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Boolean Laws Of Logic In Critical Thinking

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Have you ever noticed that there are some individuals who always seem to have the best solution to any problem? If they are believers in critical thinking this may just be the case? "Critical thinking is the careful, deliberate determination of whether we should accept, reject or suspend judgment about a claim and of the degree of confidence with which we accept or reject it.  (Moore & Parker, p.6) Critical thinking is a process used to come to the most logical of conclusions to a problem.

When we choose to think critically, we have decided to perform a process that will allow us to come to the most informed and logical of conclusions and not simply accept the opinions of someone else. Assumption, perception, emotion, language, argument, fallacies, and logic, are some of the processes used in critical thinking. It takes great effort and persistence for one to come to a logical conclusion. Our personal experiences and views may assist us in the decision making process, but in reality they would most likely hinder the use of logic and informative information when making decisions. To successfully implement critical thinking we must first be aware of our perceptual blocks to effectively perform our perceptual process.

"Who we are is how we think. Where and how we were raised may determine whether we are pessimists or optimists, conservatives or liberals, atheists or theists, idealists or realists. Our upbringing shapes our fears, which keep us from facing thoughts. It shapes our self-concept, which moves us to defend our thoughts. And it shapes our emotions, which can distort our thinking to an exceptional degree.  (Kirby, Goodpaster and Levine, p. 13)

Whenever I am thinking, I am taping into all that formulates me into the person that I am. If I were using the processes of critical thinking correctly I would be accessing information, past experiences, belie

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Logical thinking - definition of logical thinking by The Free Dictionary

logical thinking

analytic thinking. analysis - the abstract separation of a whole into its constituent parts in order to study the parts and their relations

line of reasoning. logical argument. argumentation. argument. line - a course of reasoning aimed at demonstrating a truth or falsehood; the methodical process of logical reasoning; "I can't follow your line of reasoning"

conjecture - reasoning that involves the formation of conclusions from incomplete evidence

deductive reasoning. synthesis. deduction - reasoning from the general to the particular (or from cause to effect)

illation. inference - the reasoning involved in drawing a conclusion or making a logical judgment on the basis of circumstantial evidence and prior conclusions rather than on the basis of direct observation

ratiocination - logical and methodical reasoning

reasoning backward. regress - the reasoning involved when you assume the conclusion is true and reason backward to the evidence

An engineer's fundamental skills, such as logical thinking. problem-solving and strong numeracy, are highly desirable in many other business sectors.

Located in Hall 4 of Modhesh World, the Mirror Maze is attracting a large number of kids looking to put their logical thinking skills to use to solve an exciting maze packed with adventures and thrills.

Our approach is interdisciplinary in which disciplines such as critical and logical thinking. communication, mathematics, written and oral skills, physical education and the creative arts are brought together in a way that's both challenging and fun.

This service, described as the first electronic educational and innovative service of its kind in the region, is considered as a platform for producing electronic stories that develop the principle of logical thinking and a creative tool that enables students to formulate simplified electronic stories without the requirement of software or programming knowledge.

The dominant Augustinian-Pascalian strain of Western thought has traditionally viewed distraction as an expression of human imperfection, which has therefore to be corrected in favour of concentration and logical thinking .

The competition aims to encourage ingenuity, logical thinking and creativity in computing and the right, intuitive and critical design multimedia applications.

It works on developing logical thinking skills, strategic planning and visual and spatial perception.

When will our Government use grown up logical thinking to tackle these serious problems.

You should have good spoken and written communication skills, good negotiation skills, a tactful but assertive manner and logical thinking and problem-solving ability.

During a lecture delivered at Tishreen University, al-Zoubi pointed out that Syrians' conviction comes in the context of the reasonable and logical thinking which produced many reconciliations in several areas around Damascus accompanied by the incidents of the gunmen turning themselves and their weapons in to the authorities concerned for life to go back as normal.

However, in doing so, one must construct a logical thinking process that consists of listening to all points of views, and finally, an opinion can be reached.

