- --- In ai-philosophy@yahoogroups.com, "Peter D Jones"

<peterdjones@...> wrote:>

http://en.wikipedia.org/wiki/Logic

> Nothing has got very far in AI. I was talking about logic, anyway.

>

Quote:

Logic, from Classical Greek ëüãïò (logos), originally meaning the

word, or what is spoken, (but coming to mean thought or reason) is

most often said to be the study of criteria for the evaluation of

arguments, although the exact definition of logic is a matter of

controversy among philosophers. However the subject is grounded, the

task of the logician is the same: to advance an account of valid and

fallacious inference to allow one to distinguish logical from flawed

arguments.

End Quote

I believe the original argument dealt with the inseparability of

grounding logic to physics. I still don't see how we could

possible "advance an account of valid and fallacious inference to

allow one to distinguish logical from flawed arguments" without

reference to physics and 100% of our knowledge of physics comes in

via the sensory interface.

IMO, the humble binary switch works very well for the purpose of

logic.

Q1. Do we need to contemplate tri-state or > switches to represent

our logic wrt AI?

(on, off, (on & off))?

and what would it possible mean for a switch to be both (on & off)?

How could we possible "advance an account of valid and fallacious

inference to allow one to distinguish logical from flawed arguments"

if a switch could be both?

Q2. Does the brain have the neural equivalent of these?

I think not. What would be the reason which couldn't be done by just

using more bits? The only difference gained would be time not ability

to reason.

John J. Gagne - On Thursday, August 03, 2006 6:44 AM, Eray Ozkural wrote in ai-

philosophy:

EO>> ... logic is simply a crude formalization of macro-level

causality, and locality that we observe in the world. There is

nothing else to it. But it's a cool abstraction nonetheless. <<EO

Your remark, and the various philosophical points raised on the

subject so far, prompt me to share some views on "What is logic, and

what is mathematics?" from a mathematicians point of view.

Without attempting to address the issue in its broader dimensions, we

may, without any loss of generality, consider mathematics simply as a

set of precise, symbolic, languages.

Any language of such a set, say Peano Arithmetic PA (or Russell and

Whitehead's Principia Mathematica, PM, or ZFC), is intended to

express - in a finite, unambiguous, and communicable manner -

relations between selected concepts that are external to the language

PA (or to PM, or to ZFC).

Each such language is, thus, in a Platonic sense, two-valued, if we

assume that the selected relations either hold or do not hold

externally (relative to the language).

Further, a selected, finite, number of primitive formal assertions

about a finite set of selected primitive relations of, say, a

language L are defined as axiomatically L-provable.

All other assertions about relations that can be effectively defined

in terms of the primitive relations are termed as L-provable if, and

only if, there is a finite sequence of assertions of L, each of which

is either a primitive assertion, or which can effectively be

determined in a finite number of steps as an immediate consequence of

any two assertions preceding it in the sequence by a finite set of

Logical rules of consequence.

Thus, each such language has an associated Logic that is specific to

it, and consists of the selection of the rules by which we

assign 'truth' and 'provability' values to the assertions of the

language under an intended interpretation.

So, changing the Logic of a language would yield an alternative mode

of expressing that which we intend to express and communicate by the

language. In other words, there is nothing sacrosant about the choice

of the Logical rules we opt to associate with a language.

An effective interpretation of a language L into another language,

say PM (or PA, or ZFC, or English, etc.), is essentially the

specification of an effective method by which any assertion of L is

translated unambiguously into a unique assertion of PM (or PA, or

ZFC, or English, etc.).

Clearly, if a relation is provable in L, then our Logic should be

chosen so that it should be effectively decidable as 'true' in any

interpretation of L that shares a common Logic - since a finite proof

sequence of L would, prima facie, translate as a finite proof

sequence in the interpretation.

The question arises: Is the converse true? In other words, if an

assertion is decidable as 'true' in an interpretation M of L, then

does such decidability translate into an effective method of

decidability in L?

Obviously, such a question can only be addressed unambiguously if

there is an effective method for determining whether an assertion is

decidable as 'true' in M. If there is no such effective method, then

we are faced with the following thesis:

Thesis: If there is no effective method for the unambiguous

decidability of the 'truth / falsity' of the assertions of a

mathematical language L under an interpretation M, then L can only be

considered a mathematical language of intuitive expression, but not a

mathematical language of effective, and unambiguous, communication.

What this means is that, in the absence of an effective method of

decidability of 'truth / falsity' in a mathematical language M that

is a model of, say, a mathematical language such as Peano Arithmetic,

any correlation of soundness between a PA-assertion and an assertion

in M is essentially arguable; so it is meaningless to ask whether, in

general, an assertion of PA is decidable under interpretation in M or

not (the question of whether the assertion is decidable in PA or not

is, then, an issue of secondary consequence).

