How do you master logic?

[LauricAcid]

btw, don't bother whining about the sound / valid distinction

'whining' eh? Thanks for the personal insult. I don't recall having personally insulted you, though I may be forgetting now of opportunities I had for it. Anyway, not only is your insult gratuitious, but so is your aside itself, since I have no sound/valid distinction I care to mention here.

Various -- it's an empirical question. The obvious one is that a valid argument only admits true statements. That means that arbitrary and false statements can't be used to derive valid conclusions. Beyond that, it gets fairly philosophical, but for example inductive generalization requires something more than just an observation of correlation, in order for a universal statement to be justified. You have to determine causal relations, not just static correlations.

You say that a valid arguement has only true statements. So I can't infer from that that you hold that all sequences of true statements are valid arguements. (Maybe you do, but I can't infer it from what you said.) And you've stated one criterion for a valid inductive generalization. So I suppose that the best question is: in order to form a definition of 'valid argument', what do you consider to tbe the essential property of valid arguments? Edited September 5, 2005 by LauricAcid

[DavidOdden]

You say that a valid arguement has only true statements. So I can't infer from that that you hold that all sequences of true statements are valid arguements. (Maybe you do, but I can't infer it from what you said.)

An unstructured set of true statements does not form an argument; if you mean a structured set of true statements which are connected according to rules of inference and yield a true statement, then that would be correct.

So I suppose that the best question is: in order to form a definition of 'valid argument', what do you consider to tbe the essential property of valid arguments?

That they accurately yield true conclusions. If, for example, you were to introduce a rule that says "When at least 90% of observed incidents of F(x) are true, then F(x) is universally true", that could easily lead to an false conclusion. Validity is about whether the method gives correct answers, which has to be answered empirically.

[source]

That site might serve for reference, but it would not be a good place to start, even as a place to glean some terminology.

Uh . Sorry, Eager_Logician, LauricAcid is absolutely right. I don't know why I suggested this site to a beginner. However, once you get started on the path of mathematical logic, this site will be a good reference.

[BurgessLau]

I'm not interested in elaborating upon the terms and concepts of law [...]

I might have some quibbles with the definitions of the more primary senses given by Merriam-Webster, and wish for more detail and even greater breadth in them, but they're okay for present purposes.

I have asked you for examples that would back up your claim that there is a special logic, at least for one field, law. You didn't answer.

I have asked you for a definition of logic. You vaguely referred to dictionary usages, thereby not answering the question.

I will try again, but with a broader question that -- like the two preceding questions -- will help set a context for interpreting your various comments in this thread: In terms of essentials, what is your philosophy?

[LauricAcid]

I have asked you for examples that would back up your claim that there is a special logic, at least for one field, law. You didn't answer.

There are a bunch of terms for concepts in the field of legal reasoning listed in the link I gave you. You've ignored what I posted last: I'm not here to expatiate upon the field of legal reasoning, but I only mentioned it as one area that the poster may wish to look into since he is interested in arguments. I have no interest at this time in particular discussions about those concepts or whether they even might be reducible to more basic logical concepts, as my point was simply to remind the poster that there is this field of study he may want to take a look at.

I have asked you for examples that would back up your claim that there is a special logic, at least for one field, law. You didn't answer.

I said that the primary definititions in Merriam-Webster are close enough for the purposes of this discussion. That's not too vague. If you don't have this dictionary, then not to worry since it is free online.

Meanwhile, without hectoring you for a definition, I most welcome whatever definitions of these terms you would like to present:

logic

truth

entailment

neccessary

sufficient

essential

validity

valid argument

proof

I will try again, but with a broader question that -- like the two preceding questions -- will help set a context for interpreting your various comments in this thread: In terms of essentials, what is your philosophy?

First, I am not a philosopher. Second, I'm not going to give you a standing on one foot summary of the results of whatever philosophical conclusions I've managed to arrive upon. However, aside from a philosophical appraisal, for the purpose of the context of this thread, I have found that mathematical logic is a very powerful tool and that it has greatly illumined for me the modest amount of mathematics that I've learned, as well as it provides a paradigm, though not an exhaustive one, that can be looked to in certain other situations also. Edited September 5, 2005 by LauricAcid

[LauricAcid]

An unstructured set of true statements does not form an argument; if you mean a structured set of true statements which are connected according to rules of inference and yield a true statement, then that would be correct.That they accurately yield true conclusions. If, for example, you were to introduce a rule that says "When at least 90% of observed incidents of F(x) are true, then F(x) is universally true", that could easily lead to an false conclusion. Validity is about whether the method gives correct answers, which has to be answered empirically.

So, based on that and what you posted earlier, would it be correct to say that you hold that valid arguments are exactly those that have a structured set of true statements connected by rules of inference?

But not just any structure, right? Would you state (not just give an example of) the essential property of valid structures?

And not just any rules of inference, since people sometimes do propose invalid rules, right? Would you state (not just give an example of) the essential property of valid rules of inference?

Perhaps you've answered about rules of inference when you said that valid arguments are those that accurately yield true conclusions. So, would it be correct to say that you hold that valid rules of inference are those that always yield true conclusions from true premises? But you won't countenance that valid arguments can have false premises? So do you reject that this Aristotelian syllogism is valid:

All men have gills.

All racoons are men.

Therefore, all racoons have gills .

Validity is about whether the method gives correct answers, which has to be answered empirically.

So we have to empircially verify the method? So you think we have to empirically verify the method of Aristotelian syllogisms? But, given your framework, what's to verify? To verify would be to see each time whether the syllogism gave a true conclusion? How would we empirically verify that? I suppose by empirically verifying the conclusion. But then what do we need the syllogism for if we're going to be empirically verifying its conclusions anyway?

On the other hand, we would escape that if we did not hold that the syllogistic method needs to be empirically verified. But am I correct that you wouldn't allow that the syllogistic method can be relied upon without empirical verification of its validity?

Edited September 6, 2005 by LauricAcid

[LauricAcid]

The validity of the rules of inference of formal logic, such as modus ponens, have been established by induction from long experience with them; and thus need not be reconfirmed in each particular instance.

