contact@onbelief.orgComments on Logic
Perhaps it is more productive to think of belief as computationally logical. This of course does not mean rational in the everyday sense. It just means following a set of computational procedures to produce an outcome of some kind. From an algorithmic stance our brains could be considered to have a number of logical calculators including sensory processors, a linguistic and numeric logic calculator and a procedural logic calculator. It is therefore instructive to consider in what way logic is encoded and how computational machines could one day execute logical decisions for us.
Is logic objective or intuitional ?
Pragmatically logic can be divided in to 2 presently functional states; intuitive and objective. Intuitive logic is that which we have not yet codified in a machine-executable fashion using existing technology. Objective logic is a rule set that we can presently encode and make a sufficiently complex machine operate, given novel data inputs or descriptive beliefs. The complex machine might be a mechanical calculator, slide rule, or computer. This is to some extent obfuscation of the question is logic objective or intuitional. However this classification will become increasingly more important in our society, as technology evolves. Classifying logic in this way is not the equivalent of asking whether or not the brain is like a digital computer merely whether or not human logic can be codified and thus be simulated numerically. (For a discussion of this topic see 'Is the Brain a Digital Computer?' by John R. Searle
The proposition and open sentence of linguistic logic
In a general philosophical sense propositions are defined in the Stanford Encyclopedia of Philosophy as " are the sharable objects of the attitudes and the primary bearers of truth and falsity " (see source article). On this site the wider definition is used and proposition is taken to mean the expression of a descriptive idea, since truth and falsity are merely one class of meaning. Indeed the abstractness of truth and falseness has no real bearing on much of life. Propositions could also be considered to be the meanings or "the things expressed by declarative sentences" (see source article).
Consider the linguistic expression if 'A' is taller than 'B' 'do 'C'. It is undefined and therefore not a proposition. It only develops meaning if the terms A, B and 'do C' become substituted as follows: When the dog(A) is taller than 1.5 m (b) run (do C). In the absence of description logic exists entirely in an intangible form. Consider the famous logical statement that defines mass-energy equivalence:
. That formula means nothing until we define 'e' as energy, 'm' as mass and 'c' as a constant of known size that lets us 'weigh' or equate a given amount of energy to a given mass.
The undefined expressions above could be described as open sentences. An open sentence is approximately defined as that part of a sentence that lacks a subject or an expression in which the variables are undefined. To quote wikipedia "open sentences do not express propositions". So open sentences do not describe the world. For example 'is not likely', 'can be true' are open sentences. The 'open sentence' of formal language-based logic statements seems in general to have property of useless abstraction if considered in isolation. However, when they can be applied on a case-by-case basis to descriptive beliefs they determine to a large extent the way we express ourselves. For example, 'rain is not likely today'. Despite the stylistic imperfection and vagueness of that sentence, it clearly conveys meaning.
The core of formal logic as expressed by the opens sentence is in one sense completely abstract since in the absence of description or knowledge it has no utility. By contrast, logic when applied to data has meaning. For example x is greater than y (or x>y) has no meaning until x and y are substituted with meaningful linguistic or numeric descriptors to gives us expressions such as 2 is greater then 1. Even then the logically based descriptors of number only have real meaning when applied as descriptive parameters of objects. [ This has the unfortunate real world consequence that mathematics, beyond simple arithmetic, is for most school children completely meaningless. It is taught in terms of abstract logical propositions which they will promptly ignore for the rest of their lives upon leaving school. The fact that this seems to be no concern of school teachers or the politicians who micro-manage British school curricula is regrettable. ]
Natural Language and Number
There is something obviously different about numbers and words. Or to put it another way, logical mathematical propositions are fundamentally different from those based on non-numeric or non-logically encoded language. If number is taken to be descriptive, as it is here, the difference between linguistic and numerical logic becomes obvious. Linguistic logic has ambiguity because of variability of meaning in individual words and underlying variability in the meaning of individual words when used together. Mathematical logical is different due to the lack of ambiguity associated with the meaning of number. In addition the logical operators of mathematics have a defined and invariant meaning when used consistently. The logical operators used in arithmetic can only be applied to number. We cannot add the meaning of non-numeric words. 'Two' or '2' in mathematical terms has only one meaning although it has many logical relationships such as 1+1 = 2 or 8/4 = 2. The end result, 2, always has the same meaning. (You might have heard the old philosophical joke told to me by a respondent. Question: What did God create before he created the world? Answer: Maths!)
