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What is the difference between Aristotelian Logic and Logic

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The deductive forms of Aristotelian Logic comes in 256 forms, of which 24 are valid. (Reference: Wikipedia's Syllogism)

There are 46 references, to date, of Aristotelian Logic being discussed on this forum.

I've listened to Leonard Peikoff's Introduction to Logic course. This familiarizes me with the materials contained therein, but by no means makes me a master of their content. This course does not appear in the ARI Campus courses for free.

I'm also on my first time listening to Aristotle's Posterior Analytics. This is not the translation Robert Mayhew suggests in his recorded talk Aristotle For Objectivists. I might also add, this is not the most user friendly reading of a text I've encountered either.

I've also read, several times, David Harriman's book: The Logical Leap. This is considered to be the first objective examination of inductive logic from a perspective of Objectivism, thus falls under the Latin expression "Caveat Emptor".

Logic has two essential branches per Aristotle—deductive and inductive. Aristotle deals extensively with the deductive side, and is considered the Father of Logic from this aspect. Inductive logic is referenced in the Posterior Analytics, but is not treated as exhaustively.

The law of identity, or its corollary, as Aristotle did not state the law of identity explicitly, the law of non-contradiction, serves as the fulcrum for Aristotle's development of the syllogisms.

Logic, as Aristotle derived it, is considered to have an ontological basis. The first question I would ask, following this intro, is: If there is a difference between "Aristotelian Logic" and "being logical", is there an ontological basis for such a distinction?

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From Harriman's The Logical Leap, p.34.

"Now we see the real contrast between induction and deduction.  It has nothing to do with the notion that induction is merely probable, or that induction involves some kind of arbitrary leap.

Deduction is a simple form of reasoning.  It starts with a causal connection already conceptualized and formulated as a generalization.  In other words, it starts with a complex conceptual product regarded as established and unproblematic.  The deducer is not as such concerned with the process of conceptualizing complex data.  He takes for granted from the outset that we have solved all the difficult epistemological questions involved in forming and using concepts.  He takes as a given that the conceptual faculty has been used to gain profound new knowledge, and that it has been used properly.  He then proceeds to milk the new knowledge for it's implications.

In contrast, an inductive argument is not a self-contained series of premises from which the conclusion follows as a matter of formal consistency.  The reason is that the bridge from observation to generalization is not one premise, or even a hundred premises, but the total of one's knowledge property integrated.  This is why induction is so much more difficult and controversial than deduction, and why it is not reducible to the formalism of symbols."

Edited by New Buddha

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8 hours ago, dream_weaver said:

Logic, as Aristotle derived it, is considered to have an ontological basis. The first question I would ask, following this intro, is: If there is a difference between "Aristotelian Logic" and "being logical", is there an ontological basis for such a distinction?

Can you be specific about your use of "ontological basis" in this context?  Why is it of particular significance, i.e. why not ask is there any basis for such a distinction?

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7 hours ago, New Buddha said:

From Harriman's The Logical Leap, p.34.

"Now we see the real contrast between induction and deduction.  It has nothing to do with the notion that induction is merely probable, or that induction involves some kind of arbitrary leap.

Deduction is a simple form of reasoning.  It starts with a causal connection already conceptualized and formulated as a generalization.  In other words, it starts with a complex conceptual product regarded as established and unproblematic.  The deducer is not as such concerned with the process of conceptualizing complex data.  He takes for granted from the outset that we have solved all the difficult epistemological questions involved in forming and using concepts.  He takes as a given that the conceptual faculty has been used to gain profound new knowledge, and that it has been used properly.  He then proceeds to milk the new knowledge for it's implications.

In contrast, an inductive argument is not a self-contained series of premises from which the conclusion follows as a matter of formal consistency.  The reason is that the bridge from observation to generalization is not one premise, or even a hundred premises, but the total of one's knowledge property integrated.  This is why induction is so much more difficult and controversial than deduction, and why it is not reducible to the formalism of symbols."

I've read this book it is very good and I have heard LP also speak of induction in ways which hint at the major message of Harriman's book, which is that essentially that scientific induction and all forms of noncontradictory induction are examples of the process of conceptualization itself, i.e. that induction and conceptualization are in some sense the same thing.

This has got me thinking.  Has anyone run across a reference that treats induction not as a process of enumeration but simply of "identification" (perhaps this is not very different from Harriman's formulation).