The philosophical question, of critical interest to AI, then arises:

Can the Mind be considered as the standard - albeit intuitive - model

M of PA, with the brain as its domain, and can PA be considered a

formalisation of the Mind?

The first part of the question can, of course, be easily answered

affirmatively, since every proof sequence of PA can, intuitively, be

recognised by, and thus be seen as an effective method of

decidability in, M.

The second part, on the other hand, can be answered negatively just

as easily, if we accept that the Mind can be seen to experience

hallucinations (reflecting synaptic patterns that, by definition, we

assume correspond to some aspects of the physical state of the brain)

that cannot conceivably be effectively verified in any formal

language.

However, if we adopt appropriate Theses, based on those proposed by

Church and Turing, then every instantiationally effective method, and

every algorithmically effective method, of decidability, or

computability, in the Mind corresponds to some effective method of

decidability, or computability, in the language that is used to

describe the Mind.

It follows that, if any two interpretations of such a language are

isomorphic, then the language can, in a sense, be taken as adequately

formalising that part of the activity of the brain that lends itself

to mathematical representation.

The contensious issues arise from, and reflect, the failure of

classical theory to distinguish between the representation, within a

language, of those abstract concepts (mental constructs of an

individual mind) that are individually significant within an

individual gestalt, and those of these abstract concepts that are,

further, communicable in an unambiguous and effective manner, and

which may, therefore, be termed as uniformly significant within a

collective of gestalts.

Prima facie, most of the challenges faced in unambiguously, and

effectively, communicating mathematical concepts, seem to involve

conclusions arrived at from debatable Realist premises.

Premises that, when introduced authoritatively into formal

mathematical reasoning, and into scientific discourse, permit us to

logically validate our subjective intuitive perceptions as being

reflective of some absolute Truth that - contrary to our intuitive

experience - must be of universal significance in a Utopian, Platonic

world.

We could even go further, and consider whether the relation of a

particular language (whether mathematical or not), to that which the

language seeks to express, should be the preserve of the philosophers

of the language, and of those who study the nature of the mind and of

consciousness - not of the logicians or mathematicians who simply

provide the tools for such study.

In other words, the selection of the alphabet, selection of rules for

defining well-formed words, phrases, and declarative atomic and

compound sentences, selection of primitive objects (constants) and

primitive truths (axioms), selection of the Logical rules of

deduction for assigning truth values to non-axioms, etc., in a

language should be consequent to the adequate resolution of

philosophical issues that consider the question of what we visualise

as, and how we visualise, the properties and relations between

elements of an abstract, Platonically conceived, ontology - in our

individual gestalts - that a logician or mathematician is being asked

to represent within a formal language.

Whether the above properties and relations between elements of a

chosen abstract, Platonically conceived, ontology have limited,

individual significance, or have a wider, common significance, should

also be the rightful preserve of philosophy in general, along with

questions concerning the nature of such elements, their properties,

and their relationships.

However, once we have defined a language, we should not be at liberty

to use the rules of the language to deduce the existence of new,

unique, elements of the ontology (as opposed to elements of the

language) that are not explicitly specified by the definitions of the

language.

Such deduction would amount to a creation by extraneous definition

that would contradict the premise that the language has already been

specifically defined to express selected pre-conceived, Platonic,

concepts.

In other words, the aim of mathematics should not be to introduce

such extraneous definitions into a language, but to study the

consequences of a mathematical language that, beyond the

considerations involved in its definition, are independent of the

Platonic concepts that the language was designed to express.

By this yardstick, standard interpretations of Cantor's diagonal

argument , or of his power set theorem (as also of some of the

transfinite elements of set theory) seem to extravagantly violate

this principle.

Thus, even 70 years after Gödel's seminal 1931 paper, where he

highlighted that there is an essential asymmetry between classically

true, and classically provable, arithmetical propositions, the roots

of such distinction continue, apparently, to elude effective

expression.

Now, prima facie, for any language to be termed as a language of

unambiguous, and effective, communication, the truth of its

propositions, under any interpretation, ought to be unambiguous, and

effectively verifiable independently of the domain of the

interpretation.

We are, thus, faced with the questions: Is mathematical truth

necessarily unverifiable effectively, and are there theoretically

absolute limitations on unambiguous, and effective, communication?

