LEGAL TECHNIQUES AND LOGIC JUDGE ALGODON

I. INTRODUCTION

A. History of Logic B. Civil Law vs. Common Law Tradition C. The Role of Logic in Law

A. History of Logic

The founder of Logic is Aristotle of Stagira in Thrace (384-322B.C.), son of the physician NIchomachus and became Plato s student at 18 years of age and studied with him for 20 years. Shortly afterward in 343 B.C., became a tutor for 3 years to the 13- year old Alexander, who was to rule the world as Alexander the Great. In 335 B. C., he founded his own school, the Peripathetic School of philosophers.

Aristotle s work on logic is found in his Organon (meaning the method or organ of investigation) which consisted of a number of his writings, including the following: On Interpretation -dealt with the structure of logical proposition; Prior Analytics-the doctrine of syllogism; Posterior Analytics-the logic of science and the applications of the syllogism; Topics-the logic of argument based on probable truths; Sophistical Test-dealt with logical fallacies.

Logical reasoning makes us certain that our conclusions are true and this provides us with accepted scientific proofs of universally valid propositions or statements.

B. Civil Law vs. Common Law Tradition

COMMON LAW: That which derives its force and authority from the universal consent and immemorial practice of the people. The system of jurisprudence that originated in England and which was latter adopted in the U.S. that is based on precedent instead of statutory laws. Traditional law of an area or region; also known as case law. The law created by judges when deciding individual disputes or cases. The body of law which includes both the unwritten law of England and the statutes passed before the settlement of the United States.

In the common law, civil law is the area of laws and justice that affect the leg al status of individuals. Civil law, in this sense, is usually referred to in comparison to criminal law, which is th at body of law involving the state against individuals (including incorporated organizations) where the state relie s on the power given it by statutory law.

CIVIL LAW:

Civil law may also be compared to military law, administrative law and constitut ional law (the laws governing the political and law making process), and international law. Where there are le gal options for causes of action by individuals within any of these areas of law, it is thereby civil law.

Civil law courts provide a forum for deciding disputes involving torts (such as accidents, negligence, and libel), contract disputes, the probate of wills, trusts, property disputes, admi nistrative law,commercial law, and any other private matters that involve private parties and organizations inc luding government departments. An action by an individual (or legal equivalent) against the attorn ey general is a civil matter, but when the state, being represented by the prosecutor for the attorney general, or some other agent for the state, takes action against an individual (or legal equivalent including a gover nment department), this is public law, not civil law.

The objectives of civil law are different from other types of law. In civil law: a. there is the attempt to right a wrong, b. honor an agreement, or c. settle a dispute. d. If there is a victim, they get compensation, and the person who is the cause of the wrong pays, this being a civilized form of, or legal alternative to, revenge. e. If it is an equity matter, there is often a pie for division and it gets allo cated by a process of civil law, possibly invoking the doctrines of equity. f. In public law, the objective is usually deterrence, and retribution.

An action in criminal law does not necessarily preclude an action in civil law i n common law countries, and may provide a mechanism for compensation to the victims of crime. Such a situati on occurred when O.J. Simpson was ordered to pay damages for wrongful death after being acquitted of t he criminal charge of murder.

Civil law in common law countries usually refers to both common law and the law of equity, which while now merged in administration, have different traditions, and have historically opera ted to different doctrines, although this dualism is increasingly being set aside so there is one coherent b ody of law rationalized around common principles of law.

C. Role of Logic in Law

Regardless of the professions we are in, we always use logic. We use it when we make decisions or when we try to influence the decisions of others or when we are engaged in argumentation and debate. A lawyer presents his arguments using the principle of logic to prove the tenabi lity of his position, otherwise, he will send his client to jail. Everybody uses logic since everyone possesses r eason.

II. REASONING A. Basic Concepts

1. What is Logic 2. Propositions and Sentences 3. Arguments, Premises and Conclusions 4. More Complex Arguments 5. Recognizing Arguments 6. Deduction and Induction 7. Validity and Truth 8. Arguments and Explanations

B. Analyzing and Diagramming Arguments

C. Problem Solving

1. What is Logic

1. Reasoning conducted or assessed according to strict principles of validity: " experience is a better guide to this than deductive logic". 2. A particular system or codification of the principles of proof and inference: "Aristotelian logic".

