Blog Entry - 15th January 2009 - Philosophy - Logic

This blog entry is false


I am in the course of reading a book called Impossibility.

In the opening pages, the author touches on the Liar Paradox and seeks to use it to make some rather sweeping claims about limits to the imagined powers of imagined omnipotent beings.

This blog entry sets out disorganised thoughts that occurred to me.

They are amateur in character, and only scratch the surface (if that) of the mathematical, logical, physical, philosophical and theological ideas and theories relevant to this area.


The author considers the belief that many people have, that there is a god (or there are gods). The author does not seek to prove or disprove whether there exists a god or gods (as conceived of or defined by any people); rather, the author considers a possible argument in logic against such a god or gods being omnipotent(all powerful) or omniscient(all knowing).

The author doesn't go on to say whether omnipotence or omniscience is a necessary characteristic to be a god; although, how can one say what is necessary if the idea of a god exists only as a set of characteristics we have chosen or invented?

Nevertheless, the common description of a god, the author suggests, is that of an intelligence that is omnipotent and omniscient; so, all is known to it and all is possible for it. The author then seeks to use logic to challenge the idea that anything could be omniscient; in effect, to argue that any god you might conceive of is not above logic, and so perhaps to argue that reality comes before any god.

His argument (page 7) goes thus:-

  • We are able construct the following sentence : THIS STATEMENT IS NOT KNOWN TO BE TRUE BY ANYONE
  • If this sentence were true, then a god could not know it, according to the sentence (and assuming we include that god in the class of things referred to by ANYONE).
  • If this sentence were false, then, according to the sentence, at least one person would need to know that the sentence is true; but if someone knows the sentence is true, then the sentence must be true; but if the sentence is true, then according to the sentence, no one can know it is true: there is a contradiction.
  • The author concludes that we are therefore forced to conclude that the sentence must be true, whether we start by assuming to to be true or false, because either it is true, or somone knows it to be true, in which case it is true.
  • This, the author believes, shows there must always be true statements (or facts?) which no being (including a god) can know to be true.

Now, something felt very unsatisfactory about this argument. In particular:-

  • Is the statement paradoxical, or can we actually interpret it rationally and without contradiction?
  • Just because we can conceive of a paradoxical statement, does not mean that such paradox exists in reality.
  • The sentence feels empty of content (by being self referential) so I don't personally feel forced to conclude that statement proposed above must be true.
  • I always exercise caution when anyone asserts something to be true, without a hint of doubt. Unqualified words such as This shows or there must always make me want to proceed cautiously.

What is a paradox? and other things

Before I proceed, I have set out in the appendixes to this entry some links to background concepts.

A useful discription of a paradox can be found here:

A paradox is a statement or group of statements that leads to a contradiction or a situation which defies intuition; or, it can be an apparent contradiction that actually expresses a non-dual truth (cf. Koan). Typically, either the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. The word paradox is often used interchangeably with contradiction. Often, mistakenly, it is used to describe situations that are ironic.


The main paradox examples often arise out of self-reference, so that concepts in a sentence also encompass the entire sentence.

For instance, the mind-boggling Curry's Paradox where the following examples are given:-

  • If a man with flying reindeer has delivered presents to all the good children in the world in one night, then Santa Claus exists.
  • If this sentence is true, then Santa Claus exists.

The paradox is purported to prove that Santa Clause exists, or at least to show some flaws in formal logic.

The reasoning being, again, that if we take the word sentence as pointing to the words If this sentence is true, then Santa Clause exists, and we further conclude that the overall if-sentence is true (this is the difficult bit to comprehend), then we must conclude that Santa Clause exsists.

The main thrust of my thoughts is that:-

  • One cannot create truth (or falseness) out of thin-air. These sentences could be seen as definitional sentences, rather than truth-bearers.
  • A concept of a sentence which includes itself, seems empty. How can an empty thing be a truth-bearer?

This statement is false

The author's example is perhaps a version of the simpler sentence - THIS STATEMENT IS FALSE - which creates, it is argued, a paradox, because: if it is false, then according to the sentence it is true; and if it is true, then according to the sentence it is false. This case been termed a vicious circularity or infinite regress.

An alternative formulation, to help understanding, is : The next statement is true. The previous statement is false.

