INTRO TO LOGIC Here follows the first broad public issue of the Machine Certainty Theorem (MCT), one of the deeper theorems in Information Theory. There are two fundamental aspects to any theorem or proof, the LOGICAL FORM, and the CONTENT. The logical form can be expressed with out the content by replacing the various words and phrases in the proof with variables that have no meaning. This allows the logical form of the proof to be studied independent of its actual application. Once the logical form is verified, then the variables can be replaced by the meanings they stand for, and application of the proof with its content can be studied independent of its logical form. Any proof has at least three parts. The ASSUMPTIONS, the LOGIC, and the CONCLUSION. The logical form of the proof consists of all three parts in abstract variable form, as described above. The content of the proof also consists of all three parts in the concrete form where all variables are replaced by their intended meanings. The Machine Certainty Theorem states that a space-time machine can't be certain of anything, yet a Conscious Unit can, therefore a Conscious Unit is not a space-time machine. Before I get on with the formal presentation of the Machine Certainty Theory, I would like to provide a small sample proof to explain the various parts of what you are about to see to those who have little training in formal logic. In this case I will work backwards from an actual argument in concrete CONTENT FORM, to its abstract LOGICAL FORM so that you can see how the process will be reversed when we get to the actual proof. Consider the following argument. 1. Joe is a Christian. 2. All Christians believe in Hell. 3. Therefore, Joe believes in Hell. Q.E.D. Q.E.D is Latin for Quite Easily Done this is placed at the end of the proof to demark where the proof ends and that the conclusion has been proved. (Actually QED stands for Quod Erat Demonstrandum, 'that which was to be demonstrated'.) All proofs contain three parts, the ASSUMPTIONS, the LOGIC and the CONCLUSION. The conclusion is true if and only if the assumptions are true AND the logic is valid. If either the assumptions are false or the logic is invalid, then the conclusion may be false (it could still be true though, you don't know.) For example, it is clear from the argument above, that if Joe is not a Christian, or if some Christians don't believe in Hell, then the conclusion that Joe necessarily believes in Hell becomes indeterminate, he may or may not. A properly presented proof would show all three parts, assumptions, logic, and conclusion, clearly marked so that no confusion could result. The purpose of first presenting the proof in logic form devoid of meaningful content is to verify or validate the LOGIC part of the proof. Once that is accomplished, then the proof must be presented for a second time in CONTENT form, so that the assumptions and conclusion can first be UNDERSTOOD and then their truth verified or argued. One first verifies each of the assumptions in turn. If all of the assumptions check out to be true, then the conclusion must be true if the logic is also valid. One then looks to see if the conclusion actually fits with actuality. If it does you are finished for the moment. If it turns out the conclusion is observably false, then either the logic was invalid or one or more of the assumptions was false. In the above example, there are two assumptions. 1. Joe is a Christian. 2. All Christians believe in Hell. There is one conclusion, 3. Joe believes in Hell. Normally in a more complex proof there would be more statements in between 2 and 3 which would be partial conclusions on the way to the final conclusion, but in this case the logic is so simple we go directly from lines 1 and 2 to line 3 with a logical form called Modus Ponens. Modus Ponens is a fancy Latin phrase meaning 'If A implies B, and A is true, then B is true too.' (Actually Modus Ponens means 'Mode that affirms') For example, 'If being a dog implies being an animal, and Joey is a Dog, then Joey is an animal. Modus Ponens can be compared to Modus Tolens, another fancy Latin phrase meaning 'If A implies B and B is false, then A is false.' (Actually Modus Tolens means 'mode that denies'.) For example, "If being a dog implies being an animal, and Jane is not an animal, then Jane is not a dog." Notice that if Jane IS an animal, one cannot tell if Jane is a dog or not, because it is NOT true that if A implies B, then B implies A! 1. "Joe is a Christian" can be symbolized as "J -> C" which says "If it's Joe, then it's a Christian", or "Being Joe implies being a Christian", or more simply, "Joe implies Christian". 2. "All Christians believe in Hell" can be symbolized as "C -> H" which says, "If it's a Christian then it believes in Hell", or "Being a Christian implies Believing in Hell", or just "Christian implies Hell". 