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Senast ändrad: 2014 07 02 23:24

Predicate logic — ultimate truth, or a limited, but very useful method?

(In the following text I will try to give the reader a very simplified and basic understanding of what logic is and how it works. This is no easy task, and for the sake of simplicity I will try to use everyday language instead of technical terms as far as this is possible.)


In connection with science and mathematics concepts as logic, laws of logic, inference and inference rules are often mentioned. The word logic derives from the Greek word "logos", which among other things means "reason". Wikipedia defines logic in the following way:

Logic (from the Ancient Greek: λογικη — logike) is the use and study of valid reasoning. The study of logic features most prominently in the subjects of philosophy, mathematics, and computer science.

Logic can be described as being the science of human thoughts and its laws and forms. The objects that are studied are propositions/statemants (and combinations of these) and their relations. Conventional logic is restricted to propositions, that are either true or false (like "it's raining", "this ball is red"). True/truth is usually assigned truth value 1, and false truth value 0. Such logic is called binary, since there are only two possible truth values, 0 and 1. A statement in binary logic is thus either true or false. Statements that do not meet this requirement, are not allowed in conventional logic.

For the sake of completeness, it may also be mentioned that logical systems with multi-valued logic have been developed. In quantum mechanics e.g. attempts have been made with continuous logic, where the truth scale runs continuously from 0 (absolutely false) to 1 (absolutely true). Truth value 0.5 means that there is 50 percent chance that the proposition is true, and 50 percent chance that it is false etc. The reason that these systems have received only limited interest, is that they fundamentally are based on conventional, binary logic.

The simplest form of logic is called propositional logic. There you essentially work with binary propositions and some basic relationships between propositions (logical connectives), where "and", "or", "not" and "equals" probably are the best-known. Logical expressions within propositional logic contain combinations of propositions and logical connectives, similar to mathematical expressions containing combinations of numbers (or letters representing numbers) and mathematical operations (addition, multiplication etc) and relations (equals, smaller than etc). It is possible to construct a logic algebra, called symbolic logic, where propositions and connectives are represented with letters and symbols (that two propositions are equal, i.e. have identical truth values, are often represented by an equal sign, "="). If for example the proposition A="it is cloudy" and B="it is raining", the proposition S="it is cloudy and not raining" can be written:


This expression is pronounced as "S equals A and not B", where the "¬" means "not" and the symbol "∧" means "and". Logical structures can be represented and analyzed in a very elegant and comprehensible way, using the laws of logical algebra. The logical algebra (an important form of logical algebra is the so called Boolean algebra) has gained tremendous importance in computer design and the design of computer chips and even software. And generally in all digital electronics.

Logical systems based on propositional logic (and the extended logics desribed below) are called formal systems. This concept will be explained later on.

Propositional logic is, however, a very restricted language and can only express relatively simple mathematical and other relationships (to construct an autopilot that will keep a ship on course, a problem that can be solved by propositional logic, is not on the same level as describing and understanding all aspects of what it means to be a human being). To achieve a richer language, additional logical concepts have been introduced, including among other things so-called predicates. We shall not go into details, the interested reader is referred to elementary text books on logic, but to use a metaphor, the English language would be very poor indeed without verbs (predicates). Verbs give us a much richer language, enabling us to express much more complicated propositions (Friday in Robinson Crusoe is a good example of this — "Friday hungry!"). The first extension of propositional logic is usually called predicate logic of the first order. Here the existential- and universal quantifiers are introduced, which allow you to assign properties to all individuals in a set, or to only a subset of the individuals. The language of predicate logic (of first-order) has adequate means to express and analyze a large part of the statements and arguments of mathematics and other "exact" sciences like physics, chemistry etc, and many simple arguments used in everyday life. First order predicate logic is widely used in mathematics and computer science.

The first-order predicate logic can then be extended to second-order predicate logic, third-order etc, in which each step gives an even richer language, where more and more complex statements can be expressed and analyzed. A lot of space would be needed to explain the difference between first and second order predicate logic (often just called first and second order logic). But to make it short, a second-order predicate takes a first-order predicate as an argument. Etc. In second-order predicate logic etc the predicates can have different types of attributes (attributes of individuals, attributes of attributes etc). For each new, higher order level, new types of predicates are introduced, that express properties of the lower-order predicates. This can be repeated indefinitely. For the second-order predicate logic and for higher orders the famous Gödel's Incompleteness Theorem from 1931 applies (in the following I call such systems Gödelian systems). Number theory and Euclidean geometry are examples of Gödelian systems in mathematics (thus propositional logic is not enough to formulate and analyze all statements within theses disciplines). Gödel (Kurt Gödel — 1906-1978 — was a brilliant Austrian mathematician) proved that the following is valid for Gödelian systems:

1. There exist true statements or propositions, which can be expressed in the language of the system, but that cannot be proven within the system. Thus there are truths that are impossible to prove by inference (logic) within the system itself (based on the system's axioms — premises). In other words it is impossible to decide whether certain statements are true or false. Such statements are called undecidable.
2. Moreover, Gödel showed that if a Gödelian system is consistent, i.e. free from contradictions, its consistency cannot be proven within the system itself.

