad about it, I used to do all the exercises instead of just the ones the teacher said.

[Alex K]

Isn't the situation such that we have a language and grammatical rules, and true statements are sentences which can be simplified to "true" by applying these rules a finite number of times?

A sentence containing grammatical errors would then simply not be a sentence in this language, and thus there is no truth value (because there is no statement).

It's been a long time though, and there are different approaches.
The issue of paradoxes has played a big role in the closely related field of set theory. There, great efforts have been made to exclude things like the barber paradox by definition, fpr example.by forbidding that sets can contain themselves. Might be related.

[Mudhammam]

Have you ever read up on Pragmatism, rob? They, especially the pluralists (I specifically have in mind the American philosopher/psychologist William James), have an interesting take on the value of truth and logic (being dye-in-the-wool verificationists, my understanding is that they believe any statement with meaningful content that is both satisfactory to the individual and makes a practical difference qualifies as "true.")

[Alex K]

Have you ever read up on Pragmatism, rob? They, especially the pluralists (I specifically have in mind the American philosopher/psychologist William James), have an interesting take on the value of truth and logic (being dye-in-the-wool verificationists, my understanding is that they believe any statement with meaningful content that is both satisfactory to the individual and makes a practical difference qualifies as "true.")

What does it mean to make a practical difference ifimayask?

[Mudhammam]

Have you ever read up on Pragmatism, rob? They, especially the pluralists (I specifically have in mind the American philosopher/psychologist William James), have an interesting take on the value of truth and logic (being dye-in-the-wool verificationists, my understanding is that they believe any statement with meaningful content that is both satisfactory to the individual and makes a practical difference qualifies as "true.")

What does it mean to make a practical difference ifimayask?

If I may quote from James:

"...if you follow the pragmatic method, you cannot look on any such word as closing your quest. You must bring out of each word its practical cash-value, set it at work within the stream of your experience. It appears less as a solution, then, than as a program for more work, and more particularly as an indication of the ways in which existing realities may be changed. Theories thus become instruments, not answers to enigmas, in which we can rest... The attitude of looking away from first things, principles, 'categories,' supposed necessities; and of looking towards last things, fruits, consequences, facts... Any idea upon which we can ride, so to speak; any idea that will carry us prosperously from any one part of our experience to any other part, linking things satisfactorily, working securely, simplifying, saving labor; is true for just so much, true in so far forth, true instrumentally." (italics in original)

Hope that clarifies some...off to work now!

[StealthySkeptic]

A conundrum like what you're proposing is not actually: "This statement is true. This statement is false."

The real brain-bender is: "The following statement is true. The previous statement is false."

[Alex K]

A conundrum like what you're proposing is not actually: "This statement is true. This statement is false."

The real brain-bender is: "The following statement is true. The previous statement is false."

Consider the set of all sets which do not contain themselves. Does it contain itself?