3. "Joe believes in Hell" can be symbolized as "J -> H" which says, "If it's Joe, then it believes in Hell" or "Being Joe implies Believing in Hell", or "Joe implies Hell". We can thus symbolize the entire argument as follows, and this is its logical form. We explain each part in the section below the proof. ************************************************************************ LOGICAL FORM OF THE PROOF 1. J -> C (Being Joe implies being Christian) 2. C -> H (Being Christian implies Believing in Hell) (1,2)[A] 3. J -> H (Being Joe implies Believing in Hell) Q.E.D (M.P.) A. (A -> B) and (B -> C)) -> (A -> C) ************************************************************************ In the above example there are two assumptions, lines 1 and 2, and one conclusion, line 3. The '(1,2)[A]' to the left of line 3 denotes that line 3 was derived from lines 1 and 2 using Logical Form A which is shown at the bottom of the proof below the Q.E.D. The particular Logical Form in this case is Modus Ponens, which is denoted by (M.P.) to the left of the same line. Not all logical forms have formal names, and if not, the name or its abbreviation is left out. So how does one go about checking this proof out? 1.) Well the first thing that needs to be done is to check out and verify all the Logical Forms shown below the Q.E.D, as these are the extracted GENERALIZED statements of the LOGIC part of the proof that gets you from the assumptions to the conclusion. 2.) The next thing to do is to familiarize yourself with the assumptions and the conclusion. 3.) The next thing to do is to verify each step between the assumptions and the conclusion to see that indeed the GENERAL Logical Forms stated below Q.E.D are used correctly in their SPECIFIC application to each step of the proof between the assumptions and the conclusion. The GENERAL Logical Forms will usually be stated in generic variables like A, B and C which have nothing to do with the proof. The assumptions and the conclusion and the SPECIFIC USES of the General Logical Forms will usually be stated in letters that relate to their content, such as J, C and H (Joe, Christian and Hell). Thus one needs to be able to see that the SPECIFIC use of a particular Logical Form parallels the GENERAL use of the same form to know that the general form has been used correctly. For example, GENERAL ((A -> B) and (B -> C)) -> (A -> C) SPECIFIC ((J -> C) and (C -> H)) -> (J -> H) Where ever there is an A in the general form there had better be a J in the specific form. Where ever there is a B in the general form there had better be an C in the specific form. And where ever there is a C in the general form there had better be an H in the specific form. Don't get the C in the GENERAL form confused with the C in the SPECIFIC form. They are unrelated and are the same letter only by coincidence. In the general form the C doesn't stand for anything, it is merely a place holder. In the specific form the C stands for Christian and corresponds to the PLACE HOLDER B in the general form! Now at this point it should be possible to say with perfect certainty that the proof is either logically valid or not. There is no such thing as an uncertain proof. Either it is valid or it is not valid. This can be determined with perfect certainty before anything else is known about the meaning of the variables in the proof. Remember though that just because a proof has been prooven valid, this does not mean that the conclusion is necessarily true. This would also depend on the assumptions being true, and determining the truth of the assumptions, not the validity of the logic, comprises the main body of work in verifying the conclusion of a proof. Verifying the validity of the logic of the proof is the first and easiest step and by this time in the analysis should be satisfactorily completed. So that was a lot of work, no? But, as I said, we are not done yet. Once the logic form of the proof has been verified completely as we have just done, you next need to verify the CONTENT form of the proof. This is done by replacing each specific variable in the proof with its English equivalent so that you can see what each of the assumptions and the conclusion actually say. This is done first by providing a little table that shows what each variable means, like so. J = Joe C = Christian H = Hell Then you plug them in and you get the following. ************************************************************************ CONTENT FORM OF THE PROOF J = Joe C = Christian H = Hell 1. Joe -> Christian 2. Christian -> Hell (1,2)[A] 3. Joe -> Hell Q.E.D (M.P.) A. ((A -> B) and (B -> C)) -> (A -> C) ************************************************************************ This provides a rather sparse and pared down version of what the proof is about, but it serves to convey the meaning of each of the lines. The last step would be to take up each line of the proof and expand it into a grammatically correct full English sentence and discuss it at length. Discussion of the assumptions would involve not only their meaning, but also evidence that they are true. In general there are 4 kinds of assumptions. 1.) Logical Tautologies. 2.) Definitions 3.) Observations 4.) Intuitions LOGICAL TAUTOLOGIES are always true because of their inherent logical structure. An example of a logical tautology would be, 1.) Christian or not Christian A full english expansion of this might be, 1.) Joe is either a Christian or not a Christian. You have to be careful when presenting such tautologies to make sure that your words are defined in such a way that the tautology is true. If someone has a sloppy or fuzzy definition of what it means to be a Christian, then it might be possible to be both a Christian and not a Christian! But really he would be changing meanings in mid sentence, so its a good idea to set rigorous definitions of your words that everyone can agree on before you start an argument or proof like this one. DEFINITIONS are statements that are true by definition. An example might be, 1. All Christians believe in Christ, if they don't believe in Christ then they are not real Christians. Such a statement is true only because we say it is true, it has no other basis. There may be other people who don't believe in Christ who none the less wish to be called Christians. This is not a problem, you have the right to define your words how ever you wish, just remember that what you are calling a Christian may not include others who call themselves Christians. They will no doubt complain, but their complaints will be irrelevant to your proof, because YOUR proof has to do with Christians as YOU have defined the word. If you wish to define your words in some other way, that is fine, just make sure that everyone knows what YOUR definitions are before you proceed. No one can ever say your proof about Christians is wrong because your DEFINITION of 'Christian' is wrong. OBSERVATIONS are statements that are true by observation. 