I set out below some challenges to this, based on different possible interpretations of the words:-

Challenge 1 - Just FALSE (Definitional Solution)

  • We have a set of words on the page THIS, STATEMENT, IS, FALSE, which we have to interpret.
  • Do we interpret the word THIS as being a reference to the set of words THIS STATEMENT IS FALSE?
  • Do we also interpret STATEMENT as also referring to the set of words THIS STATEMENT IS FALSE?
  • If THIS and STATEMENT are referring to the same thing, why not just say THIS IS FALSE?
  • If we were to say THIS IS FALSE, what actually are the words THIS IS adding?
  • Why not just say "FALSE"?
  • Accordingly, one possible interpretation of THIS STATEMENT IS FALSE is that it is equivalent to simply saying FALSE; it is simply a reference to the abstract concept or value of false-ness.
  • I.e. we could just interpret these words as being the same as: THIS STATEMENT HAS THE VALUE 'FALSE'.
  • Or in other words, the statement is really acting as a defintion, as opposed to a statement of fact.

This reasoning finds some parallels in in computer programming (which has sometimes been called the practice of demonstrating proofs).

In computer programming, a statement is a collection of one or more tokens (non-white-space) which the computer acts upon in a prescribed way.

In most computer languages the token false, means the abstract value of falseness, which is often represented internally in a concrete way, such as by the number 0; whereas true might be 1.

So a computer program with the statement:-

a = false;

sets the varable a to hold the value false.

So how would we represent the words THIS STATEMENT IS FALSE as a computer program - i.e. how would we implement it in a real way, in order to demonstrate it.

This very problem of implementation may give us a clue as to whether or not those words themselves disclose any meaning at all.

Let us assume the token == means is equal to, and -!- refers to the concept of not.

So we could try to represent the statement as:-

if (a == !a)
	//do something...

Here we are saying if (a is equal to NOT a) or if (a is NOT equal to a).

In computing that will always evaluate to false.

Challenge 2 - No self reference

  • Do we interpret the words THIS and STATEMENT to be referring just to the words THIS STATEMENT, and not the words IS FALSE. I.e. to the abstact concept of a STATEMENT.
  • In which case do we interpret the words IS FALSE to be independent of THIS STATEMENT?
  • In which case, the interpretation is that IS FALSE applies to an abstract THIS STATEMENT, so all we are really saying is that some abstract STATEMENT is FALSE - which is not a paradox, because we have removed the self-reference.
  • I.e. it would be the same as saying STATEMENT X IS FALSE, where THIS STATEMENT is STATEMENT X.
  • Otherwise you would in effect be saying FALSE IS FALSE, which is stating the obvious, and is perhaps a tautology.
  • The underlying proposition here is perhaps that you cannot create truth-ness or false-ness out of thin air, pulling yourself up by your bootstraps.
  • In other words, can a statement meaningfully assert something about itself, or are true and false properties of a statement, and not something a statement can create by itself?

Challenge 3 - Meaning of statement

  • The third challenge I have, which is really a variant on the first, is whether we are miss-using the word STATEMENT, as commonly defined.
  • The common definition of a STATEMENT is a collection of words asserting some fact about reality.
  • Does this mean, then, that it must refer to something (real or postulated) external to the statement itself?
  • And this is my point: if a STATEMENT refers simply to itself, then could it be argued that it is a statement without content (i.e. that refers to no thing about reality), and therefore not a STATEMENT (as defined) at all.
  • So in our example THIS STATEMENT IS FALSE, what is the factual content of the STATEMENT, before we even get to the words IS FALSE?
  • Arguably, none: the word STATEMENT does not refer to anything factual. The words THIS and STATEMENT could be seen as simply being place-holders for something factual assertion we have yet to insert; they are empty containers for or pointers to other ideas.
  • Accordingly we could view the words THIS STATEMENT IS FALSE as simply being a template we use when we have something factual to talk about.
  • Alternatively, how can you make a "STATEMENT" that has no content? To put it another way, could we argue that that something that defines itself purely by reference to itself actually cannot count as a STATEMENT at all? I.e. the no content words THIS STATEMENT are not capable of even being made a subject of true or false values?
  • Can you come up with a definition of STATEMENT which would encompass the abstract contentless sentence THIS STATEMENT IS FALSE.
  • If we are to argue that the words THIS STATEMENT IS FALSE do have some substance, that they perhaps refer to an abstract concept of STATEMENT, then perhaps we need to look at how the statement is physically implemented in our brains, or a computer, in order to be clear what we are referring to in the abstract.

Challenge 4 - Sequential Interpretation

  • In interpreting this statement, is the only meaningful way to interpret it to take a sequential approach, and then loop back, feeding in the previous value?
  • I.e.

