mac_philo

My first statement is a "put down?"

The fog is thickening. I said that while your premisses are interesting, they do not support your conclusion.

Your position seems to be that the only way I can refrain from putting you down is to accept your argument as valid.

If sentential logic violates the law of non contradiction please show me a proof (either via natural deduction, the tree method, induction, etc) that the axioms of the system imply p&-p. Otherwise I stand by my claim that your points, however interesting, do not add up to a valid argument.

Let's not get personal; if I really didn't find this interesting I wouldn't be participating.

In any case, let me just reply to two of your claims.

It is not clear to me how insuring that all Kings of France are "the same" is relevant. What is doing the heavy lifting here? A formula that has a variable reprenting one King of France, or all Kings of France, makes no assertion that they are all the same. Rather the variable's extension is just the set of things of whom the predicate "is or was King of France" is true. I'm not following the identity problem. I can imagine constructing a formula that asserts their sameness as easily as I can imagine proclaiming that formula to be false.

I also do not see what use it is to claim that axioms are 'beliefs.' There are an infinite number of logics with an infinite number of axioms; they are all beliefs? Whose beliefs?
Axioms are propositions taken as fundamental in the context of a system. I suppose there is a tortured way in which a person, obviously competent with a given system, believes that X is an axiom of Y. But the axiom being a 'belief' is opaque to me.

[ February 24, 2002: Message edited by: mac_philo ]</p>

John Page

Sorry, my mistake, thought you were being condescending, no wish to get personal on my part. I apologise.

Now, as to the subject matter:

1. I'm not trying to prove p&-p, if anything I'm trying to prove p = p'. I'm arguing that the underlying mechanisms of (most systems of)logic are contrary to the logic itself. All the systems of logic with which I'm familiar require the user to accept that two entities are equivalent. In reality, existence 'is'. A is itself and B is itself. A can be assumed to be B, or A' or some other existence. This is my interpretation of the law of non-contradiction.

2. So, how does the knowledge of existences come into our mind? Aside from the physics, it seems reasonable to suppose that we receive an image in our mind (as a result of the cognitive process)and this image persists. By comparing these images over time a process of thought extracts the persistent portion of the image. Let me call this the Memory A. By comparing many memories frequently recurring patterns are detected. For example if we see a lot of leaves, called A1, A2, A3 etc. the concept of leaf will emerge in our minds and we will learn to recognize leaves by form, fuction and so on. In this way the images of leaves are comparable to the real (external?) leaves resulting in our axiomatic leaf concept. What my point? - the human process develops axioms and the axioms of logic (truth?) are no exception. Finally, if you subscribe to the above process of cognition and concept generation, I assert that you must agree that all 'existences' have their own unique location in space, time etc. and cannot be said to 'be' each other. This is consistent with the law of non-contradiction and why I give credence to that law. In this context the concept 'equals' and 'equivalent' are shown to be internal to the mind as are any paradoxes resulting from their use.

3. From the above, what you think you know (as truth) is only what you currently believe. Truth is internal to the mind.

4. The human mind, I suggest, comprises multiple working hypotheses about the reality. These working hypotheses include religious systems, systems of logic, ethical rules and so on. In turn, these hypotheses rest on axiomatic concepts which our minds use to analyze the structure of reality. Some systems of thought will be internally consistent and some will not. With this interpretation, I hope to show you that all systems of logic are inherently internally inconsistent because they are based on the premise that one entity can be (absolutely) equal to another entity.

5. This extends even to truth functional logic. How do I classify a leaf as a leaf in order that I may subject it to predicate logic? Which leaves are more like our axiomatic leaf than others? Which truths are more similar to our archetypal A=A concept of truth than others? This is how I reason that the whole underpinning of logics, including math, rely upon a self-contradictory statement of non-contradiction.

6. As to your final point, I do not see what use it is to pretend that axioms are any more than rules of thumb, saws, the folklore of logic or beliefs. I'm not saying they're incorrect, the hunt for constant rules gives us advantages when we better understand our reality this way. If an axiom is a 'fundmental proposition' as you state then how about "God exists" as an axiom? I believe we must try and eliminate use of any a priori knowledge (beliefs, axioms assumptions, tenets, whatever).

Again, I didn't mean to get personal. I guess what seems reasonable to me is anthema to others. I'm cautious as much of secularists who have an unconsidred belief in the great god Logic as I am of religious dogma.

Looking forward to hearing your thoughts.

diana

With this interpretation, I hope to show you that all systems of logic are inherently internally inconsistent because they are based on the premise that one entity can be (absolutely) equal to another entity.

I also hope you show us that all systems are inherently inconsistent yadda yadda yadda. Please.

While I don't claim a firm grasp of formal logic, I can usually follow along all right. I'm intrigued by your proposition. Please continue.

