Formal logic treats of formal, syntactical relations between judgments with no regard for their empirical, mathematical or even transcendental, content. Its province does not include how such judgments come about in the first place. Since judgmental forms are abstractions from ordinary employments of concepts in humans' attempts to communicate and to gain knowledge of the phenomena, formal logic is not specifically concerned with the problem of how understanding is successful by means of either empirical, logical, or mathematical judgments. The investigation of the condition of possibility of such employments of concepts and of understanding--the "tracing back to the origin" of the whole process of abstraction--is the domain of transcendental logic.
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With Kant's claim that the mind of the knower makes an active contribution to experience of objects before us, we are in a better position to understand transcendental idealism. Kant's arguments are designed to show the limitations of our knowledge. The Rationalists believed that we could possess metaphysical knowledge about God, souls, substance, and so; they believed such knowledge was transcendentally real. Kant argues, however, that we cannot have knowledge of the realm beyond the empirical. That is, transcendental knowledge is ideal, not real, for minds like ours. Kant identifies two a priori sources of these constraints. The mind has a receptive capacity, or the sensibility, and the mind possesses a conceptual capacity, or the understanding.\n\nIn the Transcendental Aesthetic section of the Critique, Kant argues that sensibility is the understanding's means of accessing objects. The reason synthetic a priori judgments are possible in geometry, Kant argues, is that space is an a priori form of sensibility. That is, we can know the claims of geometry with a priori certainty (which we do) only if experiencing objects in space is the necessary mode of our experience. Kant also argues that we cannot experience objects without being able to represent them spatially. It is impossible to grasp an object as an object unless we delineate the region of space it occupies. Without a spatial representation, our sensations are undifferentiated and we cannot ascribe properties to particular objects. Time, Kant argues, is also necessary as a form or condition of our intuitions of objects. The idea of time itself cannot be gathered from experience because succession and simultaneity of objects, the phenomena that would indicate the passage of time, would be impossible to represent if we did not already possess the capacity to represent objects in time.\n\nAnother way to understand Kant's point here is that it is impossible for us to have any experience of objects that are not in time and space. Furthermore, space and time themselves cannot be perceived directly, so they must be the form by which experience of objects is had. A consciousness that apprehends objects directly, as they are in themselves and not by means of space and time, is possible--God, Kant says, has a purely intuitive consciousness--but our apprehension of objects is always mediated by the conditions of sensibility. Any discursive or concept using consciousness (A 230/B 283) like ours must apprehend objects as occupying a region of space and persisting for some duration of time.\n\nSubjecting sensations to the a priori conditions of space and time is not sufficient to make judging objects possible. Kant argues that the understanding must provide the concepts, which are rules for identifying what is common or universal in different representations.(A 106) He says, "without sensibility no object would be given to us; and without understanding no object would be thought. Thoughts without content are empty; intuitions without concepts are blind." (B 75) Locke's mistake was believing that our sensible apprehensions of objects are thinkable and reveal the properties of the objects themselves. In the Analytic of Concepts section of the Critique, Kant argues that in order to think about the input from sensibility, sensations must conform to the conceptual structure that the mind has available to it. By applying concepts, the understanding takes the particulars that are given in sensation and identifies what is common and general about them. A concept of "shelter" for instance, allows me to identify what is common in particular representations of a house, a tent, and a cave.\n\nThe empiricist might object at this point by insisting that such concepts do arise from experience, raising questions about Kant's claim that the mind brings an a priori conceptual structure to the world. Indeed, concepts like "shelter" do arise partly
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Observation and experimentation of the physical world lead to conclusions concerning physical phenomena (usually inaccurate!) But people don't want to admit that they are ignorant, so they stick hard to what they called knowledge after receiving university degrees. Nowadays, to prove as being knowledgeable, people used to cite all sorts of diplomas before their names.\nCollege Degrees evidently have some relative values in some fields, but they guarantee no true knowledge. Otherwise, what is the use of sending children to school?\nBut, true knowledge comes only from its true source, that is the knowledge taught by the Messengers of God. In order to gain this knowledge, man must firstly realize that he is completely ignorant, and starting a new and complete investigation of truth. He must use, not only his senses, but also his intellect, his intuition, his faith, his other spiritual abilities...under the guidance of the Divine Educator. This was the way of all great saints. In the past, they were called saints, because this education had not reached the great population. Now, this education is available to all, enabling us to become true men (no more saints), why we keep on denying it?