1 we store a truth-value for the sentence this statement is false.

2 we start with the value true

3 we apply true to this statement is false

4 the value becomes false

5 loop back

6 we apply false to this statement is false

7 value becomes true

  • In which case the whole algorithm outlined above is itself perfectly rational, if we take the sentence this statement is false to be an inverter (NOT-gate).
  • The potential paradox may only arise if we assume that it is all happening simultaneously, which brings us on to the next challenges.

Challenge 5 - Fuzzy Logic

  • There only exists a contradiction, if we cling to the concepts of true and false as being the only concepts we can apply to statements.
  • In fuzzy logic, there are introduced two further concepts: (1) that something can be somewhere between true and false, and (2) that everything is a subset of everything else (every concept includes every other concept, to some degree or other).
  • Accordingly, one might interpret the sentence THIS STATEMENT IS FALSE as expressing the fundamental concept of a fuzzy set, such that the STATEMENT is a member of both the set of true and false things.

Challenge 6 - Imaginary numbers and Quantum Electro-Dynamics

  • There may be something to learn from the concept of complex numbers (the square root of -1).
  • The square root of -1 is both: -1 x 1, and 1 x -1 - at the same time (using time in a loose sense).
  • Could we interpret the words THIS STATEMENT IS FALSE, as expressing the same concept in some way. It is a bit of a strain I suppose, but could we argue that the sentence expresses the concept of something that is both false and true at the same time?
  • complex numbers do seem to have a relation to reality, in that they have been successfully used to provide models for the phenomena seen in quantum mechanics, where one of the more convincing ways to explain some observations, such as interference patterns generated by a single photon, has been to assume that a photon (before it is detected) exists in multiple contradictory positions simultaneously, called a super position, and complex numbers have been a useful tool to represent and reason about these superpositions.
  • If you like, a second dimension in logical space, where multiple true statements (or positions) co-exist simultaneously.
  • I like to imagine superposition as each state of tha photon being semi-opaque, so that each occupies the same space (in a logical dimension) in some much more complex universe :-
  • Thus, could we argue that THIS STATEMENT IS FALSE is a super-position of true and false, waiting to be measured (interpreted), and each time you do, something different results.
  • I.e. by stepping outside the single dimension of true or false, the paradox is no longer viewed as a paradox.
  • Or to put it another way, even if we say that THIS STATEMENT IS FALSE it self-contradictory, does that mean that it is a paradox? The fact that we can conceive of it, and reason about it, means that it is not beyond reason?
  • I do find my own argument difficult to grasp, so I must ask myself whether that is because the argument is meaningless (in all likelihood), or whether I myself am not capable of grasping it?

Challenge 7 - Unknown Unknowns

Finally, there are the new methematical, logical, philosophical, physical, or theological ideas we perhaps have not yet thought of or even thought might exist:-

As we know, There are known knowns. There are things we know we know.

We also know There are known unknowns. That is to say we know there are some things we do not know.

But there are also unknown unknowns, The ones we don't know we don't know.

D.H. Rumsfeld

Feb. 12, 2002, Department of Defense news briefing

The Cretans and The Barber

What is possible

Lets try to ask the author's question in a way which is hopefully a little more concrete: is there anything which we can conceive of as a meaningful idea, which an omnipotent god could never demonstrate in practice (i.e. never do)? Is it the case that our concepts can only be meaningful if they can be demonstrated in some universe or other (otherwise they are just gibberish)?

In other words, do we take the author's argument to mean that an omnipotent god should be able to do everything that we are able to conceive of (that is not meaningless?), however contradictory?

There are a couple of variants on the liar paradox puzzle which get mentioned in this context.

The Cretans

Firstly Epimenides, a Cretan, stated: "All Cretans are always liars.".

Possible interpretations are:-

  • There is no paradox. Either, not all Cretans are liars; there is at least one Cretan who tells the truth; but it cannot be Epimenides. Or, Cretans are not always liars; they sometimes tell the truth, they sometimes lie; on this occasion it was a lie.
  • The statement is true; in which case Epimenides, being a Cretan, could not possibly have said those words.
  • There is a vicious circularity (paradox). To get a vicious circularity, we need to assume that the Cretans collectively are either always liars or always tell the truth, it must be one or the other. Then, if Epimendes is lying, the statement must be false, so the opposite (that "All Cretans always tell the truth") must be true (according to our assumption); but if the opposite is true, then Epimendes must be telling the truth, which means that the statement must be true, which means that Epimendes (a Cretan) must be lying, taking us back to the start of this sentence.