1. Some Christians go to Church on Sunday. It's true because it's true, go out and LOOK for yourself. It's not true by LOGICAL TAUTOLOGY, and it's not true by definition, it's true because someone went out and measured the phenomenon and reported back what he found. The certainty level of a observation is dependent on how many vias you use to make that observation, how many levels of symbols referring to referents before you come to the actual thing being observed. A person who is using radio telescope data to determine the temperature of some planet circling a sun 4 galaxies away, is on far less certain grounds, than someone looking at a thermometer in his back yard. Someone who goes out and just feels that it is hot outside is in even more direct contact. Observations of the external physical universe however can never be perfectly certain because all observers are using effects in themselves as observers to make conclusions about what must be out there. In this sense, 'making an observation' means 'to be the effect of an external cause' and THEN to logically compute back in time to what that cause might be like in order to have had the effect that one received. The fact that one received an effect might be a certainty, but the nature of what caused that effect can not be determined from the nature of the effect alone. This 'computing back from later effects to earlier causes' is always an uncertain process, because effects 'here' do not prove anything about causes 'there'. One can merely create a 'causal model' and hope for a dependable but uncertain world view. Observations of one's own conscious color forms, though, CAN be perfectly certain. If you see a color form mockup of red and green in front of you, there can be no denying that you see it. Anything it might be USED TO REPRESENT to you in the external universe might be uncertain, but the existence of the color form itself is certain. INTUITIONS are statements which one feels to be true because it violates some inner sense of propriety to think they aren't. This of course doesn't mean that they are true, but it does mean that if you can get agreement among a number of people who have the same sense of intuition, then you can proceed with your proof as if your intuitions were true, recognizing that the truth of the conclusion is only as as certain as the truth of your intuition. Even if you can't get agreement among others about intuitions, you can still have your proof to yourself and be satisfied with it as far as it goes. As an example of an intuition, Something can't come from nothing. Any given proof will have assumptions that consist of mixtures of the above 4 kinds of 'truths'. It is often enlightening to actually state next to each assumption which kind of truth it is. For example, A something is an object with a non empty quality set. DEFINITION An nothing is an object with an empty quality set. DEFINITION 0.) An object is either a something or a nothing LOGICAL 1.) Something can't come from nothing INTUITION 2.) Something exists now. OBSERVATION Q.E.D. 3.) Something must have always existed. (conclusion) In closing I would like to add that it is not clear that every argument can be put into such simple terms as I have laid out here, or that every assumption can be divided into the above 4 categories. Sometimes its takes an enormous reworking of the WORDING of an argument to make it conform to the simpler rules of logic. The English language is very complex and the simple Logical Form is often lost in more poetic forms of argument. People in fact will often try to hide bad logic in the complex nuances of the language, which is why it is important to break arguments down into raw logical form. However for the purposes of the Machine Certainty Theorem, the above discussion is relatively complete and satisfactory. The Machine Certainty Theory is VERY SIMPLE, so simple in fact that once you get it, it will be a BIG DOWN, because you will have been expecting all these fireworks to go off in your brain once you realize this 'Great Eternal Truth' of the ages. Your actual reaction will be more like, 'Duh, so what else is new.' However it is the application of the MCT to consciousness that will give you something to think about. Machines can't be certain of anything, consciousness can. Finally, I would like to remind you of a wise old saying. "At first they said it wasn't true. Then they said it wasn't important. Then they said they knew it all along. Which was true." Well, the guy who said that, was talking about the Machine Certainty Theorem, which is the grand daddy of all truths that people argue about with you until they convince themselves they showed it to YOU in the first place! At that point you know they got it. Homer