E.G. One encounters bubbling water for the first time and experiences that it burns.

The induction starts on the true premise that "The particular nature of that bubbling water in the entirety of the context burned, bubbling water of that nature in the same contexts burns".  This follows from the fact that an entity has a nature and that an entity's action i.e. what it causes are dictated by its nature.  (note here, we assume we do not have something volitional and we do not say with certainty that it must always burn, but that burning being part of its nature is something that does happen).

If we observe that some bubbling water in what we think are the exact same circumstances does not burn, we do not simply conclude that the universe acts in contradiction with itself, i.e. we do not simply conclude that non-burning is an action that contradicts with the nature of the bubbling water in the context AND it somehow just happened.  We conclude that this is an action which flows from the nature of the bubbling water in the context, but now we need to determine whether the water or the context was different (identification) or whether the action is a second type (as of yet not observed) consistent with the nature of the original which example we simply did not happen to observe the first time. [That said, if such were true we would be remiss not to urgently investigate how such seemingly diverging behaviors could emerge from such a system]

Upon further investigation we see that the water is bubbling in the first case because it is boiling whereas in the second case the water is bubbling because air is being pumped into it.  Indeed then what was missing was the identification of boiling water and its context versus water being pumped with air and its context.

This is just a germ of an idea...

Summary:

One example of A[totality of nature and context] did B is sufficient to generalize that A[totality of nature and context] does B [at least some of the time].

Yes I realize it gets more complicated with volitional entities, and with non-deterministic quantum physics, but even these can be induced up to a point, these behave within limits which are determined by their natures, which natures and limits need only be identified.

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4 hours ago, StrictlyLogical said:

This has got me thinking.  Has anyone run across a reference that treats induction not as a process of enumeration but simply of "identification" (perhaps this is not very different from Harriman's formulation).

4 hours ago, StrictlyLogical said:

If we observe that some bubbling water in what we think are the exact same circumstances does not burn, we do not simply conclude that the universe acts in contradiction with itself, i.e. we do not simply conclude that non-burning is an action that contradicts with the nature of the bubbling water in the context AND it somehow just happened.

Pardon another long quote.  From Where Mathematics Comes From, p. 16.  The below quote demonstrates that the ability to generalize from particulars - which is the very essence of Induction - is present long before an infant reaches a level of conceptual, abstract thought.

"The ability [of infants] to do the simplest arithmetic was established using similar habituation techniques.  Babies were tested using what, in the language of developmental psychology, is called Violation-of Expectation Paradigm.  The question asked was this:  Would a baby at four and a half months expect, given the presence of one object, that the addition of one other object would result in the presence of two objects?  In the experiment, one puppet is placed on a stage.  The stage is then covered by a screen that pops up in front of it.  Then the baby sees someone placing a second identical puppet behind the screen.  Then the screen is lowered.  If there are two puppets there, the baby shows no surprise; that is, it doesn't look at the stage any longer than otherwise.  If there is only one puppet, the baby looks at the stage for a longer time.  Presumably, the reason is that the baby expected two puppets, not one, to be there.  Similarly , the baby stares longer at the stage if three puppets are there when the screen is lowered.  The conclusion is that the baby can tell one plus one is supposed to be two, not one or three."

At the most basic mechanical/perceptual level, our "brain" is looking for similarities and patterns in the world, and forming expectations based on previous experiences, in order to predict the future.  This generalizing is induction.   And this ties into Rand's Benevolent Universe idea.  The Universe does in fact "make" sense.  B follows A, and the senses are valid means of obtaining objective knowledge.  Not omniscient knowledge - but objective knowledge, as opposed to subjective knowledge.  The senses are, in fact, the only means by which knowledge is gained.

This is in opposition to what can be broadly termed "Aristotelian Logic" where a priori (Analytic) knowledge is supposed to be superior to knowledge acquired by the senses (Synthetic) which can only ever be "contingent", i.e. subjective.

Dreamweaver pointed out to me awhile back the subtitle to Knapp's book:  Mathematics is About the World: How Ayn Rand's Theory of Concepts Unlocks the False Alternatives Between Plato's Mathematical Universe and Hilbert's Game of Symbols.