Nature of Logic to Critical Thinking essays Nature of Logic to Critical Thinking

Logic and critical thinking are branches of intellectual skills in which
one's objective and desire is to arrive at truth and correct knowledge.
They are intellectual aspects, when used jointly, allows our mind to
perform proper and correct reasoning.

In simple terms, logic is principles and rules that guide a man into
correct thinking and proper reasoning. Critical thinking, on the other
hand, is the ability of the mind to determine the truth based on logical
perspectives. Logic and critical thinking are essentially related forms of
knowledge. Both need each other to successfully produce the necessary end
product – the "truth". Critical thinking requires logic. The following
discussions will explain the nature of logic to critical thinking.

Jonathan Dolhenty, in his Logic and Critical Thinking, defines the value of
logic in critical thinking as

Critical thinking involves a knowledge of the science of logic,
including the skills of logical analysis, correct reasoning, and
understanding statistical methods.

To be able to critically and productively think, one requires skills in
logic because it is from one's logical views and understanding of methods
where a decision or conclusion in critical thinking is based. For
instance, logic is a very essential requirement in mathematics. To be able
to easily and quickly solve a math problem, one needs to have a good sense
of logic. Logic in mathematics includes the abilities in correct analysis
and reasoning, as well as understanding of mathematical methods and
formulas. When these are understood and instilled in one's mind, it can be
said that he has the logic in mathematics. However, having proper logic in
does not immediately mean that any math problem can be easily and correctly
solved. Critical thinking is also necessary to determine the correct
solution and answer. It is in critical thinking where logic come.

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Some Logical Basics for Critical Thinking - McCormick Philosophy

Here are some concepts, principles, and distinctions that are foundational to good critical reasoning. We will be making reference to and building on these all semester.

Belief To believe a claim is to assent to it or to have an attitude towards it such that you think it is true. It may or may not be true, but to believe it is to think that it is. So many people believed that the earth is flat. Some may still believe it. Belief is subjective because it is dependent upon an individual to possess it. Belief is mind dependent. Smith, for example, believes, "Americans never went to the moon." Smith also believes, "There are 12 months on a yearly calendar." Smith's first belief is false, although she takes it to be true. And Smith's second belief is true. Beliefs can align, or diverge from the objective state of affairs in the world. Beliefs aren't true simply in virtue of being believed. The widespread conviction that the Sun orbits the Earth before Copernicus did not make that claim true. The Earth was orbiting the Sun while lots of people on Earth had an erroneous description of reality in their heads. So claims like, "the truth is whatever everyone believes," or "it was true then that the Sun orbited the Earth" are incoherent.

Truth The truth is what is the case or what the actual state of affairs in the world is. We form beliefs about it on the basis of our information. The truth is objective—it does not depend upon people. Truth is mind-independent. It remains what it is whether we form beliefs about it or not. When people believed that the earth was flat, in fact, it was not. The truth was that the earth was (and is) spherical. The Earth orbits the Sun. Evil demon possession does not cause bubonic plague. There are truths that are not believed by anyone. And there are many beliefs that are not true. Claims like, "the truth is relative," or "there is no objective truth, only belief" are incoherent. Consider the latter. Suppose Smith says, "there are no objective truths, there are only subjective beliefs." Paradoxically, his assertion presumes that what he is saying is false. He is asserting that his claim is actually, objectively true; it is objectively true that there are no objective truths. The real state of affairs in the world is that there are no real or true states of affairs. Relativism about truth is self-contradictory and incoherent.

A Successful Argument for a conclusion (call it C) will be a set of premises or reasons (different than C) that are true and that when taken jointly would imply the conclusion C to a reasonable person who does not already believe C. A prosecuting attorney in a murder trial will attempt to give a successful argument that will convince the jury (who has assumed the defendant's innocence) that the defendant is guilty. So a reasonable person should accept the conclusion of a successful argument. Of course, people rarely hear a convincing argument and then abruptly change their minds. But reasonable people should be prepared to. If you hear an argument with premises that you believe are true, and you understand that the premises validly imply the conclusion, then you are rationally committed to accept the conclusion. Otherwise, you are being irrational. Disagreements about whether or not an argument is successful will be disagreements about whether or not all of the premises are true or whether or not the premises, if true, would imply the truth of the conclusion. This argument may not succeed in convincing you, but here is an example of an argument:

1. If there were no God, then the world would not be so well suited to our existence, our sustenance, and our survival.

2. But the world is well suited to our existence, our sustenance, and our survival.

3. Therefore, there is a God.

Whether or not an argument succeeds in convincing a particular person often depends upon her background information, the other beliefs she has, the extent to which she has critically scrutinized the premises of the argument and her underlying assumptions, and many other factors. So success varies from person to person. Determining when a person is being reasonable or unreasonable in accepting an argument or its premises can be a complicated and technical matter.