I find that view more appealing as applied not to formal logic, but rather to our most basic reasoning that must ultimately be relied upon in formalizing logic. Things such as modus ponens and the law of identity seem to be fundamental to our thought, even and especially, at a pre-formal stage. So, since at this stage we probably wouldn't want to justify modus ponens by using modus ponens, then we have to go to something presumably even more basic, such as you mentioned induction.

Yet, I wonder how much induction is required? Isn't it possible that one can understand that modus ponens is irrefutably valid by just seeing one application of it? Granted, humans may not have the super-intelligence to grasp this in one instance, and thus require greater instances to apprehend the validity. But, at least in principle, couldn't a mind apprehend modus ponens upon one instance of it? But if grasping in one instance is not rationalism, then at least it's pretty close. So one might say: As long as just one instance of observing modus ponens is required to grasp modus ponens, then rationalism is refuted since no one can grasp modus ponens without at least one empiricial experience with it. Notice, by the way, that an empirical experience with the method is different from an abstract example of it. In the later case, if one claimed that one could be given an abstract example of modus ponens but no empirical test of it and yet grasp the principle, then it seems to me that one would be asserting rationalism.

And, isn't induction itself a kind of generalized modus ponens? In that sense, we will have justified modus ponens by itself anyway.

Edited September 6, 2005 by LauricAcid

[Jamwhite]

Thanks for the recommendations.

I'm interested in this, but does this include actual mathematics such as axioms and proofs or even statements of formal systems?

It does contain a mathematical axiom (just one).

Postmodern philosophy attempts to replace induction with axioms. I have yet to figure out where this idea orginated but I believe that it was logical positivism. This article does not subscribe to that method. The article is of the nature of other philosophic writing such as OPAR and ITOE, it inductively shows the foundation of arithmetic and hints at this foundations relationship to all the higher mathematics that results from arithmetic.

Edited September 6, 2005 by Jamwhite

[LauricAcid]

It does contain a mathematical axiom (just one).

Postmodern philosophy attempts to replace induction with axioms. I have yet to figure out where this idea orginated but I believe that it was logical positivism. This article does not subscribe to that method. The article is of the nature of other philosophic writing such as OPAR and ITOE, it inductively shows the foundation of arithmetic and hints at this foundations relationship to all the higher mathematics that results from arithmetic.

Oh please do not cloud the waters like that. Modern axiomatics has nothing to do with postmodernism nor does logical positivism. Also, modern axiomatics predates Vienna Circle logical positivism. And mathematical deductive reasoning goes back centuries before logical positivism. It is true that earlier conceptions of axiomatics, primarily material axiomatics, have a more empirical direction than do modern axiomatics, but even then notions of self-evidence and the like have been important through centuries.

And, as to supposed attempts to supplant induction with axioms, do you think the Greeks did this? What inductive system did the Greeks evade by using axioms (granted the Greek concept is regarded as material as opposed to modern). And when induction did start to roll into town, what is the inductive mathematics that you think preceded modern axiomatics? Please point to mathematical proofs that are not deductive. Please point to some mathematics in centuries past that even without declared axioms did not at least produce deductive proofs from premises, even if ad hoc premises.

I must say that you have no business flinging about terms such as 'postmodern' as if to taint twentieth century mathematics as well as logical positivism by an association that you only imagine to exist.

Edited September 6, 2005 by LauricAcid

[LauricAcid]

It does contain a mathematical axiom (just one). [...] The article is of the nature of other philosophic writing such as OPAR and ITOE, it inductively shows the foundation of arithmetic and hints at this foundations relationship to all the higher mathematics that results from arithmetic.

So I take it that there are no theorems, no proofs, and no actual mathematics other than one axiom standing alone.

If I recall correctly, I read a couple of articles by Pistaturo. (I hope I recall correctly that it was Pistaturo that I read, as they were articles linked to by Speicher.) There was nothing there. Just vague talk, and what of it did make sense did so only for being so rudimentary and common-sense obvious. It was pretty much reiterating what's already been commented upon by Objectivism, but with some added hand-waving. It doesn't take much to put together a "foundation" that is not much more than saying, "See, people learn about numbers through their empirical experiences with physical objects, and then people count, and then we generate natural numbers, and then we have fractions, and then we combine numbers to make new number systems, and then we have...voila! presto!...manifolds and Fourier series and Laplace transformations! See, all empirical!"

The attack on mathematics is a strawman anyway. No one denies that humans learn to count by handling and observing physical objects. That's not at issue. But just stating truisms about human cognition does not a foundation make. And indeed it is apples and oranges. One is about human psychology and cognition. The other is about investigations of the logical and abstract relations that were first conceived in earlier counting stages but are not limited by them. The abstractions don't "contradict" reality, unless they are even taken to be assertions about reality. They're not assertions about reality like in an almanac or in a motorcycle manual. Mathematics is not the recording of data. Mathematics just doesn't work that way and not at that level, but rather in a different abstract way such as to serve not as the analogical form itself by which we compare descriptions and data. And the abstractions as expressed in the formulas do come around to represent pre-formalized mathematical concpets - from counting on up. If there is a single pre-formal mathematical conception, such as counting, that is not represented by axiomatic mathematics, then, please, say what you think it is.

And, by the way, you misunderstand the sense of 'axiom' in algebra. What algebra does in this regard is to compare the properties of number systems and algebraic systems as they are usually generalized not by adding axioms for number systems but usually by dropping axioms, since as we drop axioms we get more and more general systems. The point is in most of this is the word 'axiom' is used but not even in a foundational sense, but rather as, pretty much, a fancy word for 'premise'. One looks at the consequences of premises and thus gains understanding into the relations among number systems and algebraic structures.

As part of the network of conceptions that is mathematics, this has a quite important role, and contributes greatly to mathematical conception that bears practical fruit in the technology you use everyday, including the very computer you're reading from at this moment, even as you scoff at this mathematics as "completely floating abstraction" or whatever was that jejune description you came up with.

Edited September 6, 2005 by LauricAcid

[Jamwhite]

I must say that you have no business flinging about terms such as 'postmodern' as if to taint twentieth century mathematics as well as logical positivism by an association that you only imagine to exist.

I have a great deal of business using "postmodern" to discuss mathematics, as well as logical positivism.