In linguistic logic there is often scope for interpretation of words and word combinations. (For a more detailed consideration of this idea watch The Stuff of thought by Steven Pinker). A completely different and very large set of logical operators are needed when applied to the meaning of words rather than numbers. A definitive universal calculus of linguistic propositions of is therefore difficult if not impossible to obtain.
There is also a different comparison to be made between language and number. Consider the word 'spaghetti'. There is no logical connection between the letters of 'spaghetti'. Similarly there is no logical connection between the sequence of letters and the object it represents. It is merely an arbitrary descriptor that applies in particular natural languages. The unit of meaning is the individual word not the letters of which it is composed. One problem for those interested in logic is that we have a very large number of words in our lexicon an innumerable number of logical ways in which to combine them. Unlike number, the words of our lexicons do not have a completely definable relationship to each other.
Metaphor is an interesting example of why language is hard to define logically. As a simple exercise consider the nonsensical sentence: 'the airplane vomited'. Then compare that sentence with the statement: "The airplane disgorged its passengers'. Clearly metaphor is a very interesting subtlety of language that would seem to require its own logical definition. In more general terms we can also consider, as Steven Pinker suggests, implied meaning. For example " your exam performance was sub-optimal" could mean you were not performing at your very best on the examination day last month or it could mean in the context of a very low score "it would be a good idea to put your brain in gear before you put pen to paper .... terrible performance". Linguistic meaning and context are inseparable. Both are dependant on knowledge which in turn requires memory either of a biological or electronic kind.
Compare the word 'spaghetti' with the number 233,753,924. Both consist of either 9 letters or digits. However each digit in the number sequence is simply and precisely defined as a power of 10. Ten itself has a clear relationship to the most fundamental digits 1 and 0, which are required for all arithmetics. Very complex linguistic logic is of necessity applied to words when we communicate with each other using words. However even the act of formulating a large number means constructing a logical statement.
In number we have a much more restricted set of digits and these have a logical relationship to each other depending on the way that they are used. In the binary numeral system we have two digits (1 and 0), in the decimal system we have 10 basic meanings (0, 1,2,3,4,5,6,7,8,9) and the hexadecimal system we have 16 digits ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F,). In Roman numerals a positional logic also applies if we consider simple numbers such as IV, VIII, IX, XII.
When the rules of mathematics are correctly applied an internally constant logic is always used. The intuitional rules of the logic of natural language are however different. We cannot devise a way of multiplying 'dog' by 'cat' or dividing 'mouse' by 'hamster'. Those particular logical operators cannot be applied in the context of those descriptors. The logical operators of language are context and therefore knowledge dependant. Logic as stated on the definitions page applies to the meaning of descriptive beliefs.
Numbers as substitutes for words and perceptions in computing
When logical operations are applied to the composition of words rather than their meaning we can nevertheless produce useful outcomes. Perhaps the most obvious use of word-associated logic is Boolean Algebra using logical operators like 'AND' and 'OR'. For example on the Google search engine, the Boolean search expressed as 'brain -logic' will give the searcher a web page from wikipedia (http://en.wikipedia.org/wiki/Brain) from which, at the time of writing, the word 'logic' is absent but the word 'brain' is present. Note however that the letter string 'logic' is not absent as it is found in words like 'biological'. If we then use the 'find' function on Microsoft interest Explorer on the letter string 'logic' we discover that it appears in words like biological. When letters can be given a number equivalent or code such as the ASCII code and so be manipulated by digital computers, they can now achieve definable goals that would be impossible for a human. Clearly there are many potentially advantages in objectifying language based logic.
