John Page

Kent:

My challenge is to explain the Liar Paradox (i.e. how it comes about), not to resolve it as either true, false or some other value.

John Page

d:

If I may recap.

1. All systems of logic (currently known to me) require the law of non-contradiction.
2. The law of non-contradiction as applied to logic is self-contradicting. (From my earlier posts, with reference to a conclusion A=A, "The law on non-contradiction contradicts itself. If A cannot be anything other than A, where does this other A-thingy come from that its supposed to be equal to? What is it?" and "I'm not trying to prove p&-p, if anything I'm trying to prove p = p'").
3. The law of non-contradiction is paradoxical and therefore internally inconsistent. Here, I'm trying to persuade you that A=A is to logic as the Liar Paradox (This sentence is false) is to natural language.
4. Therefore, all systems of logic (currently known to me) are internally inconsistent.

Sorry to keep you waiting.....

P.S. I'd be interested to know what you believe the "necessary assumptions" are.

Kent Stevens

I have taken some of the ideas I have heard and decided to come up with a new system of logic. If logic is only dependent on definitions I can make up any system. If there are an infinite number of logics why not come up with my own system of logic. I call my system the Black is White logic system.

These are the given laws.
1. A = not A. Law of Non-Identity.
2. A and not A = true. Law of Contradiction.
3. A or not A = true. Law of Excluded Middle.

Therefore these derivations follow.

Black = White
Day = Night
Big = Small
Hot = Cold
I exist = I do not exist
Reasoning is good = Reasoning is bad
Induction is good = Induction is bad
Logic is true = Logic is false
God exists = God does not exist
Belief = Non-Belief
Right = Wrong
Moral = Immoral
Murder is wrong = Murder is right
Theft is wrong = Theft is right
This logic system is true = This logic system is false
They are wrong = They are right

We can make up any logical or mathematical system we like but will it be accepted by other people. Would it have any relationship to normal reality. Even if you can prove A = not A would anyone want to use this system. Logic may not be perfect but neither is science, or human reasoning.

If someone can get away from some of the basic ideas of logic they would be called mad. If they cannot tell right from wrong they would be called immoral. If they cannot tell good actions from bad actions they would be called foolish. If they cannot tell good reasoning from bad reasoning they would be called irrational.

There are some aspects are aspects of logic that are optional. But to give up on the law of identity or the law of non-contradiction is normally to give up on reason.

Though you are trying to resolve various paradoxs in logic does not mean that you will give up on logic all together and I understand that.

diana

Thanks, John.

I'm not sure what p&-p means (which you aren't trying to prove), as opposed to p = p' (which, if anything, you are trying to prove). When I speak of my lack of expertise in "formal logic," I mean I lack much experience in breaking it down to symbols and arranging it like a mathematical equasion, so if I could trouble you to explain the meaning of the symbols (the "=" part I have ).

Oh wait. It's coming back. Is this it?

p&=p : would be saying not p equals p (which you aren't trying to prove)
p=p' : would be saying p equals p prime (IOWs, "something similar to p"), which you are trying to persuade us of)

d

John Page

Kent:

I'll concoct a more comprehensive response but four quick points:

1. I'm trying to use reason to explore the fallacies of logic (not logic to explore reason). Other respondents seem to be using logic as a self-fullfilling prophecy.

2. Your use of the (a priori) "given laws" reinforces my assertion that logic is prone to the same self-fulfilling prophecies as religions.

3. In your system, The Law of Non-Identity is contradicted by a conclusion A = A (true) and by your second law. I have put forward a cognitive approach to explore this in my "The Artifice of Language" posting above.

4. I repeat, I am not trying to resolve the paradoxes. My aim is to explain how they come to exist as "logical illusions" analagous to optical illusions like the stairs that go ever upwards.

John Page

Diana:

1. The poster quoted "p&-p" as true, being a theorem of logic.
2. Actually, I think its the = symbol which causes some of the problem. Does it mean "is", "the same as", "equivalent"?

Yes, I'm saying that p = p (true) cannot be attained in reality, the closest you can get is p = p'

Cheers.

mac_philo

1. It is still opaque to me what special place you wish to claim for logic. Priests, rabbis, clerics, and whatever do not use formal systems. A formal system is a formal language; a logical calculus. Such a system has explicit rules for determining what are the expressions of the system , which expressions are well formed (acceptable), and which sequences of expressions count as proofs. By these standards even Thomas Aquinas fails.
2. I don't see what lifting this commitment to belief holds, but you still have ducked my response: there are an infinite number of logics (here are a few: sentential,predicate,K,D,T,S4,S5,Triv, Ver). Thus there are an infinite number of axioms. If they are beliefs, then someone or some group holds this infinite set of axioms as beliefs? It is still puzzling to me where this is going, or what I'll lose if I grant you this belief commitment. Constructing a proof can to some extent be a mechanized procedure; thus if my computer has a program which can prove, say, "p>(q>p", you are committed to saying that it "believes" the axioms used in that proof?
3. I don't know what else you expect me to do with the Liar paradox. These are our options: live with it or modify our practices to avoid it. It's a paradox; we can't do anything else. Tarski offers the best answer.

As to the person who said "he isn't arguing for p&-p," it matters little, since he is claiming that logic is inconsistent with its axioms. From inconsistency one can derive any formula, thus if the argument presented here holds there must be a proof that goes from the axioms of sentential logic to the theorem "p&-p".

John Page

mac_philo:

1. I'm not claiming the special place for logic, you are. Logic is one of a number of systems through which to contemplate the reality we inhabit. "Formal" systems may be more rigorous than "informal" ones, but that doesn't mean to say they're error free. To start this thread, sr asked whether axioms are proven or self-evident and I say "Neither".