After your colorful response, I went and looked up my original sources on the history of mathematics. In particular, I wish to point out an article series by Ronald Pisaturo called "Undermining Reason: The 20th Century's Assault on the Philosophy of Mathematics" in the Intellectual Activist issues: October 2000, November 2001, and December 2001.

From the November issue:

"Since the late 19th century, several important attempts have been made to defend mathematics--to establish an objective base for the science. One of the most famous of these attempts--still quite popular today, and taught in virtually all first-level abstract-mathematics courses--is the "axiomatic system" published by Giuseppe Peano in 1889."

Now I verified the dates on positivism and Comte died in 1857, with the main writing is positivism was around the 1870s - 1890s.

The Pisaturo articles critic the different attempts at grounding mathematics much better than I could. He also offers a positive theory of the foundation of mathematics which I referenced before -- here.

As Pisaturo points out, advanced mathematics has put men on the moon. The problem is all in the philosophic foundations of mathematics -- or what do we mean by the natural numbers. How do validate addition?

The answer given by Pisaturo and Marcus (in TIA July 1994 and Sept 1994) is via an induction on the concepts that give rise to mathematics, or by understanding the conceptual common denominator that seperates "some fish" from "5 fish". Now their article goes into this in depth from pre-mathematics onward.

Induction, the process of moving from up the abstraction chain, must precede deduction, the process of expanding the breadth of the chain. The problem, historically, of mathematics is that the mathematicians have failed to explain the process their minds had to go through in order to arrive at something -- the induction. I do not fault the mathematicians for trying to defend math, nor do I fault Plato for trying to defend reason. But in both cases, for similiar reasons, all the evidence presented to me shows that both systems are wrong.

My experience in mathematics has been this (for context, I was a math minor in college). The axiomatic system of mathematical foundation turned into the equivolent of "where God is dead, anything is possible". The post-doctorial work of most of my professors was in creating or analyzing new systems of mathematics that existed for no other reason then that they could. If nothing else, and there is alot else, this tells me clearly that something is very wrong is mathematics today. Pisaturo's explinations are consistent exactly with that I saw in college.

Edited September 6, 2005 by Jamwhite

[LauricAcid]

I have a great deal of business using "postmodern" to discuss mathematics, as well as logical positivism.

No you don't. Linking mathematics or logical positivism with postmodernism is arbitrary nonsense.

And I'm laughing my keester off at the thought of you only finding out about Peano through Pisaturo! Now, that's funny.

As to Comte, after I posted, I thought about him and thus edited my post to specify the Vienna Circle logical positivists. I apologize for any misunderstanding thus created. Anyway, Comte is, of course, an important chapter here, but you have not supported your assertion that modern axiomatics originates in logical positivism, whether of Comte's or others. And my point about the Vienna Circle is that this is the logical positivism most noted for its connections with mathematical logic, and it is a plain fact that mathematical logic comes decades before the Vienna Circle and that the Vienna Circle philosophers picked up on mathematical logic that had been published well before these philosophers then had an influence in return on mathematical logic. Moreover, the greatest interest for the Vienna Circle philosophers was probably not mathematical logic itself, but rather as part of larger concern with analytic philosophy, philosophy of science, and philosophy of language.

But more basically, your stab in the dark to account for modern axiomatics as stemming from logical positivism misses looking at more pertinent logicians, especially Boole. And what you miss is not only is logical positivism connected with mathematical logic, but there are important tensions between certain important logicians and philosophers and certain logical positivists. If you're sincere about finding out about this subject, and would like more than comic-book history of it just to feed tract style denunciations, then you might inform yourself about this complex intellectual history rather than just arbitrarily start declaring ludicrous things such as that modern axiomatics is postmodernism. (I grant, though, that Pierce might be pertinent here in some respects, but even here I don't know of more than an attenuated connection with postmodernism and mathematical logic. Most importantly, studies in mathematical logic simply do not depend on postmodernism and do not refer to it in the overwhelming and vast body of its most important literature.)

conceptual common denominator

Please define and explain 'conceptual common denominator', in your own words, not just retyping from ITOE, so that I may better understand this notion.

The problem, historically, of mathematics is that the mathematicians have failed to explain the process their minds had to go through in order to arrive at something -- the induction.

First, you assume that they used induction (by the way, we're not talking about mathematical induction, which is a different, and deductive, animal). Second, whatever process was going on in their mind, it's not their intellectual obligation to explain it. Psychologists can investigate such things. Mathematicians give you mathematics. Mathematical journals are not journals of psychology (though I hardly doubt that there are journals in the intersection of these fields). You're right, mathematicians don't usually jabber on about their mental processes. Do you judge an architect by his explanations, or lack of explanations, of his mental processes or by his designs? Do you judge a musician by his explanations or by his music? Do you judge a mathematician by his explanations or by his mathematics? This is not to say that such explanations are not welcome or don't contribute to our knowledge and understanding. But they are not required. Anyway, if you are interested what mathematicians have to say about mathematical thinking, then you should some of what's been written by people like Dedekind, Russell, Godel, Quine, and others. You'll not agree with much of it, but at least you'll know something (as compared to your current nothing) about the subject you now so arbitrarily declare upon.

As to induction, okay we get it. People count physical objects. From this activity are born mathematical concepts. Now, what are mathematicians supposed to do? Face east three times a day and repeat that like a prayer? Just because Objectivists like to keep being reminded of the empirical source of mathematical thinking, mathematicians are then supposed to do what exactly? Put a statement at the beginning of each proof that they recognize, like some kind of loyalty oath, that the orgins of mathematical thinking are empirical?

The axiomatic system of mathematical foundation turned into the equivolent of "where God is dead, anything is possible".

Please cite where any mathematician ever claimed that anything is possible. You're just making this stuff up as you go along, aren't you! Or is this something your read in some other Internet chat room? Really, you need to get some self-control so that you don't just type into your message editor any arbitrary garbage that floats between your ears.

The post-doctorial work of most of my professors was in creating or analyzing new systems of mathematics that existed for no other reason then that they could.

I admit that the mathematics can get pretty remote. 1. That some mathematics is far afield does not mean that all modern axiomatics is thus wrong, for heaven's sake. 2. I suspect that even investigations that are extremely remote often have some role in a greater confluence of ideas that enriches mathematics. 3. I'm not even convinced that your professors' work was as flighty as you say it is, especially since your criterion of what is grounded is ludicrously restrictive. So, would you say what where the subjects of this research? 4. If you think certain mathematics is too recondite, then you don't have to study it. But this doesn't make the mathematics incorrect, nor does it detract from the fact that other people do enjoy reading that mathematics.