The first interpretation is not a problem.

For my purposes, I am interested in the second of these three interpretations: what is possible in reality.

The third interpretation requires a very specific assumption to be made, and the challenges in the previous section might be brought to bear on this. E.g. that we can only adopt this interpretion meaningfully on a sequential basis, feeding back in the previous value; or we adopt a superposition approach.

The Barber

A similar paradox, is the Barber paradox:-

Suppose there is a town with just one barber who is male. Suppose that every man in town is clean shaven. Suppose the barber obeys the following rule: He shaves only those men (including himself) who do not shave themselves. Does the barber shave himself?

I have re-cast it as a supposed legal obligation: The barber must shave those (including himself) who don't shave themselves. The barber may only shave those (including himself) who don't shave themselves.

Again, this leads to a vicious circularity:-

  • If the barber does not shave himeself, he breaks the first rule, because he must shave those who don't shave themselves.
  • If the barber does shave himself, he breaks the second rule, because he can only shave those who don't shave themselves.

Challenge 1 - No-Content

  • Is there a "no content" challenge again.
  • Is this equivalent to : A man says that he is lying. Is what he says true or false?, where it does not say what he is lying about.
  • Self-reference is empty. I.e. can we say that a stentence can be both object and predicate?

Challenge 2 - Conflicting Rules

  • Is there anying wrong in principle about inventing conflicting rules.
  • It seems quite possible to set up two seemingly conflicting statements.
  • The question is, whether you can implement them in reality. And what conclusions you draw about reality (or your concepts) if you are (or are not) able to implement them, or how you implement them.
  • For instance, we could implement the barber example, whereby the barber has to test each rule in turn, and can only proceed if both are satisfied, so that in effect the barber would freeze - jumping from one rule to the other infinitely. I.e. again the conflicting rules are implemented in a "serial" fashion, one rule then the other.

Challenge 3 - Circularity

  • Could we argue that these are in fact circular reasoning.
  • If we set up a situation that "A is impossible, you can only do B. B is impossible, you can only do A.", and then ask "Is it possible for a god to do A." we have avoided the problem, by hiding it away in the terms "impossible" and "possible", which are left to the poor old reader to interpret.

Challenge 4 - Superposition

  • Again, could we imagine additional dimensions in logic, analogous to the superposition approach, so that we create imaginary logical values which are neither true nor false or anywhere between. I.e. find a deeper subtlety to our concepts of logic.

Back to Omnipotence (Wild Speculation)

So the question, which I think the author of Impossibility was perhaps really driving at, is whether the concept of omnipotence ecompasses the possibility of creating a universe where it was possible for a Cretan to tell the truth when it is also impossible for a Cretan to tell the truth, or to create a computer program where false = true, or a robot that can only lie and only tell the truth, or a world where the barber shaves himself and yet cannot shave himself.

Does the concept of omnipotence beat the concept of impossible, or are both concepts meaningless except in relation to a specific context or circumstance? In order to give any meaning to words like impossible you may have to be more explicit about the conditions - i.e. we are defining properties of a universe with words like "impossible, just like Einstein did when he said that information cannot travel faster than the speed of light.

Is our concept of omnipotence really about a god resolving the conflict one way or other other, or do we insist that our imagined god must be able to create a situation where a logically impossible contradiction exists.

Are we, as mere mortals, even able to conceive of what a logically contradictory situation might physically look like? For instance, as noted, the concept of superposition might be one way of designing your universe to implement it.

If you cannot think of a way for a god to do it, is that because you are not omnipotent? I.e. is it reasonable to use your inability to solve the problem, as a basis for concluding anything about the universe? Perhaps our ability to reason about these things is constrained to our physical limits, and perhaps the limits of the logical foundations of this universe? Perhaps we only think inside this universe, inside this box, so we are unable to see other possibilities for matter, and mathematics and logic.


Irrespective of your beliefs about a god or gods, and whatever you are arguing for against, I would suggest that you need to put your arguments to the most severe test, particularly if you are making bold assertions like "there must always, if you are going to be honest with yourself; and I don't feel that the author has been hard enough testing his own arguments. The same critism can be made of this blog entry of course.

All of the above feels fairly abstract and distant from reality; it does not feel substantial enough to be a basis for arguing about the putative powers of imagined (or unimaginable?) beings.

