Adrian Selby

cheers Mac, a quick point though, the flaw in Kripke's argument about mind brain identity theory, though I'd struggle to find it without a copy now, was a straightforward one, Kripke generally is awesome.

I'll keep a note of those books for when i have some money, but as a hopeless mathematician who's never studied it beyond 16 it will take some doing to get my head around a few of these concepts I'm reading on this message board that talk about different sized infinities and non standard identities etc.

Adrian

Clutch

Agreed. Interestin' fact and co-inky-dink: I love this book; I would have written it if I'd known how!

-- Richard Jeffrey's endorsement of Logical Options

[ February 13, 2002: Message edited by: Clutch ]</p>

Kent Stevens

Something like the law of non-contradiction is fairly deeply held and comes over in how our language is like. It seems weird if we say someone is both old and young at the same time. Or if we say someone exists and does not exist at the same time which would be another contradiction.

What would happen if the law of non-contradiction was false? Then both A and not A would be true simultaneously. If you apply this to the law of non-contradiction itself you get an interesting result. If the law of non-contradiction is false then this statement can exist with the opposite statement simultanously that the law of non-contradiction is true. The law of non-contradiction could be both true and false simultanously. But if the law of non-contradiction is true then we have just got ourselves into a contradiction which is false.

If the law of non-contradiction is false how is it possible to reason effectively? Say in the case where Caesar is murdered, through reasoning we deduce that Brutus murdered Caesar. But if the law of non-contradiction is false then the opposite conclusion that Brutus did not murder is could also be true. But if you conclude that Brutus did and did not murder Caesar, reason is not seem like a good way of coming up with true statements.

Some of the principles of logic are fairly basic assumptions. Other principles of logic are more open to change and depend more on the definition of the system of logic you are using.

WJ

This statement is true, is true, because the statement has an existence.

This statement is false, is also true, for the same reason of existence.

Apriori logic provides the license to say anything.

Walrus
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Truth is Subjectivity

mac_philo

1. Not only are some systems consistent with --(p&-p), there is a modal system ("Verum") in which the negation of the "law" of non-contradiction is a necessary truth.
2. What is a "principle of logic?" Please don't think I'm being dismissive or pedantic; we are talking logic after all. A logic is characterized by its axioms, rules of inference, and theorems. I think your last statement is well intentioned but gets the ordering wrong; if by principles you mean axioms, it is the axioms that define the system you are working with.

John Page

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?

If a premise of logic is that you can abstract material entities of common types by denoting them with a similar symbol then fine, but it doesn't mean that this yam is that yam - it just means they have a common language label "yam".

The above observation even applies to symbolic logic. Clearly this A is not this A, they are different physical entities separated in space on the screen before you. Symbolic equality is inferred by the observer (you) comparing the images to impute this A is equivalent to that A.

Summarily, logic and reason are not magic. They are tools through which to understand the world we inhabit. Conversely, our lack of understanding of the world around us leads our logic to be imperfect.

mac_philo

In standard logic (first order logic, propositional calculus, sentential logic) the law of non contradiction does not refute itself.
In the formula -(p&-p) there is no identity relation which can cause problems; the variable p has a certain extension which remains constant for every occurance of the variable p. It is not as though that formula asserts the existence of two seperate 'things' (like your letter "A"), but rather asserts that the conjunction of (p is true) and (p is false) is a contradiction.
If p=There have been more than one American presidents with the name George Bush
then there is no way to make sense of this alleged identity problem. P represents a state of affairs, not an object.

Even when our variables represent objects it is possible to quantify without asserting their existence. Consider the proposition:
The present King of France is bald.
A naive formalization of this proposition will end up quantifying over a variable p which asserts that there is a present King of France, but it definately can be formalized without making any such existential claim.

John Page

No deal. It doesn't matter whether p represents a state of affairs or an object (for they are both types of object, thing, phenomenon or whatever that are represented by p). The only reason logic works at all is because it violates the law of non-contradiction - how can you say that "every p has a certain extension ... constant for every occurrence of p"? This is your assumption, not a truth.

Your last example is perfect, I agree that you can formalize the proposition even if there is no King of France, bald or otherwise. This demonstrates that truth and falsity are values ascribed by us to symbolic variables so we can better understand the outside world, not intinsic values. Existence exists, period, and no two Kings of France are the same.