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To a great extent it has to do with personal choice.  Mathmatics, physics, and hard sciences are all majors that require logical and critical thinking.  Journalism, liberal arts, and arts degrees can be achieved without such logical abilities.  So while a logical thinker could excel in any of these programs, they are more likely to find frustration being surrounded by those without their abilities.  Those without the capacity for logical thought cannot excel in the sciences.  I am sure I am going to get a lot of thumbs down for this from those unable to follow the simple logic.
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Because human beings are driven by emotions as well as logic.  We see that all the time in this forum -- people hold to their beliefs out of emotion, rather than subjecting them to rational analysis or accepting such analysis from others.  In human evolution, it is virtually certain that emotions came first -- if the lion is approaching, you'd better run.  Also, doing truly objective work is damnably difficult -- even with a genius IQ, I have had to spend decades to get satisfactory answers as to how to proceed, and most people simply have neither the time nor the intellectual horsepower to approach objectivity.
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There is indeed a thread in philosophy that treats mathematical truth as representing some kind of eternal fixed world.  However, there are plenty of mathematical objects other than numbers that have a rich field of mathematics associated with them.\n\nFor example, one defines a group as being a set of elements and an operation between them, denoted '+' (a bit like 'addition', but not restricted to acting on numbers) such that:\n1) If a and b are elements in our group then there is an element c such that a + b = c\n2) There exists an element of our group O such that for every element a in our group, a + O = a.\n3) For every element a in our group, there is also an element b such that a + b = 0.\n4) If we pick three elements in our group, then (a + b) + c = a + (b + c).\n\nThis is an abstract definition, and doesn't prescribe what the underlying elements actually are.  All it says is that they must satisfy (1)-(4) to be called a group.  You can probably check for yourself that numbers do satisfy these rules, although there are many particular examples of groups where numbers don't provide enough flexibility to be considered as the underlying elements.\n\nNow, mathematicians spend a lot of time studying these, and other, mathematical objects, and there are many beautiful theorems associated with them.  These objects, and the theorems about them, should be considered as eternal as the numbers whilst at the same time being different to the numbers.
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This may be answerable through an appeal to philosophy.  Assume that all Matter is composed of intelligent organisms.  Each organism experiences inputs and makes decisions in the process of interacting with the Universe around it.  There are three "forces" behind this phenomena:\n\nThe latter force (meaning the second of two presentations):  Each organism must experience inputs one at a time; one of them must occurr after the other as the organism experiences a Choice.\n\nThe later force:  The organism may chose the first of the inputs it experiences; then the other input is delayed by the choice.  The organism may chose the last of the inputs it experiences; then the processing of the other input is delayed by the choice.\n\nThe ladder force:  This is the bit maker force: the organism must forever execute these choices one at a time as it experiences the Universe.\n\nIn physical formula, the process of choice is abstracted away and only timeless probabilities remain; so in a physical sense, time does not exist.  Time is a moral and ethical phenomena, not a physical phenomena.