Epistemic culpability When a person can be blamed or faulted rationally for believing or disbelieving some claim, we can say he is epistemically culpable. The notion of culpability is borrowed from ethics. When your friend knowingly lies to you, we would fault her. We would say that she has done something wrong. She ought not to have done it. We use prescriptive language--"should" and "ought"--instead of merely descriptive terms about the facts. She is morally culpable for not telling you the truth. Someone is epistemically culpable when she has violated some duty or responsibility to be reasonable, rational, or thoughtful. There is a presumption that people ought to be reasonable or rational, so when they fail and they could have acknowledged the justified, reasonable conclusion, then we find fault in them. If someone has thought about the topic extensively, gathered evidence carefully, and correctly applied logical inferences, then we typically find him to be epistemically inculpable, or without blame, about the resulting belief.

The Law of Non Contradiction is the fundamental logical principle. Roughly, it is not possible for a thing to both possess and not possess a property at the same time in the same way. An assertion cannot be both true and not true at the same time.

Aristotle, the ancient Greek philosopher, said: "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect." And, "an affirmation is a statement affirming something of something, a negation is a statement denying something of something…It is clear that for every affirmation there is an opposite negation, and for every negation there is an opposite affirmation…Let us call an affirmation and a negation which are opposite a contradiction."

The affirmation and the negation cannot both be true of an object. To make an assertion of the form X is P, like "the ball is blue," is to claim that the ball has the property of blue and that it is false that the ball is non-blue. If we abandon the law of non-contradiction, then there is no meaningful difference between an assertion and its opposite. That is, our assertions cease to have meaning altogether. My claim that, "Today is Tuesday," doesn't say anything unless it denies some other state of affairs like, "Today is Wednesday." The law of non-contradiction is axiomatic to reason; that is, it is one of the most fundamental principles upon which reasoning and rationality are based. It cannot be argued for (it is the principle that makes arguments possible) nor can in be plausibly denied (to deny it is to already assume it.)

Logical Possibility is built upon the law of non-contradiction. Any proposition whose opposite does not imply a contradiction is logically possible. Any description of a state of affairs that does not contain and implicit or explicit logical contradiction is logically possible. That is, if the sentence does not include a contradiction like, "Mike is a married bachelor." then the state of affairs that the sentence describes is logically possible. So it is logically possible that Mike is a bachelor. And it is logically possible that he is married. And it is logically possible that Mike (an unaided human) could fly. But it is not logically possible that 2 + 2 = 5, or that circles have sides, or that the Pythagorean Theorem is wrong, etc. Philosophers sometimes talk about logical possibilities in terms of possible worlds; there is a possible world where Arnold Schwarzenegger is the president of the United States. There is a possible world where you have super powers and can run faster than the speed of light. There are no colorless red balls, however, and no presidents who hold no political office, and no four sided triangles.

Natural Possibility The laws of nature confine the behavior of matter in our world to a subset of the logically possible worlds. The laws of nature such as the universal law of gravitation, F = MA, and e=mc2 determine the range of what states of affairs are naturally possible. So it is not naturally possible for an unaided human body to fly—the musculature, bone structure, and other physiological traits prevent it. But it is naturally possible (we think) to cure cancer. The laws of nature, which are different from the laws of logic, could have been different without logical contradiction. All natural possibilities are a subset of logical possibilities. That is, anything that is naturally possible is also logically possible, but not everything that is logically possible is naturally possible. Being able to move objects with your thoughts alone through telekinesis is ruled out by physics and not naturally possible, but there is no logical contradiction in the scenario.