2. Propositions and Sentences

Proposition refers to either the "content" or "meaning" of a meaningful declarat ive sentence. The meaning of a proposition includes having the quality or property of being either true or false

Propositional logic largely involves studying logical connectives such as the wo rds and and or and the rules determining the truth-values of the propositions they are used to join, as

well as what these rules mean for the validity of arguments, and such logical relationships between statements as being consistent or inconsistent with one another, as well as logical properties of propositions

Sentence . Sentence logic deals with sentences of a natural language that are either true or false . Sentence logic ignores the internal structure of simple sentences . Sentence logic is concerned with sentences which are compounded in a certain w ay. . A primary goal of sentence logic is to enable the evaluation of a certain clas s of arguments in natural language . In an argument, a sentence that is the argument s con

3. Arguments, Premises and Conclusions

Argument is a connected series of statements intended to establish a definite pr oposition. ...an argument is an intellectual process... contradiction is just the automatic gainsaying of any thing the other person says.

An argument is a deliberate attempt to move beyond just making an assertion. Whe n offering an argument, you are offering a series of related statements which represent an attempt to su pportthat assertion to give others good reasons to believe that what you are asserting is true rather t han false. Here are examples of assertions: 1. 2. 3. 4. Shakespeare wrote the play Hamlet. The Civil War was caused by disagreements over slavery. God exists. Prostitution is immoral.

Sometimes you hear such statements referred to as propositions. Technically spea king, a proposition is the informational content of any statement or assertion. To qualify as a proposition , a statement must be capable of being either true or false 3 major components of Argument: 1. Premise 2. Inference 3. Conclusion Premises are statements of (assumed) fact which are supposed to set forth the re asons and/or evidence for believing a claim. The claim, in turn, is the conclusion: what you finish with a t the end of an argument. When an argument is simple, you may just have a couple of premises and a conclusion: 1. Doctors earn a lot of money. (premise) 2. I want to earn a lot of money. (premise) 3. I should become a doctor. (conclusion) Inferences are the reasoning parts of an argument. Conclusions are a type of inf erence, but always the final inference. Usually an argument will be complicated enough to require inferences linking the premises with the final conclusion: 1. 2. 3. 4. 5. Doctors earn a lot of money. (premise) [FACTUAL] With a lot of money, a person can travel a lot. (premise) [FACTUAL] Doctors can travel a lot. (inference, from 1 and 2) I want to travel a lot. (premise) I should become a doctor. (from 3 and 4)

2 Types of Claims: 1. Factual Claim 2. Inferential Claim

- it expresses the idea that some matter of fact is related to the sought-after conclusion. This is the attempt to link the factual claim to the conclusion in such a way as to support the conc lusion. The third statement above is an inferential claim because it infers from the previous two statements that doctors can travel a lot. Without an inferential claim, there would be no clear connection between the pre mises and the conclusion. It is rare to have an argument where inferential claims play no role. Sometimes you will come across an argument where inferential claims are needed, but missing you won t be able to see the connection from factual claims to conclusion and will have to ask for them. 5. Recognizing Arguments we use the term "argument" to mean a set of propositions in which some propositi ons--the premises--are asserted as support or evidence for another--the conclusion. The author doesn't just tell us something that he takes to be true; he also pres ents reasons intended to convince us that it is true. This intention is usually signaled by certain indicator words. The following is a list of the more common indicator words: Premise Indicators Conclusion Indicators Since Therefore Because Thus As So For Consequently Given that As a result Assuming that It follows that Inasmuch as

Hence The reason is that Which means that In view of the fact that Which implies that

6. Deduction and Induction Deduction: A deductive argument claims that its premises make its conclusion certain.

This deductive argument is valid because the conclusion follows with certainty if the premises are true. There is no possible way for the premises to be true and yet the conclusion false Example: All mammals have lungs. All whales are mammals. Therefore all whales have lungs.

tree.jpg A valid deductive argument with true premises is a sound argument. A sound argum ent is often called a proof, but this term can be misleading. If the premises themselves are absolutely certain, then a sound argument does indeed offer proof, as in the below example: 1. All bachelors are unmarried. 2. All bachelors are male. 3. Therefore all bachelors are unmarried males. The premises are certain here because they are true by definition, and the argum ent is sound, so the conclusion is proven

Inductive: Inductive arguments do not try to establish their conclusions with certainty. In stead, an inductive argument claims that its premises make the conclusion probable. Inductive arguments canno t be valid or invalid. Instead, they are weak or strong, better or worse. And even when the premises ar e true and provide very strong support for the conclusion, the conclusion cannot be certain. The stronge st inductive argument is not as conclusive as a sound deductive argument.

Example: Most corporation lawyers are conservatives. Betty Morse is a corporation lawyer. Therefore Betty Morse is a conservative.

7. Validity and Truth

Validity: Deductive arguments may be either valid or invalid. If an argument is valid, and its premises are true, the conclusion must be true: a valid argument cannot have true premises and a false conclusion.