If nothing else, and there is alot else, this tells me clearly that something is very wrong is mathematics today. Pisaturo's explinations are consistent exactly with that I saw in college.

Oookay. So because you found that some mathematicians are too far afield what you judge to be the correct province of the subject, you've concluded that modern axiomatics is itself wrong. Yes, I can see why you don't relate to the methods of reasoning of mathematics.

/

Self-correction: In a previous post I misspelled 'Pisaturo'.

Edited September 6, 2005 by LauricAcid

[Jamwhite]

Please cite where any mathematician ever claimed that anything is possible. You're just making this stuff up as you go along, aren't you! Or is this something your read in some other Internet chat room? Really, you need to get some self-control so that you don't just type into your message editor any arbitrary garbage that floats between your ears.

I am certaintly glad that people do not resort to ad hominems when others disagree with them on this forum.

For the honest readers of this forum, those who do not approach mathematics as apart from reality. Here is a basic example of the axiomatic approach to mathematics. I am quoting from Pisaturo -- italics his (TIA Nov 2001):

Peano lists nine axioms of arithmetic. Most textbooks presenting his system reduce this list of axioms to the following five, which have become known as Peon's axioms (or postulates). For the lay reader, they are presented here in plain English rather than in Peano's technical notation:

(P1) 1 is a number.

(P2) If a is a number, then a + 1, which is the successor of a, is also a number.

(P3) No number has 1 as its successor.

(P4) If two numbers have equal successors, the numbers themselves are equal.

(P5) Event set of numbers that contains 1 and the successor of every number in the set contains all the numbers.

What I wish to call attention to here is what I consider the be the essential, epistemological, problem in the theory of a science based on axioms -- it does so by dropping its connection to reality.

The first problem concerns the nature of axioms themselves. As Pisaturo puts it, "[a]xioms must be statements about reality, not statements about concepts" [ibid]. Ayn Rand says, "axiomatic concepts identify explicitly what is merely implicit in the consciousness of an infant or of an animal." [ITOE 2nd ed, "Axiomatic Concepts"]. These "axioms" fail every basic test of axioms in Objectivist Epistemology that I can find.

Let us contrast the "axiomatic system" of mathematical foundation with Pisaturo and Marcus's. Under their foundation, mathematics needs only one axiom: 1 = 1. Here is what they mean:

"1=1" means that each thing in the group equals each other thing in he group; each "one" equals each other "one". Furthermore, "1=1" does not mean that the number "one" equals itself. This axiom is not, most fundamentally, a statement about numbers; it is, most fundamentally, a statement about units. It states that every group of one unit within a larger group of units equals every other group of one unit within that group of units.

To me, this is a much better example of an axiom because it relates numbers to units to existents. Thus, mathematics is linked to epistemology and the relationship between between metaphysics, epistemology, and mathematics has begun to be explored.

However, let us assume that the original axioms are correct. Notice how the idea of the philosophy of mathematics fades away? In a fully deductive system, there are axioms (which are taken as unquestionable self-evident propositions) and derived proofs/theoreoms/postulates/etc... In such a system, it is the goal for it to exist without reference to anything outside itself. Number, in the example above, is not linked to units or measurement -- it is a primary. Taken that way, the entire system of mathematics can be self-contained without actually referencing anything outside its own definitions. It is for this reason that I say such systems are floating, and rationalistic.

When I talk about how worthless the research of my math professors was. It is exactly to the extent that their research was based on floating abstraction that I consider it worthless.

My purpose here is simply to present to people that there has been work done to found mathematics on an inductive basis, and that for myself, I found extremely useful in understanding the meaning and purpose of mathematics.

For those who are interested, here are the The Intellectual Activists where these articles were published:

"The Foundation of Mathematics" by Pisaturo and Marcus, July 1994.

"The Foundation of Mathematics II", Sept 1994

"Undermining Reason: The 20th Century's Assault on the Philosophy of Mathematics, Part 1" by Pisaturo, Oct 2000.

"Undermining Reason: The 20th Century's Assault on the Philosophy of Mathematics, Part 2" by Pisaturo, Nov 2001

"Undermining Reason: The 20th Century's Assault on the Philosophy of Mathematics, Conclusion" by Pisaturo, Dec 2001

Also, it was either Pisaturo or Marcus who gave a lecture at the Jefferson school a long time ago about the philosophic corruption of mathematics that was sold by 2nd Renaissence. They no longer carry it, and I cannot find a copy to buy. If anyone knows of a copy, please let me know. I would love to have it.

[DavidOdden]

Perhaps you've answered about rules of inference when you said that valid arguments are those that accurately yield true conclusions.

Yes, that is correct.

But you won't countenance that valid arguments can have false premises?

Again correct: men do not have gills, and raccoons are not men.

So we have to empircially verify the method?

Yes, I believe I said that.

But, given your framework, what's to verify? To verify would be to see each time whether the syllogism gave a true conclusion? How would we empirically verify that?

You verify the conclusion, using whatever scientific means are available. That means relating the conclusion to an observation, which either supports or refutes the conclusion.

But am I correct that you wouldn't allow that the syllogistic method can be relied upon without empirical verification of its validity?

Yes; and that allows you to chose between competing syllogistic methods.

[LauricAcid]

I am certaintly glad that people do not resort to ad hominems when others disagree with them on this forum.

Yeah, I get cranky sometimes. But look at this way, I wasn't insulting you personally; just some of the mental entities that travel between your ears. .

For the honest readers of this forum, those who do not approach mathematics as apart from reality.

Right, there are the honest people and there are the mathematicians. Tidy, I must say.

Here is a basic example of the axiomatic approach to mathematics. I am quoting from Pisaturo -- italics his (TIA Nov 2001):

What I wish to call attention to here is what I consider the be the essential, epistemological, problem in the theory of a science based on axioms -- it does so by dropping its connection to reality.

And then you list Peano axioms as if they're some terrible departure from reality! They're just basic facts of natural numbers, for godsakes. They're as rudimentary as you can get. "Dropped connection from reality". This is great stuff you're coming up with here.