While I have not read the book, Hilbert's Game of Symbols, was, at it's root, a form of Rationalism, and an attempt to over come the Analytic Synthetic Dichotomy incorrectly held by many in the late 1800's and early 1900's, and still present with us today.

Edited by New Buddha

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I have not read the book just mentioned. Does it claim that Hilbert took mathematics as entirely a game of symbols? I have never seen anyone point to a quote in Hilbert's actual writings that supports that Hilbert had such a view. Moreover, a decent understanding of Hilbert's mathematics and philosophy requires at least basic familiarity with the technical methods and developments of the field of mathematical logic (I'm not necessarily referring to the author of the book, but rather to general discussion).

Edited by GrandMinnow

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11 hours ago, StrictlyLogical said:

Can you be specific about your use of "ontological basis" in this context?  Why is it of particular significance, i.e. why not ask is there any basis for such a distinction?

Aristotelian logic is commonly distinguished as ontologically based. Just being logical has been used with other schools of logic, that are not necessarily ontological.

18 hours ago, New Buddha said:

From Harriman's The Logical Leap, p.34.

The reference would suggest the difference you brought up in the other thread, if I may be so bold, is more a distinction between Aristotelian Deductive Logic, and his Inductive Logical leads, the strong case being for his deductive with a weak case made for his inductive approach.

_____

I don't know how to state the ontological basis between deductive and inductive. It is evidenced, in part, in the genus/differentia element.

_____

You stated that

22 hours ago, New Buddha said:

Catholic dogma WAS Aristotelianism/Thomism.

It wasn't until this Scholastic, deductive-logic approach to science was "thrown-off" [largely by the British Protestants/Puritans associated with the Royal Society, and the School known as Empiricism (Bacon, Hooke, Boyle, Newton, Locke, Berkeley, Hume, etc.)] that major strides in science were even possible.

Am I to draw from this that the Catholic dogma of Aristotelianism/Thomism embraced only the deductive side of Aristotle's logic, either tossing out or strongly downplaying the inductive elements?

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3 hours ago, GrandMinnow said:

I have not read the book just mentioned. Does it claim that Hilbert took mathematics as entirely a game of symbols?

No. With that subtitle, you get this mention near the beginning of set-theory:

In this fashion [previous three paragraphs] the Bourbaki's emulating Hilbert's famous axiomatization of Euclidean geometry, aimed to systematize and axiomatize mathematics. Bourbaki's influence began to wane around 1970, but much of the legacy they did not invent, but helped perpetuate, notably the foundational role of set theory in mathematics remains.

The two earlier references to Hilbert in the book are brief as well.

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1 hour ago, dream_weaver said:

Am I to draw from this that the Catholic dogma of Aristotelianism/Thomism embraced only the deductive side of Aristotle's logic, either tossing out or strongly downplaying the inductive elements?

Yes, very much so.  There is an important distinction that needs to be made regarding Aristotelianism in Catholic, Medieval Scholastic thought and 'Aristotle' in general.  The radical shift in science, which began largely with the Protestants in the 1600's, proceeded from observation, measurement and experiment and worked towards the mathematization and formulation of generalizations, theory and laws (via induction).  It can be summed up in a quote made by Laplace to Napoleon:

Napoleon: You have written this huge book on the system of the world without once mentioning the author of the universe.
Laplace: Sire, I had no need of that hypothesis.

Thomas started with the premise that God exists and, through the use of deduction, set about explaining all things in their particulars.  Whereas Aristotle differentiated between 'form' and 'substance', Thomas differentiated between 'substance' and 'accident'.  This is somwehat a gross simplification of the issues involved, but the change in Science in the 1600's was very real.

It was actually the British Empiricists non-appeal to "God" (Hume in particular) that Kant felt threatened religion, and prompted Kant's noumea/phenomena, a priori a posteriori dialectic and transcendental epistemology.

Edited by New Buddha

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On 10/19/2016 at 0:43 PM, GrandMinnow said:

I have not read the book just mentioned. Does it claim that Hilbert took mathematics as entirely a game of symbols? I have never seen anyone point to a quote in Hilbert's actual writings that supports that Hilbert had such a view. Moreover, a decent understanding of Hilbert's mathematics and philosophy requires at least basic familiarity with the technical methods and developments of the field of mathematical logic (I'm not necessarily referring to the author of the book, but rather to general discussion).

