Sherilyn Loria1

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Sherilyn L. Loria BSEM-1A RESEARCH ( *LOGIC *ENGLISH *COLLEGE ALGEBRA)

What is science?

Science is the concerted human effort to understand, or to understand better, the history of the natural world and how the natural world works, withobservable physical evidence as the basis of that understanding

1. It is done through observation of natural phenomena, and/or through

experimentation that tries to simulate natural processes under controlled conditions.

Consider some examples. An ecologist observing the territorial behaviors of bluebirds and a geologist examining the distribution of fossils in an outcropare both scientists making observations in order to find patterns in natural phenomena. They just do it outdoors and thus entertain the general publicwith their behavior. An astrophysicist photographing distant galaxies and a climatologist sifting data from weather balloons similarly are also scientistsmaking observations, but in more discrete settings.

The examples above are observational science, but there is also experimental science. A chemist observing the rates of one chemical reaction at avariety of temperatures and a nuclear physicist recording the results of bombardment of a particular kind of matter with neutrons are both scientistsperforming experiments to see what consistent patterns emerge. A biologist observing the reaction of a particular tissue to various st imulants islikewise experimenting to find patterns of behavior. These folks usually do their work in labs and wear impressive white lab coats, which seems tomean they make more money too.

The critical commonality is that all these people are making and recording observations of nature, or of simulations of nature, in order to learn moreabout how nature, in the broadest sense, works. We'll see below that one of their main goals is to show that old ideas (the ideas of scientists a centuryago or perhaps just a year ago) are wrong and that, instead, new ideas may better explain nature.

The systematic observation of natural events and conditions in order to discover facts about them and to formulate laws and principles based on these

facts. 2. the organized body of knowledge that is derived from such observations and that can be verified or tested by further investigation. 3. anyspecific branch of this general body of knowledge, such as biology, physics, geology, or astronomy. ( Academic Press Dictionary of Science &Technology)

Science is an intellectual activity carried on by humans that is designed to discover information about the natural world in which humans live and todiscover the ways in which this information can be organized into meaningful patterns. A primary aim of science is to collect facts (data). An ultimatepurpose of science is to discern the order that exists between and amongst the various facts.(Dr. Sheldon Gottlieb ina lecture series at the Universityof South Alabama)

Science involves more than the gaining of knowledge. It is the systematic and organized inquiry into the natural world and its phenomena. Science isabout gaining a deeper and often useful understanding of the world.( from theMulticultural History of Science pageat Vanderbilt University.)

Science consists simply of the formulation and testing of hypotheses based on observational evidence; experiments are important where applicable,but their function is merely to simplify observation by imposing controlled conditions.( Robert H. Dott, Jr., and Henry L. Batten, Evolution of theEarth(2nd edition))

Science alone of all the subjects contains within itself the lesson of the danger of belief in the infallibility of the greatest teachers in the preceedinggeneration . . .As a matter of fact, I can also define science another way: Science is the belief in the ignorance of experts. Richard Feynman, Nobel-prize-winning physicist,in The Pleasure of Finding Things Outas quoted in American Scientistv. 87, p. 462 (1999).

As complex as the modern world has become, it seems unlikely that most of what surrounds us is actually the result of the ancient practice of

philosophy. Everything from the structure of democratic governments to due process of law, from a physicians Hippocratic oath to computer software,

has its roots in philosophy. Sadly, philosophy as a course of study is disappearing from our nations colleges, yet its focuson analytical thinking and

problem solving is more vitally important today than ever.

Philosophy is an academic discipline that exercises reason and logic in an attempt to understand reality and answer fundamental questions about

knowledge, life, morality and human nature. The ancient Greeks, who were among the first to practice philosophy, coined the term, which means love

of wisdom. Those who study philosophy are called philosophers. Through the ages, philosophers have sought to answer such questions as, what is

the meaning and purpose of life? How do we know what we know? Does God exist? What does it mean to possess consciousness? And, what is the

value of morals?

Philosophers attempt to answer such questions through the philosophical method. The method usually begins when a philosopher examines his own

beliefs and begins to doubt their validit y. From his doubt, questions emerge. Before answering a question, the philosopher thoroughly analyzes it to
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ensure it is clearly and properly defined. This helps narrow the path to the most precise answer. Next, the philosopher proposes possible answers to

the question and provides reasoned arguments to support each one. The arguments are then critiqued by other philosophers, who may give rebuttals.

Through this process of criticism and judgment, known as dialectic, philosophers attempt to prove the rationality of their beliefs and discover

fundamental truths.

Its no coincidence that the philosophical method has much in common with the scientific method. Indeed, early science was known as natural

philosophy. Philosophers like Aristotle developed the concepts of inductive and deductive reasoning that form the basis of modern scientific study. The

roots of the physical sciences like physics and geology can be traced back to ancient philosophy.

Philosophy itself is generally considered a type of social science, like sociology or psychology. Thats because early philosophy was primarily

concerned with describing the best way to live and organize society. From that spawned many other disciplines: economics, political science, law,

linguistics, literary and art criticism, and theologyalong with sociology and psychology.

Though many of philosophys original topics have evolved into other fields of study over time, the discipline remains rich and varied. Modern philosophy

contains six main branches of thought, each with their own unique focus:

Metaphysics: the nature of reality and the universe.

Epistemology: the study of knowledge and how it is acquired.

Logic: how to develop valid arguments; includes mathematical logic.

Ethics: the study of right and wrong and how people should live.

Politics: the study of government, citizen rights and political obligations.

Aesthetics: beauty, art and artistic perception.

At first glance, it would appear that such study has little application in the real world. Yet, philosophy shapes modern existence. Unlocking the secrets

of knowledge acquisition is the primary concern of passionate educators of young people around the globe. Logic forms the basis of all computer

technology, as more precise programming commands increase computing speed and efficiency. Ethics plays a major role in medicine, law and foreign

policy. Indeed, the hottest debates surrounding the issues of our timeabortion, capital punishment, welfare, environmentalism, torture and end-of-life

careall stem from philosophical questions.