It is possible to prove, by simple logic, that if a formal system contains at least one single contradiction, then it would be possible to prove any arbitrary statement within that system, even negations of already proven statements. If geometry contained a contradiction, it would e.g be possible to prove the Pythagorean Theorem and at the same time disprove it. It would also be possible to prove that the sum of angles in a triangle is 180 degrees and, at the same time, prove that this sum is any other arbitrary value (from zero to infintiy). Etc, etc. Such a formal system would of course be totally useless as we could prove and disprove any possible proposition within the system. In the real world we never observe contradictions. Therefore we do not believe that contradictions exist in the real world. For this reason it is crucial to know whether a formal system is consistent or not.

Paragraph 2 above shows that if a Gödelian system is consistent (and therefore useful), you have to go outside the system itself, and use a higher level logical system (a metasystem), if you want to prove this. And to prove that this metasystem is consistent (otherwise you can't be sure that the first proof is valid), you have to use an even higher level formal system (a metametasystem) etc, ad infinitum. Therefore we can never be sure that formal systems, with the same or higher complexity as second-order predicate logic, are consistent. The mathematicians were deeply shocked by this discovery.

The belief that logical conclusions, derived within a Gödelian system, are valid, is thus not only based on logic and reason, but also on a certain amount of blind faith — faith in the system's consistency, which according to Gödel is impossible to prove (if you want to know more, I suggest that you read about Gödel's theorem at Wikipedia).

Logic has proven to be an excellent tool for analyzing and understanding scientific and mathematical structures. That so is the case, and why, is far from obvious. To the extent we know that our logical tools are consistent (up to first-order predicate logic), then the tool is too meager to express more than rather simple statements about reality and mathematics. And if we use a more powerful tool (like predicate logic of the second-order and above), which can express more complex structures, then according to Gödel's proof, we no longer know with absolute certainty that our conclusions are correct (in this case we can't prove that the system we use is free of contradictions). Einstein once said, in the same spirit, "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality". This could be said about Gödelian systems as well. The fundamental problem is therefore, that if we restrict ourselves to first-order predicate logic, we know that our conclusions are correct (within the system). The problem then is that we can only speak of a limited part of the total reality. And if we go to higher orders of logic, then we no longer know with absolute certainty whether our conclusions are true or not.

Moreover, even if conclusions are correct within the logical system itself, we still cannot be absolutely sure that they correspond with reality. This statement has nothing to do with anything said above (Gödel's theorem etc) and is valid for all types of logic, even propositional logic. And it has much deeper implications! Whether logic reflects reality or not, is a question that logic itself cannot answer. Only experience can provide us with any form of answer. And only in a very limited sense.

Einstein also said, "How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality". The same thing could be said about logic. How do we know that the structure of reality is isomorphic (has a one-to-one correspondence) to the structure of logic? If this is not the case, there is no reason to believe that logical conclusions are applicable on reality. Even if logic seems to give us valid solutions to many problems, we cannot be sure that this generally is the case. And logic itself cannot resolve this crucial problem.

Essentially, logic is always based on certain improvable assumptions (or premises — in physics often called postulates and in mathematics called axioms — these cannot be proven because you have to start with a small number of statements before you can begin to prove anything at all — you have to have tools to manufacture tools and you can't lift yourself in your own hair). And from these basic assumptions we come up with (prove) a conclusion, by using the laws of logic (through a process called inference). In fact, logic provides no access to ultimate truth. Or truth at all. Logic is all about implication, i.e. "if… then…". Using logic, one can show that if certain statements (premises) are true, a certain statement (conclusion) must follow with absolute (logical) necessity. Let's call the premise A and the conclusion B. Using the logical method we can never prove that B is true. What we can prove is "if A then B" (A implies B, which is written A⇒B), i.e. if A is true, then B must be true. But if A is not true, well, then we know nothing about B, even if we have proved that A implies B. And, we can never know with certainty whether A is true or not. Premises (axioms etc) are per definition improvable. If the conclusion corresponds to reality (if observations and experiments confirm the conclusion) then this supports our assumptions (axioms etc) and we hold on to them and believe them to be a sound starting point for our reasoning. If observations do not match our logical conclusion, then we suspect that one or more of our assumptions are wrong and we must change some or all of them and then test the new theory on reality. This is basically how science works.