Fundamentally, this blog entry, the liar paradox, and any discussion about the subject, is constrained to concepts and words in our minds, and it is incumbent on us, I would suggest, to at least explain in some detail how we are actually going about interpreting these types of statements. Something which I felt the original author did not go far enough with.

Furthermore, the implementation of these concepts into practical demonstrations is perhaps also what is needed (as a computer program or physical demonstration), if you are going to argue for any particular interpretation; and in this implementation process, it may be found that the origninal concept was actually meaningless (if you are not able to demonstrate it at all) or actually rational (if you are able to provide a physical implementation).

Something is paradoxical perhaps in the sense that it discloses no "reason". Contradiction I would say is not necessarly without reason, even it it produces strange results. Where I struggle with the various "paradoxes" listed above, is when they become self-referental: i.e. if you try to reason about truth or falsity in isolation from anything else, as pure abstract concepts, they seem to disappear.

Personally, I always feel that any attempt to make a definitive statement about the nature of the universe, or beyond, potentially suffers from the limitation that I am a finite human being, with finite processing cababilities; and I feel like there may be absolute limits on what ideas my mind can understand, and on the scope of my imagination, in addition to those limits which arise naturally from the fact that I do not have sufficient time to study in detail all that is known or all ideas that have been generated to date.

Accordingly, I think any attempt to prove or disprove vague qualities of even vaguer concepts such as a god, using this kind of reasoning, is going to feel unsatisfactory. You just need to read Einstein's theory of relativity to see how complex and humbling the universe is; and with that in mind, it feels unconvincing to use a few simple lines or logical puzzles to reason about imaginary all-powerful beings.

I like Bertrand Russel's quote in the Philosophy of Logical Atomism. The note of desperation at the end says it all:-

"That contradiction [Russell's Paradox] is extremely interesting. You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of words is just noise without meaning."

Appendix 1 - Relevant Links

These are some possibly relevant Wikipedia links I found interesting for background reading:-

  • Logic : "Logic is the study of the principles of valid demonstration and inference."
  • Demonstration : "A convincing demonstration (within the accepted standards of the field) that some statement is necessarily true."
  • Fact : "Generally, a fact is defined as something that is true, something that actually exists, or something having objective reality that can be verified according to an established standard of evaluation."
  • Statement : "In logic a statement is a declarative sentence that is either true or false. Strawson advocated the use of the term statement rather than proposition, such that two declarative sentences make the same statement if they say the same of the same thing."
  • Proposition : "In logic and philosophy, proposition refers to either (a) the "content" or "meaning" of a meaningful declarative sentence or (b) the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence. Propositions in either case are intended to be truth-bearers, that is, they are either true or false. The existence of propositions in the former sense, as well as the existence of "meanings", is disputed. Where the concept of a "meaning" is admitted, its nature is controversial. In earlier texts writers have not always made it sufficiently clear whether they are using the term proposition in sense of the words or the "meaning" expressed by the words. [1]To avoid the controversies and ontological implications, the term sentence is often now used instead of proposition to refer to just those strings of symbols that are truth-bearers, being either true or false under an interpretation. Strawson advocated the use of the term "statement".
  • Paradox : "A true statement or group of statements that leads to a contradiction or a situation which defies intuition; or, inversely, it can be an apparent contradiction that actually expresses a non-dual truth (cf. Koan)."
  • Liar Paradox : "This sentence is false or The next sentence is false. The previous sentence is true. These statements are paradoxical because there is no way to assign them a consistent truth value."
  • Fuzzy Logic and Fuzzy Set : "In fuzzy logic the degree of truth of a statement can range between 0 and 1 and is not constrained to the two truth values."
  • Complex Number : Mathematics involving the square root of -1.
  • Superposition and Quantum Superposition: A concept in quantum electro-dynamics, that multiple contradictory states can co-exist. "The principle of superposition states that if the world can be in any configuration, any possible arrangement of particles or fields, and if the world could also be in another configuration, then the world can also be in a state which is a superposition of the two, where the amount of each configuration that is in the superposition is specified by a complex number."
  • Tautology "A propositional formula that is true under any possible valuation"