However, the study of philosophy is not necessarily about discovering all of the answers to lifes toughest questions. Skepticism lies at the heart of

philosophy. Therefore, asking a question is more fundamentally important than answering one. In philosophy, questioning a deeply held belief or social

practice sets one onto the path of true understanding, and its this understanding that leads to meaningful personal and social change. A good

philosopher recognizes the danger of accepting knowledge at face value. Social or scientific theories may be untested or contain personal bias; trusting

them immediately could result in terrible consequences.

Today, philosophers can be found working in nearly every career field. Some are scientists developing ways to test household products without using

animals. Some are politicians and human rights activists fighting for changes in foreign policy that will alleviate war and poverty for millions of Third


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World citizens. Some are economists seeking practical solutions to economic inequality. Still others are programmers working on the cutting edge of

technology to develop faster and more efficient computer software.

Those who study philosophy also tend to lead fulfi lling and successful lives. Some of historys most well-known figures were philosophers: Socrates,

Plato, John Locke, Friedrich Nietzsche, John Stuart Mill, Adam Smith, Karl Marx and Noam Chomsky, to name a few.

The word philosophyliterally means love of wisdom;1this tells us something about the nature of philosophy, but not much, because many disciplinesseek wisdom. How does philosophy differ from these other disciplines? A brief look at the historical development of the field will help us to answer thisquestion. On the standard way of telling the story, humanity's first systematic inquiries took place within a mythological or religious framework: wisdomultimately was to be derived f rom sacred traditions and from individuals thought to possess privileged access to a supernatural (and, presumably,honest and error-proof) realm; the legitimacy of these traditions or access of these individuals, in turn, generally was not questioned. However, startingin the sixth century BCE, there appeared in ancient Greece a series of thinkers whose inquiries were comparatively secular (see "The Milesians andthe Origin of Philosophy").2Presumably, these thinkers conducted their inquiries through reason and observation, rather than through tradition orrevelation. These thinkers were the first philosophers. Although this picture is admittedly simplistic, the basic distinction has stuck: philosophy in itsmost primeval form is considered nothing less than secular inquiry itself.3However, there are now many forms of secular inquiry, so what distinguishes philosophy from them? In the beginning, there was perhaps nodistinction. But, as civilization advanced, two parts of philosophy became so powerful in their own right that they separated off, claiming for themselvesthe status of independent disciplines. Mathematics was the first, and split off very early in the game; science (or natural philosophy, as it was calledeven into the nineteenth century) was the second, split ting off much later. To modern philosophy is left whatever questions these two disciplines cannotsolve (at least at a given time), including not only questions that are traditionally thought to be beyond the two (e.g. "What is the meaning of life?"), butalso theoretical questions at their f ringes (e.g. "Can natural selection operate at the species level?") and conceptual questions at their foundations (e.g."What is science?"). Philosophy, of course, is best known for the first class of questions, which includes some of the most difficult and importantquestions there are, such as whether or not there is a god, how one can know anything at all, and how a person ought to live.Philosophy is characterized as much by its methods as by i ts subject matter. Although philosophers deal with speculative issues that generally are notsubject to investigation through experimental test, and philosophy therefore is more fully conceptual than science, philosophy properly done is not merespeculation. Philosophers, just like scientists, formulate hypotheses which ultimately must answer to reason and evidence.4This is one of the thingsthat differentiates philosophy from poetry and mysticism, despite its not being a science.5T h e B r a n c h e s o f P h i l o s o p h y

The four main branches of philosophy are logic, epistemology, metaphysics, and ethics:

Logicis the attempt to codify the rules of rational thought. Logicians explore the structure of arguments that preserve truth or allow theoptimal extraction of knowledge from evidence. Logic is one of the primary tools philosophers use in their inquiries; the precision of logichelps them to cope with the subtlety of philosophical problems and the often misleading nature of conversational language.

Epistemologyis the study of knowledge itself. Epistemologists ask, for instance, what criteria must be satisfied for something we believe tocount as something we know, and even what it means for a proposition to be true.

Metaphysicsis the study of the nature of things. Metaphysicians ask what kinds of things exist, and what they are like. They reason aboutsuch things as whether or not people have free will, in what sense abstract objects can be said to exist, and how it is that brains are able togenerate minds.

Axiologyis an umbrella term for different studies that center upon the nature of different types of value.6These different studiesinclude aesthetics, which investigates the nature of such things as beauty and art; social philosophyand political philosophy; and, mostprominently, ethics, which investigates the nature of right and wrong, and of good and evil, both in theoretical considerations about thefoundations of morality, and in practical considerations about the fine details of moral conduct.

Many professional philosophers also double as historians, researching one or another aspect of the history of philosophical thought. Even those whodo not conduct novel historical research typically see great value in the texts of thinkers as far back as the ancient Greeks, and study these texts bothfor philosophical insight and enjoyment. Arguably, history of philosophy may be considered a fifth branch of philosophy.As you can tell, the different branches of philosophy overlap one another. A philosopher considering whether people ought to give excess wealth to thepoor is asking an ethical question. However, his investigations might lead him to wonder whether or not standards of right and wrong are built into thefabric of the universe, which is a metaphysical question. If he claims that people are justified in taking a particular stance on that question, he is makingat least a tacit epistemological claim. At every step in his reasoning, he will want to employ logic to minimize the chance of being led into error by thegreat complexity and obscurity of the questions. He may very well look to some of the ethical, metaphysical, and epistemological writings of pastphilosophers to see how his brightest predecessors reasoned about the matter.Aspects of each branch of philosophy can be studied in isolation, but philosophical questions have a way of leading to other philosophical questions, tothe point that a full investigation of any particular problem is likely eventually to involve almost the whole of the philosophical enterprise.T h e D e m a n d s o f P h i l o s o p h y