A system of the type descibed in the last paragraph is called a formal system.

The end result of this logical method (often called the deductive method) is, as we have seen, not absolute truth, but a conditional statement — if the premise (premises) A is true, then the conclusion B is true. If the premise is false, we don't know anything about B, i.e. B can be either true or false. Logical "truth" therefore is about implications, i.e. "A implies B", and does not state that "B is true". The latter is outside the capabilities of logic (and science). Logic gives us the authority to say "if my conditions are met (my axioms are true), then this and that apply", i.e. then a given statement is true. Logic never ever gives us the right to say, "this statement is true" (unless the statement is constituted by the implication itself).

A high school math teacher might say, "Today we are going to prove the Pythagorean Theorem". This gives his students the impression that he is going to prove that this famous theorem is true, which is impossible, as we have seen above. Euclidean geometry is based on five axioms (improvable statements). These axioms consist of very simple geometrical statements, which we can't prove to be true, but which appear to be so obvious that it is hard to imagine how they could be wrong. Euclide's first axiom is, "All pairs of points can be connected by a straight line". As the Euclidian space is infinite and empty, it seems impossible that this statement should be false. The other four axioms appear to be equally obvious (possibly with the exception of the fifth, which is a little bit more complex). From Euclid's five axioms the Pythagorean Theorem is now proved by very elegant logical reasoning by our high school teacher (there are several different ways to do this proof) and all the students are full of admiration. But, as said before, this is no proof that this theorem is true. No, what the teacher has proven is, that "if Euclid's five axioms are true, then the Pythagorean Theorem is true" (if A so B). But the problem is that we don't know for sure if these axioms are true. They seem to be true, according to our experience, but experience is not logic. And can lead us astray. And indeed, we know today that there are other types of geometry, called non-Euclidian geometries (where we have changed Euclide's fifth axiom).

According to Einstein's General Relativity Theory, Euclidian geometry is exactly valid only in empty space, but because Earth is a rather small planet with weak gravity, Euclidian geometry is a very good and usable approximation on and in the neighbourhood of our planet (and is used for navigation, construction work etc). Out in space things can, however, be very different. The more gravity, the less valid is the Euclidian geometry. The General Theory of Relativity, which mainly is about gravity, is for this reason based on non-Euclidian geometry (Euclidian geometry can be regarded as a special case of non-Euclidian geometry). Einstein´s theory predicts the existence of many interesting things, among them black holes. Near such a hole Euclidian geometry breaks down and gives a totally inaccurate description of space (or to be more correct, space-time).

Let us return for a moment to the issue with logic in relation to physical reality (and to total reality, which might include more than matter/energy and space/time). Even if we restrict ourselves to first-order predicate logic (where we know that our conclusions are logically correct), and even if we take into account that logic only gives us implications, and accept these limitations in the logical method, one problem still remains. How can we know that logic truly reflects physical reality? It is this reality we study with the logical method as a tool, and this study is what we call "science". Even if our axioms are indeed true, that is, corresponds with physical reality, and the logical method also provides a logically "correct" conclusion, how can we know that this conclusion, that follows from axioms and logic, reflects reality. This presupposes, as pointed out above, that logic is isomorphic to physical reality (i.e. has exactly the same structure). Even if logic is demonstrably providing accurate conclusions in many cases (conclusions consistent with our observations and measurements), we cannot know if this is universally true. We may believe so, but we cannot know. A belief might be true. Yes! But belief is not certainty and belief is not logic.

This leads us to the next problem, which could be called "the essence of logic". Let us for a moment talk about mathematics, which can be regarded as a form of applied logic. In mathematics there are mainly two basic, philosophical schools, the Formalistic and the Platonic. The Formalists argue that mathematics is created by humans, because it is we who have formulated the mathematical definitions and laws. The Platonists, on their part, argue that mathematical concepts and structures in some way seem to have their own existence. Plato himself talked about the world of ideas. According to Plato, it is the ideas that have real existence. The physical reality is just a vague reflection of the incorruptible and absolutely true ideas. Man has in this perspective not created the important numbers π (pi) and e (≈ 2,718...), or the prime numbers (numbers that can only be divided by themselves and 1 — the five first prime numbers are; 2, 3, 5, 7, 11), but Man has discovered their existence, much like Columbus discovered America. These numbers existed before we knew about them. According to the Platonists, all intelligent beings in the universe will eventually arrive at the same mathematics. It's already there, just waiting to be discovered. Many of the most famous mathematicians (of which Gödel is one) have been/are Platonists.