If an axiom is a 'fundmental proposition' as you state then how about "God exists" as an axiom?

Why is this a necessary assumption?

d

John Page

Diana:

1. You said "I also hope you show us that all systems are inherently inconsistent yadda yadda yadda. Please."

Response: I'm trying to find systems that are NOT inherently inconsistent. Puhlease.

2. You said "While I don't claim a firm grasp of formal logic, I can usually follow along all right. I'm intrigued by your proposition. Please continue."

Response: I'm not using formal logic <img src="graemlins/banghead.gif" border="0" alt="[Bang Head]" /> , I think that's what mac_philo is doing. What I'm trying to do is use human reason to understand how and why logic works in the first place and how this explains phenomena such as the Liar Paradox, Russell's Paradox, negative square roots, infinity etc. I happen to think that one of the underlying causes is an illusion caused by the reflexive nature of our thought processes. (see item 2 in my previous posting.

3. I use "God exists" as my example of an axiom to be contentious in comparing religious thought and systems of logic. I'm gratified that you call this an assumption, mac_philo wasn't happy with my relegating axioms to the rank of assumptions .

Do you have a system of thought, philosophy, etc. that employs no a priori assumptions, axioms etc?

mac_philo

I think that our rift comes from a common misunderstanding between people on logic. I think you are understanding logic and its components (specifically axioms) as it pertains to thought; logic as expressing laws of thought. I am working from the understanding of logic as a set of formal axiomatic systems. From my perspective, what you are saying just is not the case by the definition of the logical connectives, though in your context what you are saying is quite possible.

Though of course you may (and apparently do) disagree with me that this has impact on truth functional logic.

That is, I think we've changed the topic. You probably disagree insofar as these issues impact the original topic, but in any case we probably agree that the essential issues here are related more to philosophy of mind or cognitive science than logic, though they may impact logic.

With that in mind I'll re-read your post and see what I think.

John Page

You're right, I disagree - I think we're right on sr's original definition of the topic. Logicians (defined as people who believe in the absolute truth of logic) use a priori axioms (such as the law of non-contradiction) to prove their beliefs.

I'm in the empiricist group with sr (caveat; at least as far as known causal reality is concerned). Let me offer an example and issue a challenge.

Example: If you look in the mirror the tendency is to say "That's me". But its not you, its an image of you and further examination reveals that your left and right hands have been swapped round etc. This is by way of an analogy to the statement A=A.

Challenge: Explain the Liar Paradox, of which one form is the sentence "This sentence is false."

I contend that you will be unable to provide this explanation unless you discard the Law of Non-Contradiction as an axiom. If you discard the latter, the key underpinning of formal logic is undone. If I'm right, truth functional statements are illusions (although not necessarily false!).

mac_philo

Logicians do not use a priori axioms to prove their beliefs.
A logician is someone who specializes in formal systems. Absolute truth, though I do not know what that is, is not tied up with being a logician.
A logician uses logic to prove things about formal systems, not to justify their beliefs. They may intuit that a formula is a theorem, and may believe so before proving it, but that is incidental, and theorems can be proved without any belief.
What would an "a posteriori" axiom be in the context of logic?

I can explain the Liar paradox without using any of the tactics offered. Just do what Tarski tells us to do: restrict the notion of truth. When analyzing the truth of a statement, do so in a metalanguage. This prevents a statement from containing any claims about its own truth, thus avoiding the paradox.

[ February 25, 2002: Message edited by: mac_philo ]</p>

John Page

Dear mac_philo:

Disagree, logicians do use a priori axioms to define their systems.

1. OK, let's probe the definition of a logician, then. If I can paraphrase slightly what you wrote, a logician applies formal systems to prove things. How is this different than a priest (rabbi, cleric, whatever) applying a religion to prove the existence of a god?

2. You say theorems can be proved without any belief. I disagree. Taking the Oxford Reference Dictionary definition "a general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths" clearly requires belief in the (a priori) accepted truths.

3. I'm not sure what an "a posteriori" axiom would be - maybe a theorem?

4. C'mon "... do what Tarski says" is not an explanation of the Liar Paradox, its merely a way of avoiding it. Try again, please.

Kent Stevens

You could deal with the Liars Paradox perhaps if you allowed yourself the right to be uncertain about something or think something is unproven neither true or false. This would mean adding another category so that we have true, false, and uncertainty. This would mean changing the law of the excluded middle but I do not think that this law is basic anyway.

diana

With this interpretation, I hope to show you that all systems of logic are inherently internally inconsistent because they are based on the premise that one entity can be (absolutely) equal to another entity.

I'm still waiting for you to show this.

And I never said I don't make certain assumptions; I do, however, try to stick with necessary ones.

d