Philosophical inquiry is very demanding, suitable only for those who possess a fair degree of courage, humili ty, patience and discipline.Doing philosophy requires courage, because one never knows what one will find at the end of a philosophical investigation. Since philosophy dealswith the most fundamental and important issues of human existence, and since these are things that most people initially take for granted, genuinephilosophical inquiry has great potential to unsettle or even to destroy one's deepest and most cherished beliefs. Genuine philosophical inquiry alsocarries the risk of isolation among one's peers, both for the unorthodox views to which it may lead one, and for the simple unpopularity of criticalthinking. A philosopher must be able to face both consequences.Doing philosophy requires humility, because to do philosophy one must always keep firmly in mind how little one knows and how easy it is to fall intoerror. The very initiation of philosophical inquiry requires one to admit to oneself that one may not, after all, have all of the answers.Doing philosophy requires both patience and discipline, because philosophical inquiry requires long hours of hard work. One must be prepared tocommit huge amounts of time to laboring over issues both difficult and subtle. People who avoid philosophy often complain that thinking aboutphilosophical questions makes their heads hurt. This is unavoidable: if the answers come easily to you, your inquiries almost certainly are superficial.To do philosophy, one must commit oneself to pain. The only difference between one who chooses to shoulder the pain and one who does not is thatthe former recognizes that there is no shortcut to truth: every advance must be fought for tooth and nail.These virtues always are imperfectly represented in any given person, which is why philosophy is best done in a community: the critical scrutiny ofother thinkers provides an often necessary check on defects invisible to one's own eyes.T h e R e w a r d s o f P h i l o s o p h y
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But if philosophy is so demanding, why should anyone even bother with it?In the first place, there is great utility in philosophical inquiry, even for someone who does not innately care about the pursuit of truth. Consider arandom handful of classic philosophical questions: What is the meaning of life? What is the nature of justice? What does it take for a belief to bejustified? Is the world we see illusion or reality? The answers to such questions cannot help but to have a critical impact on how one ought to live one'slife. Surely one should subject one's intuitive beliefs about these things to critical scrutiny, and work hard to come as close to truth as possible. Manyphilosophical questions are fundamentalto human life; the only reason it often does not seem that way is that people simply assumethey know whatthe answers to these questions are, without ever daring to make a serious inquiry.This leads us to the second reason why one ought to do philosophy: to understand is ennobling. To go through life simply assuming one understands,is not. To be sure, one can perhaps be happy, at least in the same way as a well-fed dog is happy, if one manages to make it all the way through lifewithout questioning anything. Philosophical inquiry, on the other hand, can be disquieting, offering no guarantee that your hard work will yield theconclusions you hope for. Even worse, philosophy gives you no guarantee that your investigations will yield anyconclusion at all: at the end of the day,you may find yourself not only stripped of the certainties with which you began, but also with no new ones to put in their place. If you do philosophy,you may well have to learn to live with perpetual uncertainty, while others, in their ignorance, happily profess perfect knowledge of things they do notunderstand at all. But it is clear who has the better life: far better to understand, even if the main thing you understand is the limit of your ownknowledge.And a final reason for studying philosophy is that, for all of the pains and difficulties associated with it, the pursuit and acquisition of knowledge isenjoyable. To be sure, it is a refined enjoyment, and it is often hard to see from the outside what the appeal is. But once you become immersed in it, itcarries its own immediate rewards, and it is difficult to resist becoming addicted to it. I have experienced most of the same pleasures everyone elsehas,7but in the end, none of them hold a candle to the pleasures of the mind: the sheer pleasure of studying and investigating, and sometimes evenunderstanding.

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1 From the Greek philia(friendship/love) or philos(friend/lover) and sophia(wisdom). According to the admittedly unreliable Diogenes Laertius (Livesand Opinions of Eminent Philosophers, I.VIII), the term philosopherwas introduced by Pythagoras, who preferred it to the less modest title of sophist,or "wise man."2 The standard story, wherein the Milesians literally are the first secular thinkers, cannot literally be true; it is scarcely possible that human thoughtcould have been uniformly superstitiousbroadly, yes, but not uniformlyat any time in history. However, we have no formal record of the Milesian

style of thought prior to the "official" advent of philosophy.3 To say that philosophy is seculardoes not mean that it is anti-religious, but only that it is independent of religion. If one needed to be anti-religious oreven nonreligious to do philosophy, the history of philosophy would be very slim. To say that philosophy is secular is also not to deny that there aremany thinkers, arguably including most of the first philosophers themselves, for whom it is not always clear whether they are doing philosophy ortheology: philosophy, like any other discipline, has gray boundaries.4 In saying this, am I ruling out Continental philosophy by stipulation? I don't think so, at least not across the board. To be sure, Continentalphilosophers do not writelike analytic philosophers, and I consider that a vice, but I am not sure that their rock-bottom commitments really are thatdifferent. The phenomenologists and existentialists I am familiar with seem to base their thought upon rational and evidential grounds as much asanyone else; even with postmodernists, I am not sure that the open disdain for reason and evidence is more than just talk. Am I ruling out fromphilosophy anyone whose inquiries do not ultimately rely upon reason and evidence? Yes, and unapologetically so.5 I need to make another qualification here. It is not that poetry and literature cannot overlap with philosophy. It all depends on whether we are talkingabout the mode of expression, or the source of the ideas in the first place. Camus, for instance, expressed himself very well through the medium ofnovels and plays, but the thoughts he expressed seem to have been worked out by appeals to reason and evidence. I would contrast that with, say,Whitman's poem which expresses disgust with the way astronomers dissect everything, and contrasts their efforts with the beauty of the night sky. Isuspect Whitman was just articulating feelings, and not a rationally worked out position: however deep his feelings might be, and however much gristthey might provide for a philosophical mill, Whitman still was not doingphilosophy.6 The term axiologysometimes (and perhaps more properly) is used in a more constrained way, to refer strictly to the study of questions of value; insuch usage, axiology cuts across the studies I list, functioning as a component of each rather than an umbrella term for all.7 Two I haven't experienced are drunkenness and drug euphoria. Since our advanced ability to reason is one of the very few things that (when we use

it) elevates humans above other animals, it is difficult for me to understand the appeal of immoderate use of mind-altering substances. The massappeal of drugs and overindulgence in alcohol seems to me an example of collective madness. Both are antithetical to the spirit of analytic philosophy.What is logic?

