Truth -- What's Consistency Got To Do With It?

Frege used a 2-D notation for his logic.

Many years ago, while with the Graduate Group in Logic and Methodology of Science at Berkeley, I had written a little paper with a tease of a title: "Consistency--What Is Logic Got To Do With It?"

After a recent discussion, I realized more clearly how that paper might have been making too far of a leap in logic.

Perhaps, it should have been named "Consistency--What Is Truth Got To Do With It?"  By "consistency," all along, I had meant "consistency" as understood by mathematicians and scientists, not what we understand "consistency" to be when we speak of, say, "consistent" behavior, which is a totally different concept when compared to a "consistent" theory.

In the mathematicians' definition of truth, as expounded by Alfred Tarski, one can only speak of "truth" within the confines of a mathematically "consistent" theory. Thus, one arrives, in mathematical logic, at incomplete theories that meet mathematical "truth" criteria even as they remain incomplete. In a sense, their incompleteness is more true about these theories than their "truth." Their "truth" exists within their limited, incomplete domain, which, always remains positively finite or countable or of much lower cardinality than the continuum of claims that they can neither prove nor disprove.


In the discussion about "Technology, Time, and Space", sometimes I could clearly feel you are avoiding overemphasis on consistency. ;)

Apart from joking it was necessary for being creative and innovative there.


Posted by pasparto on June 06, 2007 at 06:51 PM PDT #

In mathematics, the Tarski model for "truth" only works if a theory is consistent. On the other hand, many interesting theories, such as the ones that describe arithmetics, are incomplete, i.e. there are an infinite number of worlds of arithmetic objects that they cannot describe and there are an inifite number of larger, mutually contradictory theories in which they can be embedded.

This, by itself, shows the philosophical failure of a model of truth that arises and is identical with the mathematical model of consistency.

Posted by M. Mortazavi on June 07, 2007 at 12:32 AM PDT #

Thanks for your attention, but I know a little about Godel's theorem and Tarski's works, my previous comment was just a joke

کامنت قبلی من صرفا یک شوخی بود، نمی دانم متن انگلیسی معنای شوخی را می رساند یا نه، امیدوارم شوخی نامناسبی نکرده باشم، که در این صورت عذرخواهی میکنم ،با ارادت و احترام، پاسپارتو

Posted by pasparto on June 07, 2007 at 02:07 AM PDT #

Pasparto -

No worries ... I generally respond seriously ... even to jokes ;-) ... and that's my joke ... including all the typos and errors in my quick comments, for which I also have to apologize ;-)

Your joke stimulated other thoughts, that's why I included them.

Take care, Masood

Posted by M. Mortazavi on June 07, 2007 at 05:30 PM PDT #

What does it mean to give a self-consistent philosophical account of emotions for example? Why should self-consistency override reality and become a criterion of truth? I believe the answers to these questions lie in the great success of science which it owes, to a large degree, to the main currents and ideas of modern logic. Physical scientists need to give consistent definitions for objects of their study in order to be able to communicate with each other.

Hi Masood,

I am not too well-versed in logic or mathematics so pardon my ignorance in these matters.

But I wondered...did not the "new" physics and mathematics (relativity, quantum physics, non-Euclidean geometry etc) make consistency irrelevant..or at least not a hard-and-fast requirement? Also, with "newer" research in fuzzy thinking etc, is the necessity of consistency not undermined? Do things not depend on "frames of reference"?

Thanks Umang

Posted by Umang Kumar on June 08, 2007 at 03:47 AM PDT #

Umang -

You've raised an important issue and I've been struggling to find enough time and a good approach to write a reply. I could probably write a whole dissertation about this but I'll limit myself to some of the thesis that I may have put forward.

I may have to break this up into multiple comments ... written over multiple days.

First ... One can probably classify human "reasoning" to that which is done by the "heart" and that which is done by the "mind". Reasoning by the "heart" should not be ignored or discounted. It is the best type of reasoning we do if only we had enough time to pay attention to it. Compassion, love, care and such feelings drive the reasoning by the heart. Anger, jealousy and envy poison the reasoning by the heart.

Reasoning by the mind can probably be classified into deductive form of reasoning and the inductive form of reasoning. (What I wrote above focuses on the deductive mode of reasoning.)

We face Puzzles, in non-Euclidean geometry because Euclidean notions no longer hold. For example, from the "point" outside of a "line," you can now draw multiple distinct parallel lines. The puzzle only exists to the extent we understand geometry in Euclidean terms but if we interpret what we mean (e.g. by parallel, distinct) and where these objects (e.g. point, line) now lie, then there's no inconsistencies, so to speak.

In non-Euclidean geometry, consistency hasn't become irrelevant. It only operates with a different "language" or a higher, more abstract-level language than the Euclidean one. The language has changed, not the requirements of consistency.

A similar, congruent analysis shows that puzzles put forth w.r.t. quantum mechanics may simply be terminological ones although, in quantum mechanical puzzles, granting the logical "law of excluded middle" (either P is true or P is false) might have played a role in generating the puzzles. To the extent they displaly a critique of the law of excluded middle, these puzzles undermine the relevance of consistency.

"Fuzzy thinking" provides, simply, a very dedutive toolkit to reason about how probabilities of events are related. It is equivalent, mathematically and practically speaking, to other systems of reasoning such as neural networks or probabilistic reasoning.

In any case, deductive reasoning governs the operations of much that is used to do reasoning on probabilities.

Inductive reasoning, sometimes confused with reasoning "with" or "on" probabilities, often relies on data of sense perception or data of instrumentation used to amplify such sense perception.

Deductive reasoning poses more interesting philosophical questions, like the ones raised by Hume. Why should we believe that something will happen again (that Sun will rise again, for example) given that it has risen again and again in the past? What is the basis of that belief?

So, as you can see, your question raises many questions, and there is a lot to say here and not enough space and time to say them.

Posted by M. Mortazavi on June 12, 2007 at 05:21 PM PDT #

Fascinatingly thought-prokoving post, Masood... it 'points to' so many interesting questions, whether mathematical, philosophical or - dare I say it - epistemological (!).

I've just finished reading "Into the Silent Land" by Paul Broks. Broks is a neuropsychologist who, like Sacks, uses case studies of mental dysfunction to analyse the workings of the mind. Anyone who enjoyed Oliver Sacks' "The Man Who Mistook His Wife for a Hat" should read Broks' book.

Broks looks at, among other things, the relation between the brain's function as a practical and social survival tool, and its function as an 'intellectual' organ. His insights cast a very pertinent light on your comments about 'reasoning with the mind and with the heart'.

As ever, thank you for providing a blog where the word 'epistemological' is not out of place! ;\^)

Posted by Robin Wilton on June 13, 2007 at 07:10 PM PDT #

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