Freethought & Rationalism Archive

John Page · 05-13-2003, 08:13 AM

Quote:

Originally posted by cobrashock
Yes I know that the laws of physics is a subjective interpitation and representation of certain events. (not objective)

I'm with you Ron - now if I could only explain to myself properly how mental reality arises from a physical process we could be more objective about the way we think....

Cheers, john

Sun Dog · 05-13-2003, 07:39 PM

Quote:

Originally posted by cobrashock
It seems we have lost our original poster.

Brent1 has stated that he will be replying to this thread as soon as he finishes a term paper and some other unspecified tasks. I hope he has the opportunity to come back; I made some criticisms of his original post, and I'd very much like to hear his response to them.

Sun Dog

Sun Dog · 05-22-2003, 07:35 PM

Brent1 still hasn't gotten back to us yet, so I thought I'd respond to one of John Page's recent posts.

Quote:

Originally posted by John Page
I don't see it as a separate form of logic, just a clearer picture of what logical operations really entail. The variables are "slippery" whether we like it or not.

Here's a table using "slippery" notation that avoids violating the LOI in it represenational format by uniquly identifying each variable by a prefix - r0 for the thing being represented (so r0T stands for the actual truth) and then r1, r2 etc for other instances.

Axiom Name,Conventional, Slippery
Law of Identity,T=T, r0T=r1T
Law of Non-Contradiction,T=(T or ~T),r0T=(r1T or ~r2T)
Law of the Excluded Middle,T=~(T and ~T),r0T=~(r1T and ~r2T)

(big time snippage)

I'm not entirely sure I'm following you here, so correct me if I seem off track. But you seem to be saying that conventional logical notation somehow violates the law of identity. Would you mind explaining why you believe this to be so? I for one am having a very difficult time imagining a value of T such that T = T (as interpreted by conventional notation) is not true!

As for your slippery versions of the three laws, none of them is actually equivalent to the conventional version, and none of them are necessarily true as the conventional ones are. I can't think of a value for T such that T = T is false, but I can think of plenty of values for r0T and r1T such that r0T = r1T is false. Indeed, it doesn't seem to be possible to express the Laws of Identity, Noncontradiction, or Excluded Middle at all using slippery notation. So why use it? What advantages are there to be gained?

In fact, I see one further problem with your system. From what I've seen, you seem to want to impose a rule that no variable may be used more than once in a single chain of reasoning. Thus, T = T becomes r0T = r1T, T = (T & ~T) becomes r0T = (r1T & ~r2T), etc. Let's call this the Slippery Rule: No chain of reasoning may use the same variable more than once. Now here's the problem. The Slippery Rule cannot be formulated in a proposition that adheres to the Slippery Rule!

To see why, let's consider what the Slippery Rule might look like in conventional notation. A formal version of the Slippery Rule might look like this: "A chain of reasoning C is valid if and only if for all X and Y such that X and Y are variables used in C, it is not the case that X is the same as Y." Now try altering the statement to follow the Slippery Rule. You'd wind up with something like "A chain of reasoning C is valid if and only if for all X and Y such that Z and W are variables used in P, it is not the case that I is the same as J." I think that strictly speaking, this new version is not even a sensible proposition, since Z, W, P, I and J are never really defined. But even if they were, the point is that this new version doesn't actually place any limits on the validity of C. If X and Y are the same variable, that's just fine as far as this new statement is concerned.

So, if the Slippery Rule is to be followed, then we have a choice to make. We can either choose to follow the conventional notation version of the Slippery Rule, in which case we are contradicting ourselves because the conventional version of the Slippery Rule does not itself follow the Slippery Rule. Or, we can follow the slippery notation version of the Slippery Rule, in which case we end up acting just as if we'd never adopted the rule at all. Either way, the Slippery Rule (and by extension, slippery notation) ends up being self-defeating.

Quote:

I have no issue with logic assuming that truth values are homogenous in a similar manner to math assuming that quantities are homogenous. However, if you look at a real system for determining logic such as a computer or a human mind these systems compare and quantify - they do not magically make things identical.

I'm not even sure what you mean by computers and minds "magically making things identical", or why you think this claim is relevant here. I'm not saying that computers or minds magically make things identical. What I am saying is that when formulating logical propositions for employment in a chain of reasoning, it is perfectly legitimate to use the same variable more than once, with the understanding that this variable has the exact same meaning each time it appears.

Sun Dog

John Page · 05-22-2003, 09:23 PM

Sun Dog:

Thanks for your response. Let me say first that I'm trying to develop a model that explains (in a cognitive/phenomenological sense) how logic works.

Quote:

Originally posted by Sun Dog
But you seem to be saying that conventional logical notation somehow violates the law of identity.

Its my underlying concern - that it seems OK to say that something is only itself, unique etc. but also OK to imply that something is something else for the purposes of deciding truth.

I'm saying that conventional logical notation does not accurately represent what is going on (under some circumstances) such that the LOI might be violated.