Logic encompasses many different kinds of study, so that one might wonder what the

common thread is. Some claim that logic is the study of truth, and is thus the most basic

and fundamental science. While every science aims at truth, logic is the science of truth

itself. It tries to discover the truth about truth. Logic studies truth in the same way that

biology studies living things.

Others say that logic is concerned with thought, and tries to discover the laws of

thought. These laws do not describe the way people actually think, for that is the task of

psychology. Rather, they prescribe the way people ought to think; they describe the way

a perfect mind thinks. Logic is like ethics and morality, in that it separates right from

wrong. Logic, one might say, is the ethics of thought and belief.

Others say that logic is essentially concerned with language, in some way. Logic tries to

understand the logical form of statements, and certain structural relations between

sentences. It is certainly true that logicians spend a lot of time studying languages,

especially the artificial languages that logicians themselves have devised. Much of this

course will consist of learning how to use artificial languages. Logicians also study

natural languages, such as English, French, Mandarin and so on.

In my view, logic is concerned with all three of these, with thought, with language, and

with truth. This is possible, since these three things are closely connected. But how?

This is a controversial area in philosophy, which we shall explore a little.
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1.1 Meaning

Lets start with language. A language consists of symbols, which are usually marks on

paper or spoken sounds. (They may be other things, such as smoke signals.) These

symbols are linked, via convention, with meanings, so that language can be used for

communication. When words are spoken, the meaning is conveyed from speaker to

listener. It should be noted that language has a variety of uses, such as to express

emotions, to give commands, and to ask questions. In this course however we are only

concerned with one function of language, namely the attempt to convey information.

The first thing to recognise about language is that a sentence is the smallest piece of

language that makes up an act of communication, or conveys information. Smaller

linguistic units, such as individual words, do not communicate anything by themselves.

They do not express statements. The meaning of a word comes from its contribution to 2

the meanings of sentences that contain that word. Thus the fundamental kind of meaning

is the meaning of a sentence. Since logic is concerned with information, the only kind of

sentence we are interested in is the kind that makes a claim, or an assertion of fact. (This

is called a declarative sentence.) Declarative sentences may state mere opinions, as well

as known facts. They may also state things that are false even things that the speaker

knows to be false. A sentence is declarative if it says something that a person could

believe.

Consider, for example, the sentence I hereby resign from this committee. It does

convey information since, when uttered by (say) Janet, it tells the audience that Janet is

resigning from the committee. But it does more than this: it also performs the act of

resigning from the committee. It is in saying those words that Janet severs her ties to that

committee. For this reason, the sentence is not declarative.

In a similar way, questions are not declarative sentences, since they request information.

Even if they sometimes supply information as well, they do not merely assert (alleged)

facts as a declarative sentence does.

MODULE: Basic logic

TUTORIAL L01: What is logic?

L01.1 A preliminary definition

The term "logic" came from the Greek word logos, which is sometimes translated as "sentence", "discourse", "reason", "rule", and "ratio". Of course,

these translations are not enough to help us understand the more specialized meaning of "logic" as it is used today.

So what is logic? Briefly speaking, we might define logic as the study of the principles of correct reasoning. This is a rough definition, because how

logic should be properly defined is actually quite a controversial matter. However, for the purpose of this tour, we thought it would be useful to give you

at least some rough idea as to the subject matter that you will be studying. So this is what we shall try to do on this page.

L01.2 Logic is not the psychology of reasoning

One thing you should note about this definition is that logic is concerned with the principles of correctreasoning. Studying the correct principles of

reasoning is not the same as studying the psychologyof reasoning. Logic is the former discipline, and it tells us how we oughtto reason if we want to

reason correctly. Whether people actually follow these rules of correct reasoning is an empirical matter, something that is not the concern of logic.

The psychology of reasoning, on the other hand, is an empirical science. It tells us about the actual reasoning habits of people, including their

mistakes. A psychologist studying reasoning might be interested in how people's ability to reason varies with age. But such empirical facts are of no

concern to the logician.

L01.3 The principles of logic

So what are these principles of reasoning that are part of logic? There are many such principles, but the main (not the only) thing that we study in logic

are principles governing the validity of arguments- whether certain conclusions follow f rom some given assumptions. For example, consider the

following three arguments :

If Tom is a philosopher, then Tom is poor.

Tom is a philosopher.

Therefore, Tom is poor.


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If K>10, then K>2.

K>10.

Therefore, K>2.

If Tarragona is in Europe, then Tarragona is not in China.

Tarragona is in Europe.

Therefore, Tarragona is not in China.

These three arguments here are obviously good arguments in the sense that their conclusions follow from the assumptions. If the assumptions of the

argument are true, the conclusion of the argument must also be true. A logician will tell us that they are all cases of a particular form of argument

known as modus ponens.We shall be discussing validity again later on. It should be pointed out that logic is not just concerned with the validity of

arguments. Logic also studies consistency, and logical truths, and properties of logical systems such as completeness and soundness. But we shall

see that these other concepts are also very much related to the concept of validity.