Peon's axioms

Peano

Event set of numbers that contains 1 and the successor of every number in the set contains all the numbers.

Every set, not "event set". You probably just made a typo, but I wonder. Do you even know what that statement means?

Let us contrast the "axiomatic system" of mathematical foundation with Pisaturo and Marcus's. Under their foundation, mathematics needs only one axiom: 1 = 1.

First, the axiom that, for all x, x = x is already an axiom of identity theory as part of mathematical logic. If it were the only axiom that mathematics needs, then of course mathematicians wouldn't adopt any other axioms. Mathematicians are always looking for ways to thin the axioms. But you can't axiomatize mathematics with just x = x or just 1 = 1. Calling 1 =1 an axiom but not USING it as an axiom to derive theorems is just pandering. It's pandering to placate people who want to think there is some other mathematics that really uses real live axioms but not those mean, nasty, empty, floating axioms the mathematicians use.

Anyway, let us see what we can prove from your axiom: 1 =1.

Well, we can prove 1 = 1.

How about proving the fundamental theorem of arithmetic (which is that every natural number has a unique prime factorization)? How about proving that there is no greatest prime number? Forget about proving, how about even defining derivatives?

Yes, please show us how to do that from your axiom 1 = 1.

To me, this is a much better example of an axiom because it relates numbers to units to existents.

No it doesn't. It's a formula. We do whatever relating there's to be done with existents. And why you chose to see some relation between this formula and existents but not other formulas is just arbitrary, and no one else needs to stop relating other mathematical formulas to models, practical applications, and advanced mathematical intuitions just because you think 1 =1 relates to reality better than other formulas do.

However, let us assume that the original axioms are correct.  Notice how the idea of the philosophy of mathematics fades away?

What idea of the philosophy of mathematics fades away? The empirical concept? The Peano axioms are just simple truths about natural numbers. They're mathematical statements. Every time we make a mathematical statement we have to attach some philosophical disclaimer that the statement is about empirically observed existents? Like a warning tag under a piece of furniture: "Notice: Do not remove this tag. This mathematical statement has been tested for conformance with the process of empirical testing by examination of existents."

Would it really make a difference if the Peano axioms said instead:

(1) 1 is a counting number, which is the counting number for just a single existent.

(2) For every counting number that is the counting number of a bunch of existents, there is has another counting number to follow, which is the number of that bunch of existents plus one more.

(3) 1, which is the counting number of just a single existent, is the first counting number, and there is no counting number less than 1 since there can be no number of existents less than are counted by 1.

(4) If n is a counting number of a bunch of existents, and m is a counting number of a bunch of existents, and if n and m are both followed by the same counting number for some bunch of existents, then n and m are the same counting number for some bunch of existents.

(5) If 1, which is the counting number of just a single existent has a certain property P, and for every counting number n, which is a counting number for some bunch of existents, if n has property P implies that the number that follows n, which is the counting number for the number of the bunch of existents counted by n plus one more, has property P, then all counting numbers, which are counting numbers for a single existent or a bunch of existents, have property P.

Would that make you happy? Wouldn't that conform to your empirical mandate for mathematics?

But can't you see that we could just as easily say:

Let 'e C' stand for 'is a member of the set of counting numbers'.

(1) 1 e C.

(2) If n e C, then n+1 e C.

(3) If n e C, then n >= 1.

(4) If n e C, and m e C, and n+1 = m+1, then n = m.

(5) If P1, and if for all n e C, Pn implies Pn+1, then for all n e C, Pn.

Them's just the Peano axioms, again. You really can't relate that to reality with much more difficulty than you relate your vaunted 1 =1 to reality?

(By the way, usually, mathematicians start the Peano axioms, with 0 rather than 1, regardless of whatever historical precedent.)

[continued next post]

Edited September 6, 2005 by LauricAcid

[LauricAcid]

In a fully deductive system, there are axioms (which are taken as unquestionable self-evident propositions) and derived proofs/theoreoms/postulates/etc

In what deductive systems? Modern axiomatics? There's no requirement of "unquestionable" or "self-evident". Some mathematicians may be inclined to this, but it is not required. Actually, that's the conception of the older material axiomatics.

In such a system, it is the goal for it to exist without reference to anything outside itself.

Okay, so now you must be back to your favorite straw man. What you posted is a plain lie. But since you don't even know what you're talking about, I guess it's not a lie; you just don't have a clue what you're saying.

First, in the quote before, you described material axiomatics, which very much does concern itself with reference to what's outside the system, such as even though the Greeks were interested in the abstractions as abstractions, but the mathematics was, of course, used for mastering the physical world. So now you're even dragging poor Euclid and all the other mathematical geniuses of ancient Greece down with this absurd denunciation of yours.

Second, modern axiomatics includes a formal semantics; moreover only the most radical formalist insists that mathematics has no reference to anything outside the formulas themselves.

And so you're so mixed up you forgot what you're even attacking. If it's "free floating abstraction" you want to take a bite out of, then self-evidency is the OPPOSITE of what you started to attack. Formalism is the OPPOSITE of concern for self-evidency. You're jsut really, really mixed up. (And you're just completely uninformed about all this upon which you declare.)

Number, in the example above, is not linked to units or measurement -- it is a primary. Taken that way, the entire system of mathematics can be self-contained without actually referencing anything outside its own definitions. It is for this reason that I say such systems are floating, and rationalistic.

No, it's for you to apply the numbers as measurements. There's nothing stopping you. It's as if you're demanding that mathematics do you the favor of reminding you each day that the numbers can be used for measurement. It's a given that counting numbers can be used for measurement. We don't need the Peano axioms to tell us that. Mathematics gives you the formulas. It's your job to use them if you want to. Don't blame mathematics if you don't want to.

When I talk about how worthless the research of my math professors was. It is exactly to the extent that their research was based on floating abstraction that I consider it worthless.

Yes, I see. If you can't handle the Peano axioms, then who could blame you for rejecting anything more complicated?

My purpose here is simply to present to people that there has been work done to found mathematics on an inductive basis, and that for myself, I found extremely useful in understanding the meaning and purpose of mathematics.