Example: The Watergate Paradox

Quote:

This paradox is about Presidential lying before the Congress of the United States. The following statements were made by Jones and Nixon:

S1. Jones says �All of Nixon�s utterances about Watergate are false.�
S2. Nixon says �All of Jones' utterances about Watergate are true.�

Conventional analysis has two cases with S1 being assumed true and then false, both of which result in a contradiction:

Case 1: If S1 is True, Jones has uttered a truth that Nixon always lies about Watergate, thereby causing S2, uttered by Nixon said, to be False.
However, if S2 is False, Nixon is lying that Jones always says the truth about Watergate, causing S1 to be false because Jones uttered it.
Case 2: If S1 is False, Jones has uttered a lie and all that Nixon says about Watergate is true, thereby causing S2, uttered by Nixon, to be True.
However, if S2 is True, Nixon has uttered a truth that Jones always tells the truth about Watergate, thereby causing S1 to be true because Jones uttered it.

To understand the cause of the apparent contradiction, it is necessary to remember that language is itself a system of representation. Thus, statements S1 and S2 are claims about what one person said another person uttered about Watergate (a representation), not about their utterances about Watergate directly (the represented item).

The following shows that Jones� and Nixon�s statements can be treated independently of each other and we do not need to rely on either�s representation of the other�s representations:

rUN (representation uttered by Nixon) is Nixon�s representation of what Jones represented about Watergate;
RUJ (Jones� actual utterance about Watergate) is what Jones represented about Watergate;
rUJ (representation uttered by Jones) is Jones� representation of what Nixon represented about Watergate; and
RUN (Nixon�s actual utterance about Watergate) is what Nixon represented about Watergate.

We can conclude:

If rUN then RUJ Nixon says Jones is telling the truth
If rUJ then ~RUN Jones says Nixon is not telling the truth

This solution demonstrates that truth is relative to the statement under consideration and all statements, by their nature, are representations that require verification against what is being represented.

Quote:

Originally posted by Sun Dog
I for one am having a very difficult time imagining a value of T such that T = T (as interpreted by conventional notation) is not true!

Agreed - as long as one understands that T is a symbol that refers to the same subject every time it is used. Hopefully the Watergate example shows how applying conventional rules for propositional logic using the different notation avoids the contradiction arrived at conventional notation.

Quote:

Originally posted by Sun Dog
In fact, I see one further problem with your system. From what I've seen, you seem to want to impose a rule that no variable may be used more than once in a single chain of reasoning. Thus, T = T becomes r0T = r1T, T = (T & ~T) becomes r0T = (r1T & ~r2T), etc. Let's call this the Slippery Rule: No chain of reasoning may use the same variable more than once. Now here's the problem. The Slippery Rule cannot be formulated in a proposition that adheres to the Slippery Rule!

But the variables really are slippery, nevertheless. I'm not insisting all variables be different variables, I'm proposing a prefix notation so context/abstraction/self-reference issue are avoided (in the true spirit of the LOI

)

Quote:

Originally posted by Sun Dog
To see why, let's consider what the Slippery Rule might look like in conventional notation. A formal version of the Slippery Rule might look like this: "A chain of reasoning C is valid if and only if for all X and Y such that X and Y are variables used in C, it is not the case that X is the same as Y." Now try altering the statement to follow the Slippery Rule. You'd wind up with something like "A chain of reasoning C is valid if and only if for all X and Y such that Z and W are variables used in P, it is not the case that I is the same as J."

Try "A deduction using the Ontologic system (that's what I call it) is only valid if and only if the notation of the system makes clear which instances of symbols within the system represent the subject under study, and which are representations for the purposes of comparison under the system."
Also, here's the first Axiom for Ontologic

Quote:

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, denoted by pre-subscript �R�, and an entity that is its representational form, denoted by pre-subscript �r�:

A [ RA + rA

Or, in words: All existence comprises at least represented existences as well as existences that are representational.

I suggest you consider the meaning of your expression ..."it is not the case that X is the same as Y." If it is the case that X is the same as Y then we're right back to the LOI issue. I'm not sure what else to suggest to make it clearer but offer a computer example - the language 'C' is implemented by using a pointer to the (starting address of the) actual value of the variable in memory and one can manipulate that pointer directly. All I'm suggesting is that to be rigorous in logic we need to specifically declare each instance of a variable to denote its context within the expression to be evaluated. The 'C' compiler turns expressions like if(A,B=C,A=!A) into implementable code in a specific sequence.
I hope my Watergate example is sufficiently clear on the issues that arise if one doesn't follow the same kind of discipline in logic.

Quote:

Originally posted by Sun Dog
So, if the Slippery Rule is to be followed, then we have a choice to make. We can either choose to follow the conventional notation version of the Slippery Rule, in which case we are contradicting ourselves because the conventional version of the Slippery Rule does not itself follow the Slippery Rule.

My Slippery Rule is much more slippery than yours!

Thanks for taking the time to go through my post.

Cheers, John

cobrashock · 05-27-2003, 08:11 AM

Or could it be that A=A can never be equal because A has to be defined and that defined wording can never be identical, so the definition describes something, even a proper name? It bepends on how far you want to persuit the definition I would say.
cobrashock

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05-27-2003, 08:11 AM	#65
cobrashock Junior Member Join Date: Mar 2003 Location: Indianapolis,Indiana Posts: 27	Or could it be that A=A can never be equal because A has to be defined and that defined wording can never be identical, so the definition describes something, even a proper name? It bepends on how far you want to persuit the definition I would say. cobrashock

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