L01.4 Topic neutrality

Modus ponensmight be used to illustrate two features about the rules of reasoing in logic. The first feature is its topic-neutrality. As the four examples

suggest, modus ponenscan be used in reasoning about diverse topics. This is true of all the principles of reasoning in logic. The laws of biology might

be true only of living creatures, and the laws of economics are only applicable to collections of agents that enagage in financial transactions. But the

principles of logic are universal principles which are more general than biology and economics. This is in part what is implied in the following definitions

of logic by two very famous logicians :

L01.5 Necessity in logic

A second feature of the principles of logic is that they are non-contingent, in the sense that they do not depend on any particular accidental features of

the world. Physics and the other empirical sciences investigate the way the world actually is. Physicists might tell us that no signal can travel faster

than the speed of light, but if the laws of physics have been different, then perhaps this would not have been true. Similarly, biologists might study how

dolphins communicate with each other, but if the course of evolution had been different, then perhaps dolphins might not have existed. So the theories

in the empirical sciences are contingent in the sense that they could have been otherwise. The principles of logic, on the other hand, are derived using

reasoning only, and their validity does not depend on any contingent features of the world.

For example, logic tells us that any statement of the form "If Pthen P." is necessarily true. This is a principle of the second kind that logician study. This

principle tells us that a statement such as "if it is raining, then it is raining" must be true. We can easily see that this is indeed the case, whether or not it

is actually raining. Furthermore, even if the laws of physics or weather patterns were to change, this statement will remain true. Thus we say that

scientific truths (mathematics aside) are contingentwhereas logical truths are necessary. Again this shows how logic is different from the empirical

sciences like physics, chemistry or biology.

L01.6 Formal and informal logic

Sometimes a distinction is made between informal logicand formal logic. The term "informal logic" is often used to mean the same thing as critical

thinking. Sometimes it is used to refer to the study of reasoning and fallacies in the context of everyday life. "Formal logic" is mainly concerned with

formal systems of logic. These are speciall y constructed systems for carrying out proofs, where the languages and rules of reasoning are precisely and

carefully defined. Sentential logic(also known as "Propositional logic") and Predicate Logicare both examples of formal systems of logic.

There are many reasons for studying formal logic. One is that formal logic helps us identify patterns of good reasoning and patterns of bad reasoning,

so we know which to follow and which to avoid. This is why studying basic formal logic can help improve critical thinking. Formal systems of logic are

also used by linguists to study natural languages. Computer scientists also employ formal systems of logic in research relating to Aritificial Intelligence.

Finally, many philosophers also like to use formal logic when dealing with complicated philosophical problems, in order to make their reasoning more

explicit and precise.

Logic (from the Greeklogik)[1]

is the study of validreasoning.[2]

Logic is used in most intellectual activities, but is studied primarily in the

disciplines ofphilosophy,mathematics,semantics, andcomputer science. It examines general forms thatargumentsmay take, which forms are valid,

and which arefallacies. In philosophy, the study of logic is applied in most major areas: metaphysics, ontology,epistemology, andethics. In

mathematics, it is the study of validinferenceswithin someformal language.[3]

Logic is also studied inargumentation theory.[4]

Logic was studied in several ancient civil izations, includingIndia,[5]

China,[6]

andGreece. In the West, logic was established as a formal discipline

byAristotle, who gave it a fundamental place in philosophy. The study of logic was part of the classicaltrivium, which also included grammar and

rhetoric.
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Logic is often divided into three parts,inductive reasoning, abductive reasoning, anddeductive reasoning.

History

Main article:History of logic

The earliest sustained work on the subject of logic is that of Aristotle.[14]

Aristotelian logicbecame widely accepted in science and mathematics and

remained in wide use in the West until the early 19th century.[15]

Aristotle's system of logic was responsible for the introduction ofhypothetical

syllogism,[16]

temporalmodal logic,[17][18]

andinductive logic.[19]

InEuropeduring the later medieval period, major efforts were made to show that

Aristotle's ideas were compatible withChristianfaith. During theHigh Middle Ages, logic became a main focus of philosophers, who would engage in

critical logical analyses of philosophical arguments.

TheChinese logicalphilosopherGongsun Long(ca. 325250 BC) proposed the paradox "One and one cannot become two, since neither becomes

two."[20]

In China, the tradition of scholarly investigation into logic, however, was repressed by theQin dynasty following the legalist philosophy ofHan

Feizi.

In India, innovations in the scholastic school, calledNyaya, continued from ancient times into the early 18th century with theNavya-Nyayaschool. By

the 16th century, it developed theories resembling modern logic, such asGottlob Frege's "distinction between sense and reference of proper names"

and his "definition of number," as well as the theory of "restrictive conditions for universals" anticipating some of the developments in modernset

theory.[21]

Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such

asCharles Babbage,Augustus De Morgan, andGeorge Boole.[22]

In the 20th century, Western philosophers like Stanislaw Schayer and Klaus Glashoff

have explored Indian logic more extensively.

Thesyllogistic logic developed by Aristotle predominated in the West until the mid-19th century, when interest in thefoundations of

mathematicsstimulated the development of symbolic logic (now calledmathematical logic). In 1854, George Boole publishedAn Investigation of the

Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, introducing symbolic logic and the principles of what is

now known asBoolean logic. In 1879, Gottlob Frege publishedBegriffsschriftwhich inaugurated modern logic with the invention of quantifiernotation.

From 1910 to 1913, Alfred North WhiteheadandBertrand RussellpublishedPrincipia Mathematica[9]

on the foundations of mathematics, attempting to

derive mathematical truths fromaxiomsandinference rulesin symbolic logic. In 1931,Gdelraised serious problems with the foundationalist program

and logic ceased to focus on such issues.