A form of numeric logic can be extended to the manipulation of colour. Indeed millions of colours can be described in arbitrary brightness values by the 24 bit binary Red Green Blue (RGB colour model) or additive colour system used on television and computer screens. [Hexadecimal numbers can be used to arbitrarily and efficiently encode colours in internet pages (see examples by clicking here).] In this way arithmetic operations can be performed on the numerical codes representing colours. The colour codes can be added subtracted, divided, inverted etc. In so doing new meaning and human understanding is generated. Photoshop and computer programs like it which operate with such numbers can therefore be used to produce meaningful change to images.
More generally when numbers are used systematically as descriptors within a particular language or perceptual system they are indeed very powerful. How far machines can progress towards a simulation of human intelligence by carrying out arithmetic on the number codes assigned to words is still a matter of intense speculation. If human language is seen as an evolutionary adaptation following the evolution of perception, as it clearly must be when we compare ourselves to ants, we then see a more fundamental layer of sensory 'meaning' that is independent of language. This suggests that our memories started to evolve as stores of sensory perception rather than language. Our 'intelligent' processing was initially based on nonlinguistic analysis. Logic must then have begun 10's or 100's of millions of years before the ability to from the logician's 'open sentence' with linguistic codes was acquired.
Machines that cannot formally equate language with sensory input will NOT produce anything other than a numerically intense simulation of human reasoning. However when machine-based detection of 'redness', for example becomes connected to a sophisticated knowledge base and a linguistic a logic a more human-like characteristic will have been achieved. However the very primitive nature of present machine vision systems cannot be overstated despite the intriguing successes that many research units in this area are achieving (see for example the efforts of the University of Edinburgh in this area or a long list of groups involved in related work).
The classic 1959 paper entitled 'What the Frogs Eye tells the Frogs Brain' by Jerome Lettvin and colleagues makes it clear that neural processing of visual signals starts in the vertebrate eye. The eye does not transfer a digital image like a modern camera. Information of a kind is already extracted from the light input at the level of the retina.The pre-processed visual signal encounters yet more neural architecture in the lateral geniculate nucleus and then visual cortex at the back of the head. The visual cortex then has neural projections to brain regions where even more processing is presumably occurring. Such architecture suggest that some kind of distributed processing might be at work when we handling sensory input.
[ See The paper is 'What the Frogs Eye tells the Frogs Brain' in PDF form along with Two Remarks On The Visual System Of The Frog. For anatomical illustrations of visual processing and further discussion see Ben Best's The Anatomical Basis of Mind and in particular, Basic Cerebral Cortex Function with Emphasis on Vision, Basic Cerebral Cortex Function Other than Vision, Areas Supporting Cerebral Cortex Function]
Notwithstanding the complexity of the brain, it is amazing that we can actually 'see' anything. The fact that the worm or the mole burrows through the earth and does not need our sense of vision is understandable. However it is amazing that we have and tiny flies have very powerful optical signal processors that gives an ability to navigate the world. With the sheer complexity involved is there any wonder that the unproductive quasi-spiritual mind-brain duality musings arose? Although there is of necessity a neural correlate of 'conscious seeing' within the brain there is as yet no concrete picture of what that activity is. Emulation of human vision will be an extremely complex task especially if no single brain region decodes visual information but instead a kind of distributed processing handles a 'logic' of visual information. Development of electronics to mimic this kind of processing is far from trivial problem especially when we really do not understand the biological process (see Chips That Emulate Functions of Retina). When we then go on to consider how disease modifies the functioning of the various cortical areas associated with visual function to produces hallucinations for example the task of understanding the human condition becomes even more complex. [ See Preliminary fMRI Evidence of Visual System Dysfunction in Parkinson’s Disease Patients With Visual Hallucinations ]
The need for multiple abstract forms of human logic
If we were solitary creatures spoken language would be largely redundant. The need for language and its associated logic seems to be exclusively social. Consider the primitive language of bird song and its possible roles in the occupation of territories, feeding of young, mating and alarm. There is clearly much to be gained in reproductive terms by simple linguistic codes. There can be absolutely no doubt about the survival and reproductive advantage of developing more sophisticated communication codes both at the level of the child and adult. It seems likely that specialisation of individual human roles within communities is directly related to a logic of natural language. The cognitive abstraction of tasks and the ability to use spoken language to apportion them to particular individuals eventually made possible the development of technologically and socially complex societies.