From Godel: A Life of Logic, p. 32.  [Bracket text is mine]

Originally, Hilbert suggested the idea of a formal system for getting at the mathematical truth as a way of eliminating the possibility [of] logical paradoxes of the [Barber Paradox] sort.  The main selling point for formalization was the claim that these kinds of paradoxes stemmed from the semantic content of their expression in natural language.  Hence, if the symbols and strings of the formal system are completely meaningless, then the statements (symbol strings) should be paradox-free.  In particular, there should be no undecidable propositions.  But if that argument is the main selling point for formal systems, then why are are we all of a sudden trotting out this interpretation step and there by injecting [semantic] meaning back into the picture?  Doesn't this dictionary construction step undermine Hilbert's entire argument for formalization?

The key to resolving this apparent dilemma lies in putting the horse before the cart.  Hilbert's program involved starting with the formal system.  The second step was then to bring out the mathematical structure of concern and show how to match its objects to the strings of the formal system--that is, how to interpret the meaningful mathematical objects in terms of the meaningless formal ones.  Thus we don't begin with the semantic-laden mathematical structure but, rather start with the purely syntactic world of the formal system.  Hilbert's Program really amounted to trying to find a formal system that was above all free from internal contradictions and whose theorems were in perfect correspondence with all the true facts of arithmetic.  In essence, Hilbert didn't believe that any Russell-type paradoxes [Set Paradox, Barber Paradox, etc.] lurked in the world of mathematical truths, even though they might exist in the far fuzzier realm of natural language.,  And the way he thought we could prevent them from crossing the border separating ordinary language from mathematics was to formalize the entire universe of mathematical truth.  What Godel showed was that Hilbert was dead wrong.

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(1) Those passages don't quote Hilbert or cite any reference to his texts.

(2) The passages are from what I think might be a popularizing book [Godel: A Life In Logic] on the subject. Often such popularizations misleadingly oversimplify the subject. Without having read the book, I won't claim that it does misleadingly oversimplify, but I would caution to look out for possible oversimplifications. That set of passages onto itself might be okay yet it could stand some explanation.

(3) Anyway the passages don't say or even imply that Hilbert took mathematics as entirely a meaningless game of symbols.

(4) And not only do those passages not say or imply that Hilbert took mathematics as entirely a meaningless game of symbols, but the passages say the OPPOSITE.

/

I am not an authority on this subject; I have read only some of Hilbert's translated writings and none of his writings in German that remain untranslated to English. So my own comments may be too simple or require qualification or sharpening. For a first reference on the Internet, I would suggest:

Moreover, a few years ago, one of the contributors to the Foundations Of Mathematics Forum asked whether anyone knows of any attribution to the writings of Hilbert in which he said that mathematics is only a game of symbols. As I recall, at that time, no one did. (Posters on The Foundations Of Mathematics Forum are almost entirely scholars in the field of mathematical logic and the philosophy of mathematics.)

That said, here are some general points:

(1) Hilbert recognized the role of mathematics in the sciences. He would not regard mathematics as merely a symbol game.

(2) Hilbert may regard formal systems as subject to being taken, in certain respects, as without meaning. However, I know of no attribution in which Hilbert claimed that mathematics is merely formal systems. Moreover, Hilbert recognized that, while in one aspect formal systems are to be regarded as without meaning, in other aspects, formal systems are to receive interpretation and in interpretation we evaluate meaning.

The rough idea is that syntax onto itself is without meaning but with semantics we do evaluate meaning.

The syntax includes the formation rules for formulas and the rules for proof steps. Syntax is regarded onto itself as without meaning so that no "subjective", vague, or inexact considerations are allowed in checking whether a symbol string does obey the formation rules for formulas or whether a purported formal proof does indeed use only allowed inference rules. For example, regarding formation rules, when you run a syntax check on lines of computer program code, the syntax checker doesn't care about the "meaning" of your code (say, for example, what it will accomplish for the user of the application or whether the user will like the results, etc.) but only whether the code follows the exact rules of the syntax of the programming language.

The semantics include the interpretation of the symbols and of the formulas made from the symbols. This is meaning. The interpretation itself can be done either in formal or informal mathematics. For example:

Ax x+0 = x

This is formal string of symbols that in itself has no meaning.