The periodic table could have been different, gravity could attract at a different rate, or force could be equal to something different than mass times acceleration. If one of those different sets of natural laws were in place, then the range of what events that is naturally possible would be different. The super powers of comic book heroes, for example, such as teleporting, flying, telekinesis, super speed, super strength, and so on would probably be violations of the laws of nature, but they are not logically impossible. Miracles such as walking on water or bringing the dead back to life are violations of the laws of nature, but they are not logically impossible. Roughly speaking, your rule for figuring out whether something is logically impossible vs. naturally impossible should be this: Ask yourself, "Does this sentence describe a scenario that creates an explicit logical contradiction like a square circle or a married bachelor, or does it merely describe a situation that runs contrary to the laws of physics as we find them in our natural world? If it is the former, then it is a logical contradiction, if it is the latter, then it is a natural impossibility.

Possible vs. Probable

We should contrast events that are naturally impossible, or ruled out by natural law, with events that are statistically improbable. If we dropped a million dice onto a parking lot and all of them came up with a 1 on top, it would be statistically improbable, but it is not ruled out by the laws of physics. The odds of winning the lottery are millions to one, and if your friend said he had the winning ticket, you might say, "That's impossible!" But it violates no logical or natural laws. It's merely statistically improbable. We might also think about it this way: the laws of natural, such as the ones governing the periodic table of elements, constrain the regular behavior of matter. They also make some events improbable. Snow is Sacramento is rare because of the typical weather conditions in the winter. But the natural conditions can happen. On Feb. 5, 1976, two inches of snow accumulated in Sacramento. Such an event is clearly logically possible, and it is naturally possible. But it naturally improbable. Having it snow in Sacramento on 10 consecutive days this winter is exceedingly unlikely. We might even say that it is impossible because it is so improbable. But on the definitions we are using of natural and logically impossibility, this would be a misnomer. Every year, approximately 1 person in 500 million is eaten by sharks. So the odds of your being eaten by sharks, all other things being equal, are about 99.9999998%. It is not logically impossible; clearly there is no logical contradiction in the scenario the way there is with a four sided triangle. But it is naturally possible. There is nothing about the laws of nature that prevent such events from occurring. But is it likely? No. Is it probable? No. Is it reasonable to believe?

In probability theory, odds are depicted on a range from 0 to 1, with 1 being certain. The threshold where the odds begin to favor belief is .5. As evidence accumulates in favor of a claim and the odds increase towards 1, the claim becomes more and more reasonable to believe. The conviction that you have about the truth of a claim should be proportional to the quality and quantity of the evidence you have concerning it.

The mere logical possibility that a claim is true should not be enough to elevate it across the .5 threshold and make it reasonable. That is, possible does not imply probable. While is is possible that you will be eaten by sharks today, as we saw above, it is not probable. It is so unlikely, you should believe with a great deal of conviction that you will not be eaten by sharks. Here are more examples illustrating the point that possible does not equal probable:

It is possible that the Holocaust didn’t happen.

It is possible that wearing a raw steak hat wards off disease.

It is possible that even though you are taking birth control pills exactly as prescribed everyday you are pregnant.

It is possible that the government is watching everything you do and hiding it very well.

It is possible that Christopher Marlowe wrote all of Shakespeare’s plays.

It is possible that having sex with a virgin cures HIV.

It is possible that eating the flesh of your enemies gives you power.

It is possible that birth defects are caused by wickedness from a past life.

It is possible that the Detroit Lions could win the Super bowl.

It is possible that fever is caused by demon possession.

It is possible that the earth rests on the back of a (invisible) turtle.

It is possible that lightening is thrown by an angry Zeus.

It is possible that the moon is made of green cheese.

It is possible that the moon landing in 1969 was faked on a secret Hollywood set by NASA.

It is possible that wishful thinking can help you win the lottery

It is possible that wearing your lucky underwear will help you win the basketball game

It is possible that Santa exists.

It is possible that there are still dinosaurs.

It is possible that if you concentrate you can levitate.

It is possible that tossing spilled salt over your shoulder improves luck.