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It is mistakenly imagined by some that belief in a Supreme Being as the Creator and Controller of the universe is a mere emotional aspiration, a superstition of ancient times, irrational and illogical, and exploded by modern science. It is believed that scientists (physicists, biologists and others) have erected some theory which both refutes and replaces the traditional belief in God. Such ideas have only a very superficial grounding, and are the result of ignorance or an indifference to both the fundamentals of religious faith and the scope of the physical sciences. It is a significant fact in the history of world thought that very few people have ever made it their business to refute the existence of God. The views of the universe which are considered to be anti-religious are almost all agnostic, not atheistic, that is to say, they attempt to ignore the existence of God instead of denying it. This is true of certain views of modern science as well as of the ancient non-religious theories. The universe in which we live comprises an evident system of causes and effects, of phenomena and their results, and it is possible to discuss them indefinitely and construct theories about them, giving a superficial appearance of completeness. This is done, however, only at the expense of ignoring fundamentals or claiming that they cannot be known. If one were to search for a convincing statement based on firm principles that the existence of a Supreme Being is impossible, one would not be able to find it. \nThe reason for this state of affairs is that belief in God is at once instinctive, rational, evidential and intuitional, and it is only by deliberately neglecting to consider it that the non-religious attitude is maintained. It is instinctive in that man has an innate feeling of his own inadequacy and helplessness, which accompanies him from the cradle to the grave, a feeling accompanied by the complementary desire to seek refuge and support with a being who controls all those forces before which he feels himself inadequate. We put this feeling forward as instinctive, although it will immediately be perceived that it is also evidential. The weakness of man before all the uncountable influences over which he has no control is a fact so obvious as to require no discussion. \n\nWhat is less well grasped by some who have claims to intelligence is that belief in God is fully supported by reason and logic, the principles on which all human intelligence stands. For instance, it is a basic requirement of reason that an effect cannot exist without a cause. However hard we press our mental faculties, we cannot conceive rationally of a causeless effect, and if we wish to postulate one we can only do so by temporarily putting our reason on the shelf. Reason leads us to the conclusion that just as the elements which compose the universe are effects of certain causes, the universe itself must be the effect of a cause, a cause which is itself mightier than and outside the universe. Non-religious thinkers have to ignore the origin of the universe and postulate something existing in the beginning without any known cause. This postulate is essentially non-rational and therefore unscientific, but it is a necessity for those thinkers who have unconsciously or deliberately decided not to consider fundamentals. Of these there are even some who openly proclaim their refusal to discuss or admit any metaphysical concept. This kind of attitude, however, can only be upheld by abandoning reason. Reason itself guides us inexorably to the conclusion that there is an ultimate cause, the Cause of causes, beyond this universe of time, space and change; in fact, a Supreme Being. \n\nAnother of the basic demands of reason is that diversity cannot exist without a fundamental unity. Whenever the human mind is confronted with diversity, it immediately sets to work to synthesise it into unities, then to synthesise these unities into higher unities and so on until it can go
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It relates to humans not being rational - one of the axioms of rationality is that preference is transitive - that is, if I like A better than B, and B better than C, then I must like A better than C.\n\nUnfortunately, in studies, humans often create circular rankings like the one you describe when asked to make each judgment independently - particularly when they don't think about any given judgement in depth.\n\nMore information should be available by researching "decision making" and "reasoning", depending on the timescale of the judgments - decision making pertains to snap decisions, whereas reasoning tends to involve choices with multiple seconds or minutes of thought.
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You miss, however, that I never said free will implied deity, only that lack of free will implies lack of deity.\n\nEstablish this in formal logic in the following manner:\n\nA = A deity exists\nB = Free will exists.\n\nIF (A) THEN (B).\n\nNotice that I use IF, not IIF (if, not, if and only if).  I assert that Deity is sufficient cause, but not necessary cause.  There could be a scientific principle of which we are not yet aware that could create free will without the necessity of a deity.\n\nHowever, since I establish deity as a sufficient cause (really, what kind of deity would create perfectly deterministic if stochaistic thinking beings?), if free will can be disproven, then deity is disproven by modus tollens:\n\nIF A THEN B. (axiom)\nNOT B. (given)\nTHEREFORE NOT A. (modus tollens)\n\nThis can be shown to be Modus Ponens via the contrapositive:\n\nIF A THEN B. (axiom)\nIF NOT B THEN NOT A. (contrapositive).\nNOT B. (given)\nTHEREFORE NOT A. (modus ponens)\n\nThis also means that the deific hypothesis is testable -- it can be falsified.  Show that there is no free will, and you show that there is no deity.\n\nHow to test free will?  Can free will exist?  We've already discussed this in some detail.  If you suppose the three axioms of science are valid (and to invalidate any of the three, you invalidate the ability to have any knowledge at all: Math & Logic are valid, observations are valid if based on direct sensation or math & logic, the supernatural either does not exist OR does not interfere with the natural), then free will can rationally be discounted.  