The development of logic since Frege, Russell, andWittgensteinhad a profound influence on the practice of philosophy and the perceived nature of

philosophical problems (seeAnalytic philosophy), andPhilosophy of mathematics. Logic, especially sentential logic, is implemented in computerlogic

circuitsand is fundamental tocomputer science. Logic is commonly taught by university philosophy departments, often as a compulsory discipline.

Topics in logic

Syllogistic logic

Main article:Aristotelian logic

TheOrganonwasAristotle's body of work on logic, with thePrior Analyticsconstituting the first explicit work in formal logic, introducing the

syllogistic.[23]

The parts of syllogistic logic, also known by the nameterm logic, are the analysis of the judgements into propositions consisting of two

terms that are related by one of a fixed number of relations, and the expression of inferences by means ofsyllogismsthat consist of two propositions

sharing a common term as premise, and a conclusion which is a proposition involving the two unrelated terms from the premises.

Aristotle's work was regarded in classical times and from medieval times in Europe and the Middle East as the very picture of a fully worked out

system. However, it was not alone: theStoicsproposed a system ofpropositional logicthat was studied by medieval logicians. Also, theproblem of

multiple generalitywas recognised in medieval times. Nonetheless, problems with syllogistic logic were not seen as being in need of revolutionary

solutions.
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Today, some academics claim that Aristotle's system is generally seen as having little more than historical value (though there is some current interest

in extending term logics), regarded as made obsolete by the advent of propositional logic and thepredicate calculus. Others use Aristotle

inargumentation theoryto help develop and critically question argumentation schemes that are used inartificial intelligenceandlegalarguments.

Propositional logic (sentential logic)

Main article:Propositional calculus

A propositional calculus or logic (also a sentential calculus) is a formal system in which formulae representing propositions can be formed by

combiningatomic propositionsusinglogical connectives, and in which a system of formal proof rules allows certain formulae to be established as

"theorems".

Predicate logic

Predicate logic is the generic term for symbolic formal systems such asfirst-order logic,second-order logic,many-sorted logic, andinfinitary logic.

Predicate logic provides an account of quantifiersgeneral enough to express a wide set of arguments occurring in natural language. Aristotelian

syllogistic logic specifies a small number of forms that the relevant part of the involved judgements may take. Predicate logic allows sentences to be

analysed into subject and argument in several additional ways, thus allowing predicate logic to solve theproblem of multiple generality that had

perplexed medieval logicians.

The development of predicate logic is usually att ributed toGottlob Frege, who is also credited as one of the founders of analytical philosophy, but the

formulation of predicate logic most often used today is the first-order logic presented inPrinciples of Mathematical LogicbyDavid HilbertandWilhelm

Ackermann in 1928. The analytical generality of predicate logic allowed the formalisation of mathematics, drove the investigation ofset theory, and

allowed the development of Alfred Tarski's approach tomodel theory. It provides the foundation of modernmathematical logic.

Frege's original system of predicate logic was second-order, rather than first-order.Second-order logicis most prominently defended (against the

criticism ofWillard Van Orman Quineand others) byGeorge BoolosandStewart Shapiro.

Modal logic

In languages,modalitydeals with the phenomenon that sub-parts of a sentence may have their semantics modified by special verbs or modal particles.

For example, "We go to the games" can be modified to give "We should go to the games", and "We can go to the games"" and perhaps "We will go tothe games". More abstractly, we might say that modality affects the circumstances in which we take an assertion to be satisfied.

The logical study of modality dates back toAristotle,[24]

who was concerned with thealethic modalitiesofnecessityandpossibility, which he observed

to be dual in the sense of De Morgan duality.[citation needed]

While the study of necessity and possibility remained important to philosophers, little logical

innovation happened until the landmark investigations ofClarence Irving Lewis in 1918, who formulated a family of rival axiomatizations of the alethic

modalities. His work unleashed a torrent of new work on the topic, expanding the kinds of modality treated to includedeontic logicandepistemic logic.

The seminal work ofArthur Priorapplied the same formal language to treattemporal logicand paved the way for the marriage of the two subjects.Saul

Kripkediscovered (contemporaneously with rivals) his theory offrame semanticswhich revolutionised the formal technology available to modal

logicians and gave a newgraph-theoreticway of looking at modality that has driven many applications incomputational linguisticsandcomputer

science, such asdynamic logic.

Informal reasoning

The motivation for the study of logic in ancient times was clear: it is so that one may learn to distinguish good from bad arguments, and so become

more effective in argument and oratory, and perhaps also to become a better person. Half of the works of Aristotle'sOrganontreat inference as it

occurs in an informal setting, side by side with the development of the syllogistic, and in the Aristotelian school, these informal works on logic were

seen as complementary to Aristotle's treatment ofrhetoric.

This ancient motivation is still alive, although it no longer takes centre stage in the picture of logic; typicallydialectical logic will form the heart of a