Now, following Gödel's interpretation of his own reasoning, and

conclusions, in his 1931 paper, standard interpretations of classical

mathematical theory seem to implicitly imply that, even in a

mathematical language as basic as formal Peano Arithmetic, the most

fundamental of our intuitive mathematical concepts cannot be

expressed, and communicated in a complete, unambiguous, and effective

manner .

The extent to which such interpretations are accepted as setting

implicit, and, apparently absolute, limitations on unambiguous, and

effective, communication, depends, no doubt, on individual

psychological, and philosophical, predilections.

Thus, although professional mathematicians and mathematical logicians

might succeed in placing such limitations in comfortably abstract,

albeit counter-intuitive, perspective, such comfort may be denied to

other disciplines that primarily need to express their fundamental

concepts and observations in an unambiguous, and effectively

communicable, manner.

For instance, seekers of extra-terrestrial intelligence, or those

striving to replicate human intelligence in controllable artifacts,

could find such acceptance disturbingly constricting, and

philosophically disquieting.

This is an issue of increasing interest to scientific disciplines

that look to mathematics for providing a language of reliable, and

verifiable, external expression and communication. Ignoring the issue

seems to obscure the raison d'etre of a mathematical language - to

communicate unambiguously, and effectively.

Thus, if mathematics is to serve both as a language that expresses

all possible abstract (including Platonic) mathematical concepts in

formal languages, and also to serve as a language of precise

expression and unambiguous communication, then, particularly in the

latter case, we may need to specify effective decision procedures for

determining whether, or not, a proposition of a formal language of

mathematical communication, L, is to be termed as true or not under

each given interpretation M.

In other words, we need to reflect deeper upon the way we choose to

perceive the nature of Intuitive Knowledge, and more particularly the

nature of factual, or intuitive truth.

By Intuitive Knowledge we refer loosely to that body of pro-active

knowledge that stems directly from our conscious states, in contrast

to our reactive Instinctive Knowledge, which stems from, and lies

within, our sub-conscious and unconscious states.

Now one may, when attempting to express mental concepts within a

language, argue reasonably - as Gödel does - in Platonistic terms,

and define intuitive truths as characteristics of relationships that

are assumed to exist in some absolute sense (that is, even in the

absence of any perceiver) between the objects of an external ontology

(both of which are also taken to exist in some absolute sense).

However, for effective communication, we may need an alternative view

of relationships as belonging to individual perceptions that we

consciously construct, and selectively assign, to abstract objects

(that themselves are individual conceptual constructs) of an abstract

ontology (that is similarly an individual conceptual construct).

In other words, each individual perception can, reasonably, be

assumed to be a subjective, abstract, construct, which is based on a

unique, one-of-a-kind, never-to-be-repeated, consciousness of an

individual experience.

An intuitive truth is, then, essentially, a constructed, space-time

localised, individual factual truth (we shall, henceforth, use the

terms synonymously). It corresponds to a subjectively constructed

characteristic of the expression of an individual perception.

Loosely speaking, it corresponds to a characteristic of the way we

construct an expression for that which we select as common to a

series of subjective perceptions, rather than to a characteristic

that we discover of an objectively observed something.

The distinction seems significant. Platonic concepts, when introduced

into the interpretations of a formal language of communication, could

permit us to misleadingly validate our subjective intuitive

perceptions of individual factual truth as being reflective of some

absolute Truth that must be of universal significance in a Utopian,

Platonistic world. The Biblical Tower of Babel can, arguably, be seen

as illustrating the extreme possibilities of such Platonistic beliefs.

If we accept that, in formal languages, a selected set of axiomatic

truths is expressed as a set of Axioms (or Axiom schemas), then the

selection criteria should reflect that the Axioms are readily

accepted by any perceiver as faithfully referring to some significant

factual truths, pertaining to the expression of abstract constructed

elements of the constructed ontology under consideration, as

perceived and conceived by the perceiver.

For the most basic, and intuitive, of our scientific languages,

namely Number Theory or our Arithmetic of the natural numbers, we

take the commonly accepted selection of such axiomatic truths as the

classical set of Peano's Postulates, first expressed in semi-

axiomatic format by Dedekind in 1901.

The challenge, then, is to express these in a formal language such as

Peano Arithmetic, PA, along with a suitable set of Logical Rules of

Inference by which we can assign unique formal truth-values

algorithmically to as many well-formed propositions of the theory as

possible that are not Axioms.

The concept of formal algorithmic truths is, thus, merely the result

of the application of a set of Rules of Inference for effectively

assigning such formal truth-values algorithmically to various logical

permutations and combinations of axiomatic truths as (finite and

infinite) compound assertions (which, ideally, should not introduce

any new axiomatic truths that are not already implicit within the

Axioms).

Regards,

Bhup