What have we learned so far from your mathematics? 1 = 1. Edited September 6, 2005 by LauricAcid

[LauricAcid]

Jamwhite, it would be more productive for you to say what you think is good mathematics, and show developments that realize this value, rather than to denounce mathematics that you don't understand. Nothing is above critique, including mathematics, but a meaningful critique is not just to blindly and wildly throw mud (or worse) in every direction, which is not only an injustice to that which you know nothing about but is an injustice to your own philosophy as well.

[LauricAcid]

that allows you to chose between competing syllogistic methods.

What do you mean by competing syllogistic methods? Do you mean different Aristotelian syllogistic forms? Or do you mean an entirely different set of syllogistic forms other than the Aristotelian system?

You rejected the men, gills, and raccoon syllogism due to the falsity of the premises. But I take it that you do see the form as special:

All M are G.

All R are M.

Therefore, all R are G.

So, would it be correct to say that you consider the form itself valid, but not arguments that use the form but with other than true premises?

You say that we have to empirically check the conclusion of each syllogism. This raises two points:

1. If we're going to have to empirically check each conclusion, then there's no point of using a syllogism. If we have to empirically check each conclusion anyway, then the syllogistic method becomes just and empty exercise.

2. As far as I can tell, you are rejecting the Aristotelian view that, if the premises of a valid syllogism are true, then the truth of the conclusion necessarily follows.

3. If, as I take you to assert, the essential property of valid forms is that they are ones that "accurately" yield true conclusions, but the conclusions need to empirically checked anyway, then what does it mean for a form to be "accurate" other than it has an observed success rate? But if the success rate were less than 1, then a form (such as the presumed valid form AAA) will have at least once yielded a false conclusion from true premises. But has anyone ever encountered such a thing? And, if the form fails just once, then, again, what's the point of using it since you hold that we should empirically check each conclusion anyway?

It is impossible for me to conceive that a syllogism such as AAA could fail, just as it is impossible for me to concieve that an absurdity is true. To allow that AAA could fail by yielding a false conclusion from true premises is as absurd as to allow that the laws of identity could fail. I don't see how you hold the laws of identity, which I take it you do, but allow that AAA could fail by yielding a false conclusion from a true premise.

Edited September 6, 2005 by LauricAcid

[DavidOdden]

What do you mean by competing syllogistic methods? Do you mean different Aristotelian syllogistic forms? Or do you mean an entirely different set of syllogistic forms other than the Aristotelian system?

I mean any method of logic at all.

But I take it that you do see the form as special:

All M are G.

All R are M.

Therefore, all R are G.

So, would it be correct to say that you consider the form itself valid, but not arguments that use the form but with other than true premises?

I think it's a way of compressing the structure of a valid argument, but a more accurate statement would be expressed in terms of subsets, i.e. "mammal" is implicit in the concept "man".

1. If we're going to have to empirically check each conclusion, then there's no point of using a syllogism. If we have to empirically check each conclusion anyway, then the syllogistic method becomes just and empty exercise.

You are not paying attention to what I am saying. I never said that you have to empirically verify each conclusion for the conclusion to be valid. You need to verify the method -- the law of inference. That was the point I made earlier, and you riase this question about how one determines the validity of a logical method. I answered the question -- that doesn't give you license to impose a requirement of independent verification on every other form of derived knowledge. If the method is empirically proven to be valid, you may use it to integrate existing knowledge in order to derive new knowledge. There is no obligation tor redundantly check that the conclusion is indeed correct. You should check the conclusion if you have come to possess new knowledge that suggests that the previous conclusion needs refinement.

2. As far as I can tell, you are rejecting the Aristotelian view that, if the premises of a valid syllogism are true, then the truth of the conclusion necessarily follows.

I do't see what could possibly make you think that. Unless you are specifically focusing on the concept of "necessity". Do you mean that in some metaphysical sense, as though logic were simply "out there"? If so, and if it really is the case that Aristotle held that logic is "out there", then I would reject that position. I don't do Aristotle, so I'll leave it to e.g. Burgess or some other person who can speak more authoritatively on Aristotle's beliefs.

Maybe I'll come back to the third point later -- gotta work some.

[DavidOdden]

Picking up where I left of...

3. If, as I take you to assert, the essential property of valid forms is that they are ones that "accurately" yield true conclusions, but the conclusions need to empirically checked anyway, then what does it mean for a form to be "accurate" other than it has an observed success rate?

It means that the form does always result in accurate description of reality, as determined empirically. It’s not just that it has some success rate, rather it has a “certain” success rate (i.e. no reason to disbelieve).

And, if the form fails just once, then, again, what's the point of using it since you hold that we should empirically check each conclusion anyway?

If the form were to fail, you would not use that form again; obviously you would seek the cause of the failur and correct it by providing a revise form. As I mentioned before, you are simply wrong that you need to independently verify all conclusions. That leads to an infinite regress.

It is impossible for me to conceive that a syllogism such as AAA could fail, just as it is impossible for me to concieve that an absurdity is true.

Nor can I imagine it, again obeying the requirement that the premises be true. It is impossible for me to understand why you would think I held otherwise. If you’d like to find a more useful grounds for fighting, I’d suggest such things as the relation between universal and existential quantifiers (that A does not entail I) or the correct use of =>.

[LauricAcid]

I mean any method of logic at all.

So your term 'competing syllogistic system' refers to any method of logic at all. It's true that 'syllogism' is sometimes used that way, but overwhelmingly, 'syllogism' has been used at least in the last several decades to mean the three-line argument forms in which each line is one of the AEIO statement forms. So, we have to work with your non-standard use here. Rather inconvenient.

I think it's a way of compressing the structure of a valid argument, but a more accurate statement would be expressed in terms of subsets, i.e. "mammal" is implicit in the concept "man"

(*) All humans are primates. All primates are mammals. Therefore, all humans are mammals.

If the concept mammal is "implicit" (what's the definition of 'implicit', by the way?), then after whatever empirical verification went into forming the concepts mammal and human, do we keep empirically verifying? If so, wouldn't that entail that we hold some possibility that we'll encounter a human that is not a mammal? I recognize the Objectivist sense in which the concept human is arrived upon inductively. Objectivist concepts are supposed to be formed by looking for essential properties, and the Objectivist definition of human is that the essential properties are that of being rational and an animal, and one might encounter a rational animal that is not a mammal. But once the concept has been formed, I fail to see the need for empirical verification that the conclusion of (*) follows from the premises. Moreover, your explanation now mentions sets, so what is your definition of 'set'?