The problem with the Formalistic school is to find a materialistic explanation as to why mathematics is so useful when it comes to describing the physical reality (here you can read more about this — this article is not yet translated). That a randomly generated universe should have a logical and mathematical structure is not what you would expect. The Platonic discipline, also seems difficult to reconcile with a purely materialistic perspective. Where would the mathematical structures exist in the sterile universe of materialism? Before Big Bang, i.e. before our universe, in what sense did these structures exist? And how were they "stored"? "Where" were they? One answer could be that these structures simply reflect the properties of the material objects in the universe. But if so, the next question will be why all objects in our universe are connected to a very few basic numbers (natural constants) such as pi, e, etc. (which describe all objects in our universe), and why the structures and dynamics of these objects are isomorphic to the rather simple laws of logic?

We can now summarize our reasoning. Although first-order predicate logic allows us to draw firm conclusions within a logical formal system itself, how can we, as I asked before, be sure that these conclusions also apply to the physical reality? If this logic is something man himself has created, well, then the answer must be that we cannot be sure. If our logical conclusions should apply to reality, this would be unexpected and very unlikely. If instead logic in a certain sense has its own existence, i.e. is something that man has discovered and not invented, how can this be consistent with a purely material universe? Why would a randomly generated universe follow the simple laws of logic? If this is a coincidence, it must be highly improbable, not to say miraculous. On the other hand, if an intelligent Creator brought forth the universe, we would expect the universe to be mathematically and logically describable, in the same way as human machines and structures are logically describable (it is, after all, we ourselves who have ensured that they are well-structured and logically constructed). To the materialist, the immense usefulness of logic must be extremely troublesome, while, to the believer, it comes quite naturally. The fact that logic and mathematics so incredibly well reflect the physical properties of the universe suggests, according to many prominent physicists and mathematicians, the existence of a superior Intelligence behind everything.

Atheists often argue that science and logic are not only methods to examine and describe the physical reality, but are also expressions of the ultimate truth. All meaningful questions are in this perspecitve answered by logic and science. If the absolute truth lies outside the domains of science and logic, well, then they seem to be prepared to abandon the truth rather than abandon the scientific and logical method. The method has thus become more true than reality (which is exactly what characterizes an ideology), and science has been degraded from a method for unbiased investigatation of reality, to a slave under the materialistic ideology. That's why I nowadays talk about "the predicate logic people", that is, the people that argue that predicate logic can express all there is a need to express. What cannot be expressed with predicate logic, can in this perspective possibly not exist, and is thereby dismissed per definition as nonsense or pseudo phenomena. Like the infamous Master of Balliol, who was omniscient, because what he did not know was not knowledge.

My name is J-W-TT.
There's no knowledge but I know it.
I am Master of this College,
What I don't know isn't knowledge.

In general the predicate logic people deny that this is what they really mean, but still their reasoning and argumentation show that this is exactly what they mean. They cannot, even in theory, for a brief second, do the thought experiment, "what if the universe is created by a supernatural Creator, what would be the consequences in terms of scientific possibilities to explain the origin of the universe?" or "assume that there are real phenomena which are not contained in the straitjacket of the logical method, what would that lead to?". They know (or think they know) in fact that the universe cannot have been created, and that phenomena that cannot be understood by the help of logic, are impossible. They cannot exist. They know this with 100 percent certainty! How do they know that? Well, because if that were the case, we cannot logically explain the origin of the universe, and what we cannot explain logically cannot exist. Or rather, should not exist. In this way they have raised logic and human reasoning to the measure of all things. It is, for example, in this spirit that atheists often refer to Occam's Razor — according to which one must always choose the simplest possible explanation and not impose unnecessary concepts. This they do for instance when trying to dismiss creation. As if the absolute truth would be dependent on philosophical and logical principles!!! Occam's Razor is sometimes useful, when we talk about scientific models. As these are not about absolute truth, we might as well choose the simplest one of these usable models (if several models can explain the same phenomena). However, there is nothing to say that the absolute truth is equal to the easiest option. On the contrary. Within science we have seen, again and again, that so is not always the case. Sometimes the truth is rather simple. Other times the truth is very complicated.