It is possible that opening an umbrella indoors or breaking a mirror is bad luck.

It is possible that conceiving in the spring produces boy babies.

It is possible that swinging a wedding ring on a string in front of a pregnant woman's stomach will reveal the sex of the baby.

Some of these will seem outrageous to you, and you won't be tempted to believe them, or even argue that we should be agnostic about them. But some of these might strike you as being more plausible than others. Some of these might seem to have more evidence in their favor than others. And it might strike you that there have been or there are people who believe some of these things, and believe them with great conviction. And we are reluctant to conclude that some of those people are irrational or wrong.

The answer is that under some circumstances, some of these claims could be reasonable (but not true) for some people. If you have limited access to information, or the best sources that are available to you misrepresent the truth, and you have tried your best to gather and evaluate the evidence, then you might not be at epistemic fault for believing what you believe. Ptolemy, the ancient Greek astronomer, believed that the Sun orbited the Earth on the basis of the best information available in his day. He was justified, but his conclusion was mistaken. But most of these examples above will strike you as merely logically possible, but not even close to being reasonable, probable, or justified to believe.

A necessary truth is one whose opposite implies a contradiction. It is a proposition that must be true without exception. Sometimes, it's a proposition that is true in virtue of the meanings of the concepts involved. A = A is a necessary truth, as is "Bachelors are unmarried," and "Triangles have three sides." Given what the terms mean, the sentence cannot fail to be true, whereas some sentences, like contingent truths (see below) can either be true or false depending on what events unfolded.

A contingent truth is one that can be true or it can be false without logical contradiction. "The Washington monument is 555 feet tall," is true, but it could have been false if the stone mason carving the pinnacle block had opted to make it a foot taller. Contingent truths are assertions whose opposite would not have violated the law of non-contradiction. "The Washington Monument is not 555 tall," is not contradictory. "This triangle does not have three sides," is contradictory.

A priori truths are truths that can be known without an appeal to experience. They are true by definition or in virtue of the meanings of the words involved. "A square is a four sided figure," is an a priori truth. We do not need to count the sides of objects that are squares in the world to know that it is correct. "Mammals have warm blood," is another a priori truth.

A posteriori truths are truths that we discover and know on the basis of experience. "McCormick is 6' 1" tall," is a fact that can only be discovered by experience. It cannot be known by conceptual analysis the way "bachelors are unmarried" can. "Interest rates were at a 40 year low in July" is another example of an a posteriori truth.

Outline of logic

Outline of logic

Outline of logic

The following outline is provided as an overview of and topical guide to logic:

Logic – formal science of using reason. considered a branch of both philosophy and mathematics. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and through the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes. to specialized analyses of reasoning such as probability. correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct (or valid ) and incorrect (or fallacious ) inferences. Logicians study the criteria for the evaluation of arguments.

Contents Foundations of logic Philosophical logic Informal logic and critical thinking Deductive reasoning Theories of deduction Fallacies

Other types of formal fallacy  · List of fallacies

Formal logic

Mathematical logic, symbolic logic and formal logic are largely, if not completely synonymous. The essential feature of this field is the use of formal languages to express the ideas whose logical validity is being studied.

Symbols and strings of symbols Logical symbols

Main articles: Table of logic symbols and Symbol (formal)

Logical connectives

Strings of symbols Types of propositions Rules of inference Formal theories Expressions in an object language Expressions in a metalanguage Propositional and boolean logic Propositional logic

Main article: Propositional logic

Boolean logic Predicate logic and relations Predicate logic Relations

Main article: Mathematical relation

Mathematical logic Set theory Metalogic Proof theory

Proof theory – The study of deductive apparatus.

Model theory

Model theory – The study of interpretation of formal systems.