course incritical thinking, a compulsory course at many universities.
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logichttp://en.wikipedia.org/wiki/Epistemic_logichttp://en.wikipedia.org/wiki/Epistemic_logichttp://en.wikipedia.org/wiki/Epistemic_logichttp://en.wikipedia.org/wiki/Arthur_Priorhttp://en.wikipedia.org/wiki/Arthur_Priorhttp://en.wikipedia.org/wiki/Arthur_Priorhttp://en.wikipedia.org/wiki/Temporal_logichttp://en.wikipedia.org/wiki/Temporal_logichttp://en.wikipedia.org/wiki/Temporal_logichttp://en.wikipedia.org/wiki/Saul_Kripkehttp://en.wikipedia.org/wiki/Saul_Kripkehttp://en.wikipedia.org/wiki/Saul_Kripkehttp://en.wikipedia.org/wiki/Saul_Kripkehttp://en.wikipedia.org/wiki/Kripke_semanticshttp://en.wikipedia.org/wiki/Kripke_semanticshttp://en.wikipedia.org/wiki/Kripke_semanticshttp://en.wikipedia.org/wiki/Graph_theoryhttp://en.wikipedia.org/wiki/Graph_theoryhttp://en.wikipedia.org/wiki/Graph_theoryhttp://en.wikipedia.org/wiki/Computational_linguisticshttp://en.wikipedia.org/wiki/Computational_linguisticshttp://en.wikipedia.org/wiki/Computational_linguisticshttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Dynamic_logic_(modal_logic)http://en.wikipedia.org/wiki/Dynamic_logic_(modal_logic)http://en.wikipedia.org/wiki/Dynamic_logic_(modal_logic)http://en.wikipedia.org/wiki/Organonhttp://en.wikipedia.org/wiki/Organonhttp://en.wikipedia.org/wiki/Organonhttp://en.wikipedia.org/wiki/Rhetorichttp://en.wikipedia.org/wiki/Rhetorichttp://en.wikipedia.org/wiki/Rhetorichttp://en.wikipedia.org/wiki/Dialectichttp://en.wikipedia.org/wiki/Dialectichttp://en.wikipedia.org/wiki/Dialectichttp://en.wikipedia.org/wiki/Critical_thinkinghttp://en.wikipedia.org/wiki/Critical_thinkinghttp://en.wikipedia.org/wiki/Critical_thinkinghttp://en.wikipedia.org/wiki/Critical_thinkinghttp://en.wikipedia.org/wiki/Dialectichttp://en.wikipedia.org/wiki/Rhetorichttp://en.wikipedia.org/wiki/Organonhttp://en.wikipedia.org/wiki/Dynamic_logic_(modal_logic)http://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computer_sciencehttp://en.wikipedia.org/wiki/Computational_linguisticshttp://en.wikipedia.org/wiki/Graph_theoryhttp://en.wikipedia.org/wiki/Kripke_semanticshttp://en.wikipedia.org/wiki/Saul_Kripkehttp://en.wikipedia.org/wiki/Saul_Kripkehttp://en.wikipedia.org/wiki/Temporal_logichttp://en.wikipedia.org/wiki/Arthur_Priorhttp://en.wikipedia.org/wiki/Epistemic_logichttp://en.wikipedia.org/wiki/Deontic_logichttp://en.wikipedia.org/wiki/Clarence_Irving_Lewishttp://en.wikipedia.org/wiki/Wikipedia:Citation_neededhttp://en.wikipedia.org/wiki/De_Morgan_dualityhttp://en.wikipedia.org/wiki/Logical_possibilityhttp://en.wikipedia.org/wiki/Necessityhttp://en.wikipedia.org/wiki/Alethic_modalityhttp://en.wikipedia.org/wiki/Logic#cite_note-23http://en.wikipedia.org/wiki/Aristotlehttp://en.wikipedia.org/wiki/Linguistic_modalityhttp://en.wikipedia.org/wiki/Stewart_Shapirohttp://en.wikipedia.org/wiki/George_Booloshttp://en.wikipedia.org/wiki/Willard_Van_Orman_Quinehttp://en.wikipedia.org/wiki/Second-order_logichttp://en.wikipedia.org/wiki/Mathematical_logichttp://en.wikipedia.org/wiki/Model_theoryhttp://en.wikipedia.org/wiki/Alfred_Tarskihttp://en.wikipedia.org/wiki/Set_theoryhttp://en.wikipedia.org/wiki/Wilhelm_Ackermannhttp://en.wikipedia.org/wiki/Wilhelm_Ackermannhttp://en.wikipedia.org/wiki/David_Hilberthttp://en.wikipedia.org/wiki/Principles_of_Mathematical_Logichttp://en.wikipedia.org/wiki/Analytical_philosophyhttp://en.wikipedia.org/wiki/Gottlob_Fregehttp://en.wikipedia.org/wiki/Problem_of_multiple_generalityhttp://en.wikipedia.org/wiki/Quantifiershttp://en.wikipedia.org/wiki/Infinitary_logichttp://en.wikipedia.org/wiki/Many-sorted_logichttp://en.wikipedia.org/wiki/Second-order_logichttp://en.wikipedia.org/wiki/First-order_logichttp://en.wikipedia.org/wiki/Logical_connectiveshttp://en.wikipedia.org/wiki/Atomic_propositionshttp://en.wikipedia.org/wiki/Propositional_calculushttp://en.wikipedia.org/wiki/Legalhttp://en.wikipedia.org/wiki/Artificial_intelligencehttp://en.wikipedia.org/wiki/Argumentation_theoryhttp://en.wikipedia.org/wiki/Predicate_calculus


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Argumentation theory is the study and research of informal logic, fallacies, and critical questions as they relate to every day and practical situations.

Specific types of dialogue can be analyzed and questioned to reveal premises, conclusions, and fallacies. Argumentation theory is now applied

inartificial intelligenceandlaw.

Mathematical logic

Mathematical logic really refers to two distinct areas of research: the first is the application of the techniques of formal logic to mathematics and

mathematical reasoning, and the second, in the other di rection, the application of mathematical techniques to the representation and analysis of formal

logic.[25]

The earliest use of mathematics andgeometry in relation to logic and philosophy goes back to the ancient Greeks such asEuclid,Plato,

andAristotle.[26]

Many other ancient and medieval philosophers applied mathematical ideas and methods to their philosophical claims.

Kurt Gdel

One of the boldest attempts to apply logic to mathematics was undoubtedly thelogicismpioneered by philosopher-logicians such asGottlob

FregeandBertrand Russell: the idea was that mathematical theories were logicaltautologies, and the programme was to show this by means to a

reduction of mathematics to logic.[9]

The various attempts to carry this out met with a series of failures, from the crippling of Frege's project in

his GrundgesetzebyRussell's paradox, to the defeat ofHilbert's programbyGdel's incompleteness theorems.