In the modern world the learning of logic by individuals is at its most powerful in the context of description or data. Aspiring design engineers, scientists, doctors, law makers do not therefore attend non-numeric formal philosophical logic courses to be qualified to work in their profession. They do not undertake courses in the Logic of Belief Revision, Hybrid Logic, Fuzzy Logic, Linear Logic, Connexive Logic, Epistemic Logic, Modal Logic, Temporal Logic, Informal Logic, Second-order and Higher-order Logic, Propositional Dynamic Logic and so forth. The fact that such systems have been formally described by logicians however points to the overall complexity of human logic that we intuitively develop through education to become legally competent and professionally functioning adults.
In practice the medical student, for example, learns an extremely complex diagnostic or fault finding logic based on observation and natural language and also acquires a very large and diverse data set to make the logic useful. The qualified doctor doctor then becomes intuitively aware of the the type of reasoning that she should test without recourse to the formal statements of a philosophy encyclopedia. Intuitive logic of the kind used by humans in most waking situations probably stems from the auto-recollection or auto-association of sensory input with existing stored information. (For an elaboration of this notion watch Brain science is about to fundamentally change computing by Jeff Hawkins). In this way human logical abilities may be formed by pattern recognition. By learning the logic of what patterns or movement 'mean' within images we apply logic of comparison with other situations. In natural language the logic or rules or patterns in one situation can be applied analytically in other conditions.
Education in critical thinking as well as the accumulation of descriptive beliefs in medical, legal, financial domains for example therefore potentially has great value. It remains to be determined what will be the productive outcome of the very formalised systems of logic that can be read about in sources such as Wikipedia and the Stanford Encyclopedia of Philosophy. At present these attract much interest in computer-based Artificial Intelligence research and so hold out the prospect of changing what machines can do for we humans (see also the very readable What is Artificial Intelligence? by John McCarthy). It was only in 1854 that George Boole gave us Boolean Algebra which has now become the basis for all computer calculation. The prospects for further objectification of human logic therefore seems a tantalising prospect.
In situations such as engineering design, mathematical and numerically encodable procedural logic can be used proactively or predictively to facilitate development of the appropriate descriptive model of a wall, a roof structure, a bridge, or a machine. In that sense the logic of applied mathematics has objectivity. The objectivity comes not only from the logic of the computer encoded equations but also in the measured or descriptive numeric values used. For the engineer who generates complex models or computer-based models or representations of material objects the logic of advanced mathematics needs to applied. If mathematical logic is either inadequate or absent there is a limit on the creativity of the designer, architect or computer programmer. Formal logic is now an indispensable part of creativity in these particular activities. Despite the fact that intuitions form the essence of creativity there is a limit to what complexity and innovation an architect, for example, can introduce without recourse to the structural engineer's calculations or logic. (We could of course have strikingly innovative buildings and bridges collapse into abstract formlessness)
The logic that we use in everyday speech is intuitive in the sense that we do not consciously consult a rule before formulating a sentence. However these rules exists in abundance. They are of course arbitrary as they vary between languages. The more educational time is devoted to such natural language rules the more embedded those arbitrary rules become within us. It is an irony that the wrote learning of all kinds both in linguistic and procedural logic would seem to produce a feeling of intuition rather than a set of rules and so gives our consciousness the illusory detached quality by which the dualist's are enchanted. Consider the procedural logic of touch typing at a keyboard or of driving a car. When the agent becomes 'educated' in these matters she feels as if the semi-automatic implementation of the procedures are in operation. The experienced driver's main conscious focus could for drift way from clutch pedal and gear stick to a favourite song on the radio and so have a more diffuse conscious state with partial recognition of multiple simultaneous inputs and motor outputs. Patterns of learned limb movement, of optical and auditory recognition have a logic associated with them. However the logic might be one of automated pattern recognition through memory embedding.