Such discounting can be done as follows:\n\nThe mind is an emergent, if still not understood, property of the physical and energetic nature of the human being.  (nonsupernatural causation -- a soul would be supernatural).\n\nThe state of any physical system can be stochaistically determined from its previous state and the current inputs.  However, the past is 'past' and therefore it is determined.  The consequence is that I can predict the next state with great precision but in a few thousand states, the precision may be nearly nill, and within a million states, it is as close to zero as you can get.  (mathematical principle).\n\nThe stoichasm is a consequence of quantum randomness.  If quantum randomness is redefined as an input to the process (ie: I'm going to kill that guy but a virtual electron/positron pair form in my brain with the cascade effect of me changing my mind), then the stoichaism is removed.  The situation is now deterministicly based on the current state and the inputs.\n\nDeterminism is incompatible with free will.\n\n-----\n\nI agree that to have perfect knowledge of state and inputs is impossible.  However, the feasibility of a computation does not change its computability.  RSA encryption starts with taking two prime numbers of significant size and multiplying them together, then ditching the primes.  The security of RSA is based on the infeasibility to factor the multiple.  But I can give you an algorithm, perfectly deterministic, for factoring that number:\n\nN = P*Q.  Discard P & Q.\nMAX = sqrt(N).\nfor COUNTER 1 to MAX:\n.. if N / COUNTER is integer\n.... P = COUNTER\n.... Q = N / COUNTER\n.. endif\nendfor\n\n100% guaranteed this algorithm will work.  It just happens that it'll take 10^85 years to run it on a 4096-bit key.\n\n\n-------------\n\nCogito Sum, I in no way object to your choice to reference me as you have.  Quite the contrary, I consider it a significant honor and compliment that you consider my view sufficiently curious to bring it up specifically.\n\n------------\n\nAs I understand your added comment (sorry for delay, went to lunch), you are saying that since I cannot know the state of your mind without you reporting it, I cannot perform the calculations at all.  However, this overlooks the idea that whether you or I can detect it, the state DOES exist.  For example, if I fill an olympic sized swimming pool w
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Yes, the concept is illusion, you need action to make it real.  And some concepts are fallacious; you can understand it but it is not truth.  Others concepts may be truth for some but not for you.   For example, if it goes against your basic values.   Imagine a capitalist trying to live as a socialist, for lack of a better example.
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Rules (constraints) can be forman and informal.\nThe formal rules are the ones set by institutions, like the laws, central banks, etc.\nInformal rules are those non written like morals, codes of conduct, etc.\n\nIn all societies, people impose constraints upon themselves to give structure to their relations with others. Informal rules come from socially transmited information and are a part of the heritage that we call culture. Culture can then be defined as the "transmission from one generation to the next, via teaching and imitation, of knowledge, values, and  other factors that influence behavior".
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The centrality of goodness demands that it come from One Source.  Are there absolutes?  Is truth a thing that can be relative?\n\nYes, there are absolutes.  Truth is not relative.  What or Who determines truth though?\n\nYou really should find out.
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Your question cannot be answered.\n\nReason and faith are unrelated.  More, they are unrelatable.\n\nIt is a confusion of terms to consider the vernacular "belief", implying an awareness of facts which are subject to confirmation and disproof, with "belief" in the context of faith and religion, where the tenets are merely stated.\n\nEven within the technical arena, even within that most exacting of disciplines, mathematics, there remain elements that are the subject of belief without proof and without the possibility of proof (Kurt Godel).  \n\nFaith and reason are disjoint, and the tools of one are inadequate for a study or analysis of the other.  Therefore your question is improper, from a logical viewpoint, and the concept of a rational explanation of faith has no meaning.
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It is important because the different interpretations will be encountered as they try to apply their knowledge.  It's like teaching the use of a comma in a sentence is to separate a list, but then leave out the other uses of a comma.  Just knowing how to multiply isn't enough to get an understanding of when to multiply.  \n\nSo much of school and math in particular is sets of seemingly aribitrary rules.  But one of the main points of math is that the rules are, in fact, logical and not arbitrary.  So, if students learn that the mathetical operations are aribitrary rules and not steeped in a logical base, it gets very difficult for them to understand math at all.  That is to say that teaching more than one interpretation of mathematical operations can allow the students to see the logic behind the operation as opposed to memorizing the "hows"
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To answer the non-answer (#6): Solving this kind of question requires EXACTLY the sort of thinking that is important in logic.
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I also could not find anything on the web. try buying the book, or borrowing it from your local library, and getting the information you need direct from the source: The Science of Correct Thinking - Logic.
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I did not take this specific course in college however I have studied it extensively since I have been practicing. I honestly believe it supplies more answers than many of the theorist were able to provide.  It of course is used in conjunction with theories in practice but there is honestly an area you miss out if you don't cover it.  Try reading Molecules of Emotion, I can't think of the author, she writes in a way that is very down to earth and yet really explores the impact of emotion on neurochemistry.  From their it is easier to expand into the concepts of behavior.