Computability theory Classical logic Non-classical logic Modal logic Concepts of logic

History of logic Logicians Literature Lists See also

External links

General reference  · Culture and the arts  · Geography and places  · Health and fitness  · History and events  · Mathematics and logic  · Natural and physical sciences  · People and self  · Philosophy and thinking  · Religion and belief systems  · Society and social sciences  · Technology and applied sciences

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Boolean laws of logic in critical thinking

One area of mathematics that has its roots deep in philosophy is the study of logic. Logic is the study of formal reasoning based upon statements or propositions. (Price, Rath, Leschensky, 1992) Logic evolved out of a need to fully understand the details associated with the study of mathematics. At the most fundamental level, mathematics is a language and it is a language of choice and must be communicated with great precision. (Wheeler, 1995) The idea of logic was a major achievement of Aristotle. In his effort to produce correct laws of mathematical reasoning, Aristotle was able to codify and systemize these laws into a separate field of study. The basic principles of logic center on the law of contradiction, which states that a statement cannot be both true and false, and the law of the excluded middle, which stresses that a statement must be either true or false. The key to his reasoning was that Aristotle used mathematical examples taken from contemporary texts of the time to illustrate his principles. Even though the science of logic was derived from mathematics, logic eventually came to be considered as a study independent of mathematics yet applicable to all reasoning. (Kline, 1972)

Logic serves as a set of rules that govern the structure and presentation of mathematical proofs. (Fletcher, Patty, 1996) Since proofs are constructed with the English language, mathematical logic seeks to break down mathematical reasoning for a clearer understanding. By using statements or propositions that are either true or false, the English language becomes the building blocks for a mathematical language. In ordinary English, new propositions are formed from existing propositions. There are three ways that one can form new proposition, which stem directly from the basic principles of Aristotle. One can connect two propositions with the word "and." The word "or" can also serve as a connecting word. Forming negations is another way to build new propositions and involves making an assertion that a given statement is false. (Fletcher, Patty, 1996) While Aristotle was the first to focus on the idea of logic, the efforts of Richard Dedekind and Georg Cantor also contributed greatly to this study. The intuitive reasoning of these two mathematicians with regard to sets, led to paradoxes. Paradoxes are those statements that appear to be both true and false. However, logic makes the assumption that this cannot be the case. With the work of Dedekind and Cantor, a new type of scientist, the mathematical logician, came into existence. (Fletcher, Patty, 1996)

On the whole, logic is a way to improve one's critical thinking skills by not just looking at a problem, but studying the problem and implementing strategies to find a solution. It involves both inductive and deductive reasoning. Inductive reasoning is the process by which a general conclusion is arrived at by making limited observations. On the other hand, deductive reasoning is the process where one proceeds carefully from definitions and established facts to arrive at a possible conclusion. (Wheeler, 1995) Other aspects of logic include assertions, open sentences, simple statements, compound statements, conjunctions and disjunctions. (Wheeler, 1995) One way to summarize the study of logic is to use a Truth Table. A Truth table is used to summarize the possibilities of the statements one is studying to determine whether they are true or false. By using the Truth Table, one is able to replace statements or predictions with letters and symbols. For example, if you have one simple statement such as, "It is raining," then there are only two possibilities. The statement can be replaced with a simple variable to simplify the calculations. The statement can be true or it can be false. The possibilities are multiplied as a new statement or proposition is considered. Logic is about being non-contradictory, being rational, and being consistent. It is not related to personal beliefs. It is simply a means to cause us to think and apply our critical thinking skills. Logic may even force us to change our opinions and our beliefs once we have rationally thought through a particular proposition. In logic one creates formal languages for reasoning, then sets aside the reasoning and proofs to follow the rules of the new language. New derivations of purposed formulas, and new conclusions may occur, or the rules of logic can lead to derive the truth-value of a formula in some kind of logical arithmetic. (Christer, 1998)

In order to visualize this idea of logic; consider the following example:

Nathan likes all red things.

For all X, if X is red then Nathan likes X.

The house is red.

The color of the house = red.

Because of this "for all X" we can replace X with "the house" to get,
If the house is red then Nathan likes the house.
This we can write as,
If the color of the house = red, then Nathan likes the cottage.
We know that the color of the house is red, so one can conclude,
Nathan likes the house. (Christer, 1998)
While this is just a simple illustration, one can see how logic can be applied and used to simplify the process by which a conclusion is arrived.