2. If there are an infinite number of logics, as you claim, the probability is some of them will conflict, in which case one of them will have reached invalid conclusions. True? If so, logic is not a panacea.

3. You ask "or, what I will lose if I grant you this belief commitment." If you grant that axioms are necessarily a priori beliefs you will have opened your mind to freer thought and access to new hypotheses about reality. What you may lose (according to Kent) is your sanity.

4. We could get into your example of 'whether computers can believe' in another thread. (Except to observe that both computers and humans can utter gibberish, so I'm not sure this topic would progress the current debate).

5. What do I expect you to do about the Liar Paradox? If you continue to subscribe to systems of logic with contradictory axioms, nothing. I'm suggesting that the apparent paradox may stem from inadequate understanding of the axioms of logic.

6. You said "From inconsistency one can derive any formula, thus if the argument presented here holds there must be a proof that goes from the axioms of sentential logic to the theorem "p&-p"." Wow! I think you're trying to say that if the axioms of logic were inconsistent p&-p is indeed true. However, I think what you're agreeing is that the axioms of sentential logic may be inconsistent.

7. You ae relying on a formal systems approach. Please consider the definition of a formal system as rigorous, internally consistent and not self-contradictory. Formal systems are driven by their axioms. Therefore, if a formal system is internally inconsistent or self-contradictory then either a) the formal system has been applied inappropriately or incorrectly or b) the axioms are mistaken.

8. Are you happy with my definition of an a posteriori axiom?

Cheers

John Page

Kent:

Please forgive my edit of the recap above. The Black and White logic system does not represent what I'm trying to get across. I'm not trying to prove A = not A, I'm trying to say that the law of non-contradiction is self contradictory i.e. the basic truth operation of saying the A is A assumes there must be two A's to compare. So, when you respond that I'm asserting A = not A you are agreeing with my premise that a logical operation compares two dissimilar entities.

Regarding a new logical system that requires no axiomatic assumptions, the following is the first of (currently) eight rules for the system called "Ontologic", being the logical relations between things that are.

_________________________________________

A Priori Assumptions. It is assumed that the existences of the author and the reader are sufficiently common as to render ontologic intelligible.
_________________________________________

Axiom #1, This ontology exists. An ontology is a representational system and, for a representational system to exist, there must be an entity being represented, ‘R’ and an entity that is its representational form ‘r’:

A [ AR + Ar

Where:
A represents the concept “All Existence.” In relation to set theory, this is equivalent to the Universal Set of Universal Sets. While A could be replaced by a symbol representing “the domain of existence under consideration” it is, in fact, All Existence that is contemplated by ontology as “The study of what there is”.
[ represents “comprises at least”
AR represents “all existences that are represented”
+ represents “as well as”
Ar represents “all existences that are representational”

In words, “All existence comprises at least represented existences as well as existences that are representational.”

If an idealist reader cares to state there are no represented existences this can be refuted by observing that the word ‘Dog’ is not an actual or conceptual dog but a symbol representing a dog, and similarly the represented ‘dog’ need not be a real dog but a representation of some other dog or Dog. If a materialist reader cares to state there are only represented existences, this can be refuted by observing that the word ‘Dog’ is not a dog but a symbolic representation of a dog. Ontologic’s Axiom #1 does not predicate material and non-material existences, only represented and representational, showing how Idealist (Ar ) and Materialist (AR) views can be represented within existence. Finally, if a Nihilist reader denies the existence of Axiom #1, I would ask how it came to be that its existence could be denied.

________________________________________________

Any comments? Refutations?

PJPSYCO

How about this:

The law of non contradiction is not really a law it is derived. I refered to it as the basis of usiversal definition. Basis means that it is something that we get from the fact (of universal definition in this case). Universal defition is derived from the facts of consistancy and relevence.

1) If the universe is consitant we can know things for sure. Communication is relevant(we can talk about things that are and they will be).

2) If the universe is not consistant we can not know things for sure. Communicaiton is irrelevant(there is nothing to communicate, since it is all most likely false).

Hypothetical 1 is experimentally true, and also the only hypothetical that would make any sence to talk about since as hypothetical 2 says talking(the spreading of knowledge) is futile since your most likely wrong.

An extention of 1 is that things can be given consistant defintions A=A. Noting that this means my A and your A are the same, your A will be the same as your other A (saying you don't change A at some time), or your A and your A at the same time both are the same.

The constant definition is called the objects universal definition.

Failure of this rule is if you say at a certain time A has defintions B and C, if these definitions are mutually excusive(not the same, and can't be put together). Then we can infer that they are not both defintions of A. In this case the A that is being presented cannot exist due to its lack of consistancy. It may be that you are talking about idea M and N and are just confusing them together into idea A, or perhaps you are making it up, but either way A can not exist as long s it has concepts B and C together.

ie the Basis of Universal Definition, the concept that forces objects(relevant ideas) to be consitant within them selves.
As I state it: an object can have one and only one defintion and that defintion cannot be subject to change.

This then leads to where proofs come in. Concept A is brought up. Using other defined concepts you proove that concept A is consitant. Everything is assumed to be true unless inconsistant with something else.

Example: ( claims are independant givens are true for both)
Given: we have a universal definition on blue,and orange(wavelength(w), intensity(i), etc.), and sky(the object parrellel to the ground made of gasses - it has a more complicated defintion, but it should be well understood by all for the sake of the example)
Claim 1): the sky is blue
Proof: spectrographic analysis of the sky yeilds the fact that it has the all characteristics of blue in it. Therein verifying proof
Claim 2): the sky is orange
Proof: spectrographic analysis of the sky yeilds the fact that it does not have the characterisits of orange in it therein making the second claim invalid due to it's inconsitancy with the truth.

[ February 26, 2002: Message edited by: PJPSYCO ]</p>