Both the statement of Hilbert's program and its refutation by Gdel depended upon their work establishing the second area of mathematical logic, the

application of mathematics to logic in the form ofproof theory.[28]

Despite the negative nature of the incompleteness theorems, Gdel's completeness

theorem, a result inmodel theoryand another application of mathematics to logic, can be understood as showing how close logicism came to being

true: every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege'sproof calculusis enough

to describethe whole of mathematics, though not equivalentto it. Thus we see how complementary the two areas of mathematical l ogic have

been.[citation needed]

If proof theory and model theory have been the foundation of mathematical logic, they have been but two of the four pil lars of the subject.Set

theoryoriginated in the study of the infinite byGeorg Cantor, and it has been the source of many of the most challenging and important issues in

mathematical logic, fromCantor's theorem, through the status of theAxiom of Choiceand the question of the independence of thecontinuum

hypothesis, to the modern debate onlarge cardinalaxioms.

Recursion theorycaptures the idea of computation in logical andarithmeticterms; its most classical achievements are the undecidability of

theEntscheidungsproblembyAlan Turing, and his presentation of theChurch-Turing thesis.[29]

Today recursion theory is mostly concerned with the

more refined problem ofcomplexity classes when is a problem efficiently solvable? and the classification ofdegrees of unsolvability.[30]

Philosophical logic

Philosophical logicdeals with formal descriptions of natural language. Most philosophers assume that the bulk of "normal" proper reasoning can be

captured by logic, if one can find the right method for translating ordinary language into that logic. Philosophical logic is essentially a continuation of the

traditional discipline that was called "Logic" before the invention of mathematical logic. Philosophical logic has a much greater concern with the

connection between natural language and logic. As a result, philosophical logicians have contributed a great deal to the development of non-standard

logics (e.g.,free logics,tense logics) as well as various extensions of classical logic(e.g.,modal logics), and non-standard semantics for such logics

(e.g.,Kripke's technique of supervaluations in the semantics of logic).

Logic and the philosophy of language are closely related. Philosophy of language has to do with the study of how our language engages and interacts

with our thinking. Logic has an immediate impact on other areas of study. Studying logic and the relationship between logic and ordinary speech can

help a person better structure his own arguments and critique the arguments of others. Many popular arguments are filled with errors because so many

people are untrained in logic and unaware of how to formulate an argument correctly.

[edit]Computational Logic

Main article:Logic in computer science
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Logic cut to the heart of computer science as it emerged as a discipline:Alan Turing's work on theEntscheidungsproblem followed fromKurt Gdel's

work on theincompleteness theorems, and the notion of general purpose computers that came from this work was of fundamental importance to the

designers of the computer machinery in the 1940s.

In the 1950s and 1960s, researchers predicted that when human knowledge could be expressed using logic withmathematical notation, it would be

possible to create a machine that reasons, or artificial intelligence. This turned out to be more difficult than expected because of the complexity of

human reasoning. Inlogic programming, a program consists of a set of axioms and rules. Logic programming systems such asPrologcompute the

consequences of the axioms and rules in order to answer a query.

Today, logic is extensively applied in the fields ofArtificial Intelligence, andComputer Science, and these fields provide a rich source of problems in

formal and informal logic. Argumentation theoryis one good example of how logic is being applied to artificial intelligence. TheACM Computing

Classification Systemin particular regards:

Section F.3 onLogics and meanings of programsand F.4 onMathematical logic and formal languagesas part of the theory of computer science:

this work coversformal semantics of programming languages, as well as work offormal methodssuch asHoare logic

Boolean logicas fundamental to computer hardware: particularly, the system's section B.2 onArithmetic and logic structures, relating to

operatives AND, NOT, and OR;

Many fundamental logical formalisms are essential to section I.2 on artificial intelligence, for examplemodal logicanddefault logicinKnowledge

representation formalisms and methods, Horn clauses in logic programming, anddescription logic.

Furthermore, computers can be used as tools for logicians. For example, in symbolic logic and mathemat ical logic, proofs by humans can be computer-

assisted. Usingautomated theorem proving the machines can find and check proofs, as well as work with proofs too lengthy to be written out by hand.

[edit]Bivalence and the law of the excluded middle

Main article:Principle of bivalence

The logics discussed above are all "bivalent" or "two-valued"; that is, they are most naturally understood as dividing propositions into true and false

propositions.Non-classical logicsare those systems which reject bivalence.

Hegel developed his owndialectic logic that extendedKant's transcendental logic but also brought it back to ground by assuring us that "neither inheaven nor in earth, neither in the world of mind nor of nature, is there anywhere such an abstract 'eitheror' as the understanding maintains. Whatever

exists is concrete, with difference and opposition in itself".[31]

In 1910Nicolai A. Vasilievextended the law of excluded middle and the law of contradiction and proposed the law of excluded fourth and logic tolerant

to contradiction.[32]

In the early 20th centuryJan ukasiewicz investigated the extension of the traditional true/false values to include a third value,

"possible", so inventingternary logic, the firstmulti-valued logic.[citation needed]

Logics such asfuzzy logichave since been devised with an infinite number of "degrees of truth", represented by areal numberbetween 0 and 1.[33]

Intuitionistic logicwas proposed byL.E.J. Brouweras the correct logic for reasoning about mathematics, based upon his rejection of the law of the

excluded middleas part of hisintuitionism. Brouwer rejected formalisation in mathematics, but his studentArend Heytingstudied intuitionistic logic

formally, as didGerhard Gentzen. Intuitionistic logic has come to be of great interest to computer scientists, as it is aconstructive logicand can be

applied for extracting verified programs from proofs.

Modal logic is not truth conditional, and so it has often been proposed as a non-classical logic. However, modal logic is normally formalised with the

principle of the excluded middle, and itsrelational semanticsis bivalent, so this inclusion is disputable.

"Is logic empirical?"

What is theepistemologicalstatus of thelaws of logic

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