But with a semantics that specifies the domain of natural numbers and interprets 'A' as 'all', 'x' as a "pronoun", '+' as the operation of addition, '0' as the natural number zero, and '=' as identity, we have the interpretation:

zero added to any number is that number

Of course, that example is so simple as to make the method seem silly; with more complicated formulations we see the advantage of the method.

(3) Also, Hilbert distinguished between the contentual and the ideal in mathematics.

Most basicially, the contentual is the the finitary mathematics of "algorithmic" operations on natural numbers. This was later articulated as the formal system PRA (primitive recursive arithmetic), though Hilbert's own earlier work was in a different but akin system. Such operations on natural numbers can be mutually understood as operations on finite strings of symbols.

The ideal are the infinitary notions of set theory that is used to axiomatize real (number) analysis, as with analysis we regard infinite sequences, etc.

Hilbertian formalism ("Hilbert's program") is:

The finitary is "safe" and unimpeachable. But, while the ideal may itself be without contentual meaning, it is used as a formal framework for deriving formal theorems (that are later interpreted as generalizations regarding natural numbers and also for real analysis). Then, we wish to know whether the finitary mathematics can prove that the infinitary mathematics is consistent (without formal self-contradiction).

It is Hilbert's hope and expectation of such a finitary proof of the consistentency that was proven by Godel to be unattainable. With regard to Hilbert's program, Godel's second incompleteness theorem reveals that there is no finitary proof of the consistency of the theory of natural numbers (generalizing beyond Godel's particular object theory, say, for example, first order Peano arithmetic) let alone of real analysis.

/

Now let's look at some of the passages from that book:

(1)  "getting at the mathematical truth "

Truth pertains to meaning. If Hilbert was concerned with "getting at the mathematical truth", then he could not have regarded mathematics as merely meaningless symbol manipulating.

(2)  "the statements (symbol strings) should be paradox-free.  In particular, there should be no undecidable propositions "

I don't know all that's intended there, but (un)decidability is a separate (though in certain ways, related) question from consistency (consistency being a formal counterpart to "paradox-free").

(3)  "how to interpret the meaningful mathematical objects in terms of meaningful formal ones"

I might say that what are meaningful or not are not objects but instead formulas (or even notions). In any case, again, we see that Hilbert is indeed concerned with meaning. Notions about ideal objects may not be meaningful, but notions about contentual objects are meaningful. PRA has an immediate and "concrete" meaningful interpretation. Then other systems give rise to abstract infinitary notions that don't have such concrete meaning but are "residue" of said formal system that provides theorems regarding generalizations with finitary mathematics and real analysis (which is the theory of the real number calculus used as the basic mathematics of the sciences). Hilbert hoped further that finitary mathematics would prove the consistency of infinitary mathematics - but that's the part proven by Godel not to be possible.

(4)  "Hilbert didn't believe that any Russell-type paradoxes [Set Paradox, Barber Paradox, etc.] lurked in the world of mathematical truths, even though they might exist in the far fuzzier realm of natural language"

Hilbert would have easily known that the Russell paradox can occur even in a formal system (most saliently, Frege's system). Formalization itself does not ensure consistency.

(5)  "And the way he thought we could prevent them from crossing the border separating ordinary language from mathematics was to formalize the entire universe of mathematical truth.  What Godel showed was that Hilbert was dead wrong."

Hilbert hoped for (indeed, expected) a consistent and complete formal axiomatization of the arithmetic of natural numbers and of analysis. By 'consistent' we mean that there is no formal sentence of this system such that both the sentence and its negation can be proven in the system. By 'complete' we mean that for every formal sentence of this system, either the sentence or its negation can be proven in the system, thus that the sentence is decidable, i.e. that there is an algorithm to decide whether there exists a proof in the system of the sentence (for example, we could keep running proofs until we reach one that either proves the sentence or proves its negation). Godel proved that that expectation was wrong. But this does not entail that formal axiomatizations are not still of great value and interest, as indeed the vast amount of "ordinary" analysis is formalized in any of various formal systems.

Edited by GrandMinnow

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GrandMinnow,

I think you are missing the point of the OP, or at least how I'm framing my responses to it.

It's about the difference between the pre- and post-Baconian approaches to science.  And more broadly, the difference between Rationalism and Empiricism (edit: I place Hilbert symbolic logic squarely in the former).