Additionally, we should ask whether such mathematical statements as that all prime numbers greater than two are odd are merely empirically true. If the oddness of prime numbers greater than two is merely empirically true, then we do have to keep checking each prime number, even though we have a mathematical proof that would save us having to do this. So, in what sense does Objectivism even recognize the authority or use of such mathematical proofs if they don't ensure that we don't have to keep empirically checking their conclusions anyway? On the other hand, if it is recognized that mathematical proofs do provide such surety, then how can Objectivism deny the distinction between contingent and necessary truth?

Anyway, no one has yet responded to my request for an explanation of the Objectivist concept of essential beyond the few sentences in ITOE; and I suspect that such terms as 'implicit' will need to be defined in terms of the concept essential.

You are not paying attention to what I am saying.

No, I am paying very close attention to what you are saying. I'm giving your statements quite a bit of thought. It seems you are not paying close attention to what you are saying:

I never said that you have to empirically verify each conclusion for the conclusion to be valid. You need to verify the method -- the law of inference. That was the point I made earlier, and you riase this question about how one determines the validity of a logical method. I answered the question -- that doesn't give you license to impose a requirement of independent verification on every other form of derived knowledge. If the method is empirically proven to be valid, you may use it to integrate existing knowledge in order to derive new knowledge. There is no obligation tor redundantly check that the conclusion is indeed correct. You should check the conclusion if you have come to possess new knowledge that suggests that the previous conclusion needs refinement.

First, I said nothing about the validity of the conclusion. Rather, I mentioned the truth of the conclusion. But more importantly, here's where our conversation was:

I asked, "So we have to empirically verify the method? But, given your framework, what's to verify? To verify would be to see each time whether the syllogism gave a true conclusion? How would we empirically verify that?" [emphases added]

You responded to the first question with, "Yes, I believe I said that." And you responded to the rest with "You verify the conclusion, using whatever scientific means are available. That means relating the conclusion to an observation, which either supports or refutes the conclusion." [emphases added]

So I asked you if, according to your view, we have to empirically verify the method, and you answered that yes we do, and I asked if that entails empirically verifying each time [emphasis added] and how we do that, and, to that question - which is, how do we verify each time - you said that we empirically verify the conclusion by observation.

If you wish now to modify or qualify those responses of yours, then that's quite fair. But you are incorrect to claim that I didn't pay attention to what you posted, since I paid extremely accurate attention to what you posted. By the way, this is now the second time you've made this kind of mixup in our conversations in this forum. The first was in the discussion about Russell's paradox when you got mixed up about 'abribtrary'.

As to new knowledge requiring refinement of the conclusion, any such knowledge would then have to require refinement of the premises. So your point is not relevant. Actually, it is very relevant but in the opposite way you think it is. Showing empirically that a conclusion fails in an argument form such as AAA doen't show empirically that the argument form is invalid; instead it shows that one of the premises is false. Now it is true that an argument form can be proven invalid if we show that a false conclusion follows from true premises. But that's not what we're talking about here. Argument forms such as AAA are valid and any false conclusion from them does not contradict the validity of the form but rather the truth of the premises.

you riase this question about how one determines the validity of a logical method. I answered the question -- that doesn't give you license to impose a requirement of independent verification on every other form of derived knowledge.

No, I imposed no such requirement. I mentioned NOTHING OF THE KIND. First, 'independent' is not a word that I used, let alone that I made no such claim of a requirement for verification of "every other form of derived knowledge" since 'derived knowledge' is not a term I've used, and , most importantly, I made no assertion about any concept even remotely similar to "every other form of derived knowledge".

The need for verifying forms is your claim, not mine. I asked whether having to empirically verify a form such as AAA, as you've argued that we do, entails having to empirically verify each conclusion of arguments of the form, and you granted that it does (though you now claim that you didn't). And you mentioned that if one form is not upheld then we go to another 'syllogistic method', and I simply asked you what you mean by 'competing syllogistic method'.

And if "competing syllogistic methods" are "any logical method at all", as you defined, and if you claim that syllogistic methods must be empirically verified, then it is you, not me, who is claiming that all logical methods must be empirically verified. As to "every form of derived knowledge", since this is your concept, not one mentioned by me, then I can't opine whether any logical method at all is the same, in your system of concepts, as every method of derived knowledge.

Anyway, your explanation now just brings this back full circle. If syllogisms such as AAA need to be empirically verified, but now you deny that that entails having to empirically verify each conclusion, then just what empirical verification is required? That we empirically verify only some of the conclusions? If so, then fine, but that would give even less confidence in the form than from empirically verifying all the conclusions, so that not only has confidence in the form been reduced from the surety logicians give it, but it's not been demonstrated that confidence in the form is any more empirically justified than confidence in any empirical statement that has been tested as often as the form has been tested. Perhaps one might argue that a form, say AAA, is more tested than a particular empirical statement that is the conclusion of the form, so this does gives us an empirical basis to infer the statement rather than testing it directly. But that leaves a lot more statements much less sure than I, at least, I am willing to have my degree of confidence diminished. For example, I am not so willing to have my degree of confidence diminished that, all poodles are animals since all poodles are dogs and all dogs are animals.

I wrote that, as far as I can tell, you are rejecting the Aristotelian view that if the premises of a valid syllogism are true then the truth of the conclusion necessarily follows. You wrote:

I do't see what could possibly make you think that. Unless you are specifically focusing on the concept of "necessity". Do you mean that in some metaphysical sense, as though logic were simply "out there"? If so, and if it really is the case that Aristotle held that logic is "out there", then I would reject that position.

I don't know what your definition of '"out there" in a metaphysical sense' is, so I can't respond to the question. As to Aristotle:

"[...] 'being necessary' means-that it is impossible for the thing not to be."

"[...] the expressions 'it is not possible to belong', 'it is impossible to belong', and 'it is necessary not to belong' are either identical or follow from one another; consequently their opposites also, 'it is possible to belong', 'it is not impossible to belong', and 'it is not necessary not to belong', will either be identical or follow from one another."