The whole thing reminds me of a three-year old child who's got one of those boxes, where the lid contains differently shaped holes — round holes, square holes, triangular holes etc. The box includes a number of blocks with different shapes, and the task is to put the blocks through the lid into the box. It is a great educational toy for young children, training them in detecting different geometric shapes. So it's all about putting the right block in the right hole. If you choose the wrong block it won't work. Sometimes the blocks fit so precisely, that a small plastic hammer is included in the set to hammer the blocks through the holes. It is not uncommon to see that the little three-year old throws a tantrum when the round block doesn't go through the square hole. I have myself watched little kids go loose on the device in wild fury with the little hammer, when trying to bang the wrong block through the hole. And if you exert yourself enough, and possibly pick up a real hammer, it might just work. Or you can get yourself a knife and carve away bits from the blocks or the holes.

Paul Johnson writes in his book Intellectuals (Phoenix Press 2000, pg. 201) on the philosopher Bertrand Russel:

In fact the notion of Russell carrying philosophy into the world is quite misleading; rather he tried, unsuccessfully, to squeeze the world into philosophy and found it would not fit.

Isn't it just in this way that many materialists make use of the logical method? We can call it "Pete 3-year-old-method". If the reality does not enter through the "holes" of logic, well, then you become either angry and hit and hit until it fits, or appears to fit. Or, you "carve" away a piece of reality for it to fit through the "holes" of logic. The "Pete 3-year-old-method" is the foundation of the philosophy of materialism and positivism, which the mathematician L K Frank once remarked (indirectly) during a conference in cybernetics (systems theory). He gave the following excellent summary of the program of positivism:

The concepts of purposive behavior and teleology [goal-oriented behavior] have long been associated with a mysterious, self-perfecting or goal-seeking capacity or final cause, usually of superhuman or super-natural origin. To move forward to the study of events, scientific thinking had to [my emphasis added] reject these beliefs and these concepts of teleological operations for a strictly mechanistic and deterministic view of nature.

I.e., the reason for the rejection of belief in the supernatural is not that its existence has been disproved, but that it must be rejected in order to provide a comprehensive materialistic explanation of the world. For reality to be contained in the sterile landscape of logic to one hundred percent, one must simply peel away those parts of reality that don't squeeze through the narrow "holes" of logic. Materialism is thus about a circular reasoning, and does not constitute a valid logical conclusion.

The materialist's confidence in human reasoning and belief in the unlimited capabilities of logic, is really extremely contradictory. Especially for those materialists who claim that logic is something created by man himself. The foundation for materialism is the theory of evolution, according to which we developed through a struggle for life and death, in which the genes that gave immediate survival benefits were rewarded (survival of the fittest). Why would a blind evolution, that can't look ahead and plan, but only spread genes that increase the likelihood that an individual will be able to obtain offspring, give rise to creatures capable of understanding the outermost and innermost structures of the universe? How can we, in the evolutionary perspective, be so confident that our conclusions about the universe are correct? Or that our brains are at all capable of understanding the universe and its origin? Why has evolution given us the ability to create theories like algebraic topology and differential geometry (which are necessary to understand space and time)? However, if Man were created by an intelligent Creator, to the image of this Creator, that is, with many of the Creator's qualities, then there is every reason to expect that our intellect has the capacity to understand the Creation. But even then we cannot be sure of being able to understand everything.

The famous Christian philosopher and apologist C S Lewis wrote:

If the solar system was brought about by an accidental collision, then the appearance of organic life on this planet was also an accident, and the whole evolution of man was an accident too. If so, then all our present thoughts are mere accidents — the accidental by-product of the movement of atoms. And this holds for the thoughts of the materialists and astronomers as well as for anyone else's. But if their thoughts — i.e., of materialism and astronomy — are merely accidental bi-products, why should we believe them to be true? I see no reason for believing that one accident should be able to give me a correct account of all the other accidents (C S Lewis, God in the Dock, Eerdmans, Grand Rapids, MI, pp 52-53).

I believe logic in general and predicate logic in particular to be extremely useful and valuable tools within their limited areas of expertise. I have no intention, with what I have written above, to declare logic as in any way incompetent or not useful. Rather the opposite! But to use a tool outside its given area of competence, as the materialists do, is to do this tool an injustice. It's like seating a talented painter artist, who cannot play the piano, in front of a grand piano in a crowded concert hall, point a gun at him and order him to play Rachmaninoff's 3rd Piano Concerto. It would be to make a fool out of him in front of thousands of people. To try to use logic as a tool for examining the outermost boundaries of reality, and then argue, that what logic cannot see or understand or describe, cannot exist, that is to do logic a great disservice. And besides that, a person who comes with such claims is himself (or herself) extremely illogical and unscientific! Or perhaps abysmally ignorant!


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© Krister Renard