Appendix 2 - Reality and Mind

Some working assumptions I am making:-

  • There is reality, which simply is. You have no choice in the matter; reality forces itself on you.
  • Whatever we think or believe, however fervently, devoutly, does not change reality.
  • You can only know reality relatively; relative to something else in reality.
  • You may be able to manipulate reality, but only in accordance with the laws of reality.
  • A heavily filtered, modified and selective view of reality is presented to you through your eyes and other senses, and the mental apparatus for processing such sensory data and modelling reality.
  • There is no such thing as true or false, in reality, just what is. true and false are simple concepts we use to judge other concepts against reality. They are not the only concepts we have for judging our thoughts.
  • The words concept, idea, notion refer to some internal grouping of components: images, experiences, actions, procedures, etc (there is no limit to what the components of a concept might be).
  • The concept of a dog may be made up of lots of components: images and memories of dogs we have known.
  • Our minds support concepts.
  • How they do this is something we still actively seek to understand.
  • Concepts can be much wider than reality.
  • We can create new concepts from existing concepts.
  • Concepts are malliable - we can change them, add to them.
  • We have words, made of letters (or logograms).
  • Words themselves are essentially nothing, and require interpreting.
  • The interpretation process is something our brains do, but is by no means perfectly understood. It involves linking to concepts, it seems.
  • We combine those words together to make sentences; which can lead to more complex concepts.
  • The interpretation is something we bring to the words and sentences, from what is in our mind (which has a variety of sources); it is not there in the words themselves; they are just marks on a page.
  • It is this interpretation process which is the important thing, and it is primarily personal to each of us, but it depends on evolved mental structures, rearing, education, experience, etc. We start our interpretation from somewhere.
  • If we cannot interpret a sentence, then it is either meaningless, or the problem lies with limits or failings or mistakes on our part.
  • A statement is a sentence which asserts something about reality, or a part of it.
  • Not only must a statement be interpreted, but also it must be related to, mapped on to, judged against, tested against reality.
  • The rigour, method and honesty of this judging process is a matter of much controversy among human beings, for whom polical ambitions, power struggles, and other personal agendas often take precedence,
  • We use the concepts of true or false express something about the relationship between a statement (and its associated process of interpretation) and reality.
  • There are other concepts we use as well, to test statements against reality, such as multi-valued fuzzy logic, and the concepts of complex numbers and superpositions if states, and perhaps other concepts we have yet to imagine. I.e. true and false are not the only concepts by which we can relate statements to reality.
  • The concepts of true and false, mult-valued logic, or complex numbers, are bound tightly to the interpretation process and process of mapping on to realty, for otherwise how are you to establish true or false or any other measure?
  • Relativity seems to me to suggest that this mapping of concepts to reality is also a definitional process. I.e. true is not a property, but a definition by reference to a process by which we map a statement to reality. E.g. for a statement such as "The sky is blue.", we might define sky as the the upwards direction we look in when stood on the ground, and we might chose to point a photon detector in that direction, and we might define blue as photons detected at a given wavelength, and therefore define true in this instance as the detection of one or more photons in that range. So The Sky is Blue is only meaningful and true by reference to the process of verification.
  • If we cannot map it, we call it an Axiom. In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths..
  • Axioms are tools for making guesses about aspects of reality we are unable to see; their truth is indirect, based on the verification of statements we deduce from those axioms.
  • The fact that the human mind is able to come up with axioms that may bear no relation to reality is an interesting feature of the human mind.
  • The concept of valid or invalid expresses something about a statement's relationship to other statements, through the medium of certain rules of interpretation.
  • E.g. given certain starting statements, do we have enough information to make a further given statement? The classic example is:-
All men are mortal.
Socrates is mortal.
Therefore, Socrates is a man.
  • This is considered to be invalid because the first two statements say nothing themselves about whether Socrates is a man. All men are mortal does not rule out cats or dogs etc. Socrates could be my cat.
  • The process of extracting additional statements from existing statements itself is a difficult one to capture, involving as it does creativity, lateral thinking etc.
  • The process of relating additional statements back to existing statements, of double checking, is discipline of logic and logical deduction.
  • Logic is our tool for creating and handling statements (and their underlying truth verification processes.
  • One way understanding logic is the concept of demonstration.
  • I.e. if you wish to assert something, can you demonstrate" it? Can you realise it, show it, have confidence in it?
  • So, in some sense, whilst logic may appear to be abstract, it is also rooted deeply in the world, and in one's ability to bring about or realise ones assertions, propositions, and theories, in the world, such as by actually manipulating and arranging matter (building a bridge) and making it happen, or procedurally, by showing that given assertion A, and following a set of rules, we can get to assertion B.
  • The links between physical and procedural demonstration, form some of the work of those linking our concepts of information and matter.


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