Anyway, you say you don't see how I can possibly think that you don't hold the Aristotelian view that the if the premises of a valid syllogism are true, then the truth of the conclusion necessarily follows. Well, the Aristotelian view is that with a valid syllogism (actually, as I understand, for him syllogisms were just what we now call the 'valid syllogisms') the truth of the conclusion necessarily follows from the truth of the conclusion, which means that it is impossible for the conclusion to be false if the premises are true. So it would not be even relevant to empirically test the validity of such forms as AAA. But you hold that we should empirically test such forms. So now you can see how I could possibly think that you don't hold the Aristotelian view that if the premises are true then the truth of the conclusion necessarily follows. If you'd like to qualify or modify what you've posted so that you actually do agree with the Aristotelian view, then that is, of course welcome, but then for your position to be consistent you'd have to admit that it is not pertinent to empirically test the validity of such forms as AAA, as well as an Objectivist who agreed with the Aristotelian view in this context would have to show how to reconcile that with the rejection of the distinction between statements that are necessarily true and those that are true but not necessarily true.

It means that the form does always result in accurate description of reality, as determined empirically. It’s not just that it has some success rate, rather it has a “certain” success rate (i.e. no reason to disbelieve).

If the success rate leaves us no reason to disbelieve, then would we need to continue empirically testing? Or, perhaps, you hold that at some point we may have realized that enough empirical testing had been done so that we;ve moved from having a reason to disbelieve to having no reason to disbelieve. So what would have been reasons to disbelieve the validity of, say, AAA? A lack, at sometime, of empirical testing, perhaps? But again, if one understands that with valid forms, all conclusions necessarily follow from the premises, then no empirical testing is even pertinent, and if empirical testing ever were pertinent, then, though we may need to test less after we've already tested quite a bit, we can still test and we still may find that the form fails to be valid, in which case, even though we did not have reason to disbelieve the validity of the form, we would have been better to do so. I think think that, fortunately, all of that silliness is avoided by just understanding how the forms are valid.

As I mentioned before, you are simply wrong that you need to independently verify all conclusions.

I did not claim that we need to empirically verify each conclusion. As I mentioned, I asked you about this specifically, with the exact words "each time", and you responded "You verify the conclusion, using whatever scientific means are available." So please don't blame me for your own answer.

If you’d like to find a more useful grounds for fighting

Asking for explanations of assertions and then showing the problems with those assertions and explanations is fighting? Only on a broad definition of 'fighting'.

As to other topics in logic, such as you suggested, if you have something to say about them, then I'm sure not stopping you.

Edited September 7, 2005 by LauricAcid

[BurgessLau]

LauricAcid,

In post 23, I asked you to give examples of forms of the logic of law. In post 24, you acknowledged my question -- and avoided answering it. Instead you referred me to another site.

Also in post 23, I asked you to define "logic." Again in post 24, you acknowledged my question -- and avoided answering it. Instead you referred me to another site.

In post 29, I asked you to identify your philosophy. In post 30, you acknowledged my question -- and avoided answering it.

Keeping in mind that my previous, open-ended questions might have been too difficult, I will ask a simpler question, a closed-ended one, that is, one requiring only a "yes" or "no" answer. Answering this question will help set a context for your many comments in preceding posts. My question is:

Are you an Objectivist?

This forum's purpose is to promote trade among Objectivists, so, if you are not an Objectivist, you could set context for your voluminous comments by describing your philosophy, at least in its basic principles.

In part quoting from your post 37, I can say that I hope you will "[p]lease define and explain ... in your own words, not just ..." offer references to other sites, "so that I may better understand ..." the context from which you are operating.

[LauricAcid]

BurgessLau,

If you are dissatisfied with my answers already given, then so be it. Beyond that, I am interested in the topic of this thread, not in fulfilling your own personal curiosity as to whatever philosophic views I may hold. If you have substantive comments about the matters that have been touched on in these discussions, then I will respond per my time and interest. Otherwise, if you wish to waste your time hectoring me to satisfy your own agenda of philosophical profiling, then that's your choice.

[logicaalroy]

Hi Laura,

I am new to this forum but not new to the study of Logic and philosophy which is why I joined this forum. I desired a place to post things on Logic and teach those that want to learn and perhaps learn something new myself from those with more knowledge than me. I must say BergessLau is correct. You stated that legal reasoning is a different type of reasoning. Where did you get this notion? Is the LSAT exam in the United States mostly comprised of Deductive logic? Or would you say that this is the legal resoning you mentioned? Sounds like nonsense, which is why Burguess asked for examples that differ from deduction. Legal reasoning consists of both deduction and induction at times. There are basically only two types of LOGIC as taught in colleges: DEDUCTIVE and INDUCTIVE. All of these so called types of logic fit in one of those two categories; modal logic =deductive, predicate logic=deductive, classical = deductive, etc. There is also a hybrid called ABBDUCTION which usually combines feelings with fact such as deontic logic ( ie. moral reasoning). Where does this so called Legal reasoning type fit in? Deductive isn't it? Yep!

BurgessLau,

If you are dissatisfied with my answers already given, then so be it. Beyond that, I am interested in the topic of this thread, not in fulfilling your own personal curiosity as to whatever philosophic views I may hold. If you have substantive comments about the matters that have been touched on in these discussions, then I will respond per my time and interest. Otherwise, if you wish to waste your time hectoring me to satisfy your own agenda of philosophical profiling, then that's your choice.

[LauricAcid]

The field of legal reasoning applies its own approaches to special topics that are in the intersection of law of logic. I have not opined that logic in its many forms and applications does not reduce to inductive logic or deductive logic. However, certain fields of study may concern themselves with the application of logic in the context of certain considerations especially important in those fields. Again, if one wishes to argue that these applications, practices, and conventions reduce to more basic logic, then I have no interest in arguing against that. But the field of legal reasoning does exist and has special things to say about certain kinds of arguments. If this does not interest you, then that's fine with me; though I don't see how I can be faulted for simply reminding someone who expresses an interest in logic especially applied to argumentation that there is a field for logic especially applied to legal argumentation. As to examples, I provided a link that mentioned several examples. The examples are abundant in just the names of principles in the course outlines and one could do an Internet search on 'legal reasoning' to find all kinds of things about the subject.

My poster name is 'LauricAcid' not 'Laura'.