This post was a spin-off from here:

Edited by New Buddha

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I wished only to address the specific matter I posted on.

Now though you claim that symbolic logic is rationalism. Perhaps you mean that certain philosophies or ways of regarding symbolic logic are rationalism. The use or study of symbolic logic itself does not require such philosophical commitments. Also, I'm wondering whether you know that symbolic logic is used in the field of computer science that provides you with the programs and systems that you're using right now on your computer; indeed that some of the most important logicians and mathematicians involved in twentieth century computing used symbolic logic and notions in mathematical logic integrally with the advent and development of the modern computer. Moreover, probably the most (or at least, among the most) important mathematical problems with the greatest economic or practical importance today is one that is in the scope of mathematical logic and comes from theoretical questions in this field (if I recall, there's something like a million dollar prize for the solution).

Edited by GrandMinnow

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On 10/24/2016 at 2:36 PM, GrandMinnow said:

I wished only to address the specific matter I posted on.

Here's a quote from the Kurt Godel on the Stanford Encyclopedia:

Of course I do not claim that the foregoing considerations amount to a real proof of this view about the nature of mathematics. The most I could assert would be to have disproved the nominalistic view, which considers mathematics to consist solely in syntactical conventions and their consequences. Moreover, I have adduced some strong arguments against the more general view that mathematics is our own creation. There are, however, other alternatives to Platonism, in particular psychologism and Aristotelian realism. In order to establish Platonic realism, these theories would have to be disproved one after the other, and then it would have to be shown that they exhaust all possibilities. I am not in a position to do this now; however I would like to give some indications along these lines. (Gödel 1995, p. 321–2).

This "nomainalistic view" was central to much of the late 19th and early 20th Century philosophy (not just Hilbert's Program).  Logic and Philosophy were were pretty much inseparable.  Boole, Peirce, Frege, Russell etc., were not just addressing "mathematics".

And it is the issue of nominalism which is central to this post.

A quote from Russell:

Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great work springs. Remote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos, where pure thought can dwell as in its natural home, and where one, at least, of our nobler impulses can escape from the dreary exile of the actual world.

Edited by New Buddha

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As I've as much as said (and as you even just quoted me), I don't intend or claim to address what is central to your posts. My scope has to been to address specific matters only. These are:

(1) I've explained how it is not correct (or at least that there has not been given here good reason) to claim that Hilbert's view was that mathematics is merely a game of symbols (or merely to consist of syntactical conventions).

(2) The use, study, or appreciation of symbolic logic do not require a commitment to rationalism.

Now you put in bold that Godel claimed to have disproved nominalism as he describes it as the view that mathematics consists solely in syntax and its consequences. Whether Godel is correct that he disproved nominalism or whether his is a fair characterization of nominalism, I don't opine. In any case, his claim, even if correct, does not vitiate my two points above.

I'll reiterate: My own points have been specific and limited. You've replied with certain quotes that have not vitiated my points (whether they were even intended by you to vitiate) and your descriptions about what you take to be the main point of discussion do not control the purpose of my own specific and limited points. So I don't see what purpose is achieved by the quotes you mention as specific replies to me.

Edited by GrandMinnow

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Are you denying the Positivism of the 1800's?  Or do you just not understand it's relevance to the post?

42 minutes ago, GrandMinnow said:

Whether Godel is correct that he disproved nominalism or whether his is a fair characterization of nominalism, I don't opine.

Well, maybe you should "opine".

I've really seen no evidence that you understand the issue at hand.  Or, you believe that the history of Logic is somehow irrelevant to the discussion of Logic.

Edited by New Buddha

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I don't claim to understand (or not too understand) or to be interested in (or not to be interested in) any issues other than the very specific issues I've given definite information about.

I'll try ONE MORE TIME: I don't intend to address any topic in generality or any of its particulars other than those I've mentioned or that may arise as I am motivated to do so. I was interested to add clarification and explanation on, so far, two particular points. This does not suggest that I need to address every aspect, or even the main aspects, of the subject matter.

I am generally familiar with logical positivism, Godel's philosophy, and the history of logic. But nothing about them or anything you've said or quoted vitiates the particular points I've made.

I don't get your need now to hector me into discussing a general subject matter on your own terms when all I've done or claimed to have done is address some particulars that needed clarification.

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