One important ingredient in both critical thinking and mathematical reasoning is logic. (Wheeler, 1995) While its roots are deep in philosophy, logic has valid applications in the area of mathematics. There are several areas of logic, which include everyday logic, formal logic, Boolean algebra, and many other areas that are considered as some type of logic. (Christer, 1998) To completely discuss and fully explain logic, one would need several volumes. However, through the efforts of Aristotle and others, mathematicians today have a way to produce correct laws of mathematical reasoning and establish rules that provide structure and govern the presentation of mathematical proofs. (Fletcher, P, Patty, C.W. 1996)

Contributed by John Stockstill

  1. Christer's pages of logic, math and reasoning. (1998) World Wide Web.
  2. Fletcher, P. Patty, C.W. (1996) Foundations of higher mathematics. Boston: PWS Publishing Company.
  3. Kline, M. (1972) Mathematical thought from ancient to modern times. New York: Oxford University Press.
  4. Price, J. Rath, J.N. Leschensky, W. (1992) Pre-algebra, a transition to algebra. Lake Forest: Macmillan / McGraw - Hill Publishing Company.
  5. Wheeler, E.R. Wheeler, R.E. (1995) Modern mathematics for elementary school teachers. New York: Brooks / Cole Publishing Company.


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Jonathan Dolhenty, Ph.D.Logic and Critical Thinking. Available at

  • Truth is the object of thinking.
  • Some truths are obvious; others are difficult to acquire.

  • Some judgments we make are simple; some judgments are complicated.

  • Some arguments, whether made by us or others, may be straightforward and easily understood; other arguments may be complex and consist of a series of smaller arguments, each needing to be critically examined and evaluated.

    • Every object of knowledge has a branch of knowledge which studies it.
    • Planets, stars, and galaxies are studied by astronomy.
  • Chemistry studies the structure, composition, and properties of material substances and the transformations they undergo.

  • The origin, evolution, and development of human society is the object studied by sociology.

  • Economics, biology, geography, and grammar all have objects of knowledge which they investigate, describe, and try to explain.

    • Critical thinking involves knowledge of the science of logic, including the skills of logical analysis, correct reasoning, and understanding statistical methods.
  • Critical thinking, however, involves more than just an understanding of logical procedures.

  • A good critical thinker must also understand the sources of knowledge, the nature of knowledge, and the nature of truth.

    • The object of knowledge involved in the science of logic is "thinking," but it is "thinking" approached in a special way.
  • Generally speaking, logic is that branch of knowledge which reflects upon the nature of "thinking" itself.

  • But this may confuse logic with other branches of knowledge which also have the nature of "thinking" as a part of their specific object of investigation.

    • Logic doesn't just deal with "thinking" in general. Logic deals with "correct thinking."
    • Training in logic should enable us to develop the skills necessary to think correctly, that is, logically.
  • A very simple definition would be:
  • Logic is the subject which teaches you the rules for correct and proper reasoning.

  • A more complete and "sophisticated" definition of logic, you can define it this way:

  • Logic is the science of those principles, laws, and methods, which the mind of man in its thinking must follow for the accurate and secure attainment of truth.

    • “Natural Logic" or Common Sense
  • We all have an internal sense of what is logical and what is not, which we generally refer to as "common sense."

  • This "natural" logic we have learned from the moment of birth, through our personal experiences in the world and through our acquisition of language.

    • Scientific logic is simply our natural logic trained and developed to expertness by means of well-established knowledge of the principles, laws, and methods which underlie the various operations of the mind in the pursuit of and attainment of truth.

    • Logic as a science:
    • The science part is the knowledge of the principles, laws, and methods of logic itself.
    • Logic must be put into action or else the knowledge provided within the science of logic is of little use.
  • We can speak of the "art" of logic, that is, the practical application of the science of logic to our everyday affairs.

  • Logic as a science and an art
    • Logic is not intended merely to inform or instruct.
  • It is also a directive and aims at assisting us in the proper use of our power of reasoning.

  • In this sense, we can speak of logic as both a science and an art, a practical art meant to be applied in our ordinary affairs.

    Why Study Logic?
    • Aim: To develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own.
  • Benefits: an increase in confidence that we are making sense when we criticize the arguments of others and when we advance arguments of our own.