On the question of identity, identity being use of a naming label, let me pose a question to you. Consider the set of all sets that are not members of themselves; is not the apparent contradiction in this statement mere confusion that the "set of all sets" has to be a member of itself? I am asking you to agree that the fundamental device underlying logic necessarily violates the law of non-contradiction - we run into paradoxes when unjustified substitutions are
made.

mac_philo

It's enjoyable to read your opinions. I think they are interesting but don't see how they relate to your rejection of the law of contradiction.
I think if you define 'intrinsic value' we'll no longer have any quarrel. What are you speaking of? What is this nebulous, intrinsic, truthful way in which logic should 'latch onto' the world?

You say that my claim that p has a constant extension in each formula (or set of formula) is an 'assumption' and not a 'truth.' Well, are we talking about formal logic or not? What would it be for it to be True that p's extension is constant? That it is so is merely a fact about the system; a fact about how we construct and employ formal logic.

Truth is a semantic concept. (a&b) just means that a is true and b is true. What does a is true means? It means a is true. Truth is an undefined primitive.

I suspect that this 'intrinsic' notion is doing some work in your opinion on Russell's paradox. Here is what you said: "is not the apparent contradiction in this statement mere confusion that the set of all sets *has* to be a member of itself?"
(Perhaps you aren't referring to Russell's paradox, which is different: Consider the class of all classes that are not self-membered; is this class self-membered? Either answer is a contradiction.)


The resolution is not to 'see' whether or not the set of all sets 'needs' to be self membered, it is to ascertain what faulty axioms and rules of inference have allowed us to create a paradox such as this. For Russell that invovles restricting the use of 'diaganol' arguments and setting up a heirarchy of types so that we don't quantify over sets in this naive way.

[ February 22, 2002: Message edited by: mac_philo ]</p>

John Page

OK, mac_philo:

1. Your first sentence is a put down so I'm ignoring it.
2. Intrinsic Value. This means a value contained within, belonging to, as opposed to a conferred value. For example, colors are conferred upon objects by the human organs of light perception - an object may appear green but it is not made of "green". In my posting I refered to truth and falsity as values that exist in our minds, not within the objects or situations themselves. So, "This sentence is true" or "That knight is true" are semantically incomplete.
3. If you so define an extension of p as constant I'm not going to argue with you, you can devise any system you want. However, in a real world application, how can your system (of logic, whatever) validly assume that all Kings of France are the same? One needs to be clear about what is an axiom and needs to be defended as such and what is a working assumption.
4. Agreed, truth is a semantic concept, and see point 2. above.
5. OK, I quoted a different natural language example of Russell's paradox than you. Your mistake is to give me two choices by ascribing a truth value to the question "Is this class self membered?". Russell's hierarchy of types doesn't work because the world doesn't fit into neatly ordered mutually exclusive classes - 'typing' an entity is a process of finding a definition that fits best. I believe the source of the paradox is related to ambiguity in the term 'class' and the implied recursion in its definition. You must make a distinction between an 'instance' of an object and its abstract 'type'. In your definition of Russell's paradox the word 'class' is an instance of a collection of sets or classes but you then try and treat it at a different level of abstraction by having it either belong to itself or not! This is exactly my beef with how the law of non-contradiction is expressed - A=A. Now, A=A may be a 'truth functional statement' but taken literally it is a self-contradiction.
6. The Artifice of Language. Language comprises only adjectives. Words can be classified into nouns, verbs and adjectives but, literally, they are all labels that describe things - objects, events, properties etc. The human mind groups words to fit our perception of reality's structure. However, few people seem to be aware of this and most that I speak to can't handle it. I know what you mean when you type -(p&-p) but its just another language and assumes that all the objects represented by p are homogenous. Mathematics has the concept of negative numbers but can you go find me 'minus three pigs'? Let's not be fooled into believing that all systems of logic are true because some systems appear internally consistent when applied to limited range of parameters.
7. Axioms in general. An axiom is a statement of belief. A system of logic is a group of coherent and (hopefully) non-contradictory axioms that can be applied to reality on a repeatable and reliable basis. One of my pet projects is to use an Existential approach to derive ontological axioms forming a system called "Ontologic". The aim is to provide a more reasoned basis for symbolically representing reality. Unless we have such an underpinning, I believe it will be impossible to bridge the gap between cognitive science and higher level thought.

Apologies if this is over top, I think very emotionally sometimes and feel strongly that the axioms of logic sometimes look very much like the a priori precepts of religion. Anyone want to collaborate on the Ontologic project and its relative, Comparison/Detection Theory?


[ February 22, 2002: Message edited by: John Page ]</p>