Creativity and the associated understanding from which it emerges is in that sense is either the formulation of new rules or the application of those that we already have. The highly practiced architect has embedded within her a large body of 3-dimensional language rules that allow her to read the prose of a design drawing and use that language to develop new prose. The musical composer has a highly developed though intuitive mathematical rule set expressed through the language of instrumental voices which fortunately is able to transcend the logic from which it is derived. In such ways logic is related to pleasure as well as physical survival. In this way the mathematician and scientist can even come to see the structure of maths and scientific theory as beautiful.
Automation of Logic
If survival and pleasure are both related to logical rules of many kinds then automated numerical simulation of such rules could have profound effects on us and our societies. The question then arises as to how far computer systems will progress towards 'intelligence' in the human sense.
Automated or computer based grammar checking is one interesting aspect of numerical analysis applied to natural language which we already routinely use. Although as yet imperfect, grammar checking is surprisingly effective despite the fact that the methods employed are different from that which we humans appear to use. Will the day ever come however when a machine (computer) will go further and pass the Turing Test and converse naturally with us? Wikipedia says "in order to pass a well designed Turing test, the machine would have to use natural language, to reason, to have knowledge and to learn". Clearly the machine would to have knowledge in the sense that we know it, so that the context in which the words were applied could be known. If the Turing test could be past by a machine this web site would become redundant. We would simply converse with the machine.
As yet machine 'intelligence' can only be applied in very precisely defined situations such as database applications, chess playing, automated design, and industrial machine-based product inspection using vision systems. In the matter of chess playing solutions, creativity of a kind, has effectively been designed. In a more practical sense the electronics industry already has gone some way in electronic design automation (or EDA). [ Go to http://www.freepcb.com/ and run the demonstration to see automated logic in operation for yourself.]
Can a machine demonstrate recognisable creativity however? Clearly if a chess playing computer, using the deterministic rule set of the game, can formulate a novel winning strategy then puzzle solving or creativity in one sense that we commonly used is being applied. (This still applies despite the fact that a human does not apply the enormous number of calculations that are used by machines.) It is indeed difficult to describe a logical difference between novelty and creativity.
Given a set of customer specified input requirements could a machine develop a pleasing and practical new building design? If an automated system could win the RIBA Stirling Prize for architectural design then an ancient form of human creativity would indeed have been encoded. In music, automation of composition to give pleasing results would surely be recognised as creative even amongst non-chess players.
Logic will become even more important in the future
Consider a future situation in which the machine could monitor your baby as you slept, wake you up in the morning, control the contents of your fridge and food cupboards, order varied but healthy food supplies for you, regulate the temperature, lighting and shading of your house according to your preferences, run your bath, clean your house, mow your lawn, lock your house, communicate burglar and fire alarms, stun unrecognised intruders, inspect your house for maintenance, water your flowers, plan your garden, remind you about appointments, pay your bills, make a tax return on your behalf, read a selection of newspaper articles that you might find interesting, find you a new job, order new videos for you, play a large variety of games against you, compose original music for you, design your new house, remotely regulate your car driving, record all of your movements, estimate the calorific value of your physical activity with motion sensors, monitor your weight, physiology and behaviour to determine whether or not you were becoming unwell or even bored, record, transpose and analyse everything you said and wrote, introduce you to potential partners and friends, send social messages to your friends about your activities, choose your holiday destinations, and arrange your cremation when the implanted sensors had detected your death.
In such a situation our understanding of the nature of human logic and belief would become more, not less, important. The methods we use to revise beliefs would then be of particular importance as they are at the essence of human learning and computational methods. Not surprisingly the logic of belief revision is now an area of active interest for philosophical researchers.
On the Nature of Belief
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