The validity of an argument depends, however, not on the actual truth or falsity of its premises and conclusions, but solely on whether or not the argument has a valid logical form. The validity of an argument is not a guarantee of the truth of its conclusion. A valid argument may have false premises and a false conclusion. The corresponding conditional of a valid argument is a necessary truth (true in all possible worlds) and so the conclusion necessarily follows from the premises, or follows of logical necessit y. The conclusion of a valid argument is not necessarily true, it depends on whether the premises are true. T he conclusion of a valid argument need not be a necessary truth: if it were so, it would be so independen tly of the premises. For example: Some Greeks are logicians; therefore, some logicians are Greeks. Valid argument; it would be self-contradictory to admit that some Greeks are log icians but deny that some (any) logicians are Greeks.

All Greeks are human and all humans are mortal; therefore, all Greeks are mortal . : Valid argument; if the premises are true the conclusion must be true.

Some Greeks are logicians and some logicians are tiresome; therefore, some Greek s are tiresome. Invalid argument: the tiresome logicians might all be Romans (for example).

Either we are all doomed or we are all saved; we are not all saved; therefore, w e are all doomed. Valid argument; the premises entail the conclusion. (Remember that this does not mean the conclusion has to be true; it is only true if the premises are true, which they may not be!)

Premise 1: Some men are hawkers. Premise 2: Some hawkers are rich. Conclusion: S ome men are rich. This argument is invalid. There is a way where you can determine whether an argu ment is valid, give a counter-example with the same argument form.

Counter-Example: Premise 1: Some people are herbivores. Premise 2: Some herbivor es are zebras. Conclusion: Some people are zebras. (This is obviously false.) Note that the counter-example follows the P1. Some x are y. P2. Some y are z. C. Some x are z. format. We can now conclude that the hawker argument is invalid. Arguments can be invalid for a variety of reasons. There are well-established pa tterns of reasoning that render arguments that follow them invalid; these patterns are kno wn as logical fallacies.

Truth: 1. Coherence - Is a statement true when it aligns with, is consistent with and d oesn't contradict other true statements? 2. Correspondence - Is a statement true when it corresponds to something in the real world? 3. Foundationalism - Can certain statements be asserted as true in and of themse lves, self-evidently? Will truth then deductively follow from these assumptions? 4. Pragmatic accounts - Is a statement true when it proves to be useful or pract ical? () 5. Consensus - Is a statement true when enough people argue or believe that it i s true? ()

6. Deflationism - Is truth not an actual property of a statement at all, but som ething else? Can 'true' or 'the truth' ever be predicated in a meaningful, non-redundant way? ()

8. Argument and Explanation Argument Types: 1. Deductive Argument - asserts that the truth of the conclusion is a logical consequence of the premi ses 2. Inductive Argument - asserts that the truth of the conclusion is supported by the premises

Argument is an attempt to persuade someone of something, by giving reasons or ev idence for accepting a particular conclusion. The general structure of an argument in a natural languag e is that of premises (typically in the form of propositions, statements or sentences) in support of a claim: the conclusion. Many arguments can also be formulated in a formal language. An argument in a formal language sh ows the logical form of the natural language arguments obtained by its interpretations.* arguments obtained by its interpretations.*

Distinguish arguments from explanations.

Argument Explanation

(1) expresses an

inference does not usually express an inference

(2) offers evidence, grounds or reasons offers an account why

(3) goes from well known statements to statements less well known gives less well known statements why a better known statement is true

(4) draws a logical connection between statements describes a

causal connection

(5) has the purpose to establish the truth of a statement has the purpose to give an account of something

Definition of an Argument and an Explanation Argument has a number of different definitions. Essentially, it is a line of logi c that is presented in order to support the veracity of a statement. Argument has combative connotations, but an argument does not have to be belligerent. Explanation is used to clarify and explicate a statement. Its aim is to make the listener understand the statement rather than persuade him to accept a certain point of view. Example of an Argument and an Explanation Argument one person wants to convince the other person that it is going to snow t omorrow. He will cite predictions from the weather station, as well as the clouds visible on the horiz on, the damp chill in the air, and the squirrels furiously hiding their nuts. Explanation one both people agree is it going to snow tomorrow because, they say, there is a cold front coming in and the air feels damp. In both cases, the example of snow is used, but note that the argument is trying to convince someone of the truth of their statement, whereas with the explanation, it is not a matter of if the statement is true, but why it is true. Uses of Arguments and Explanations Arguments arguments are used in a variety of professional and academic applicatio ns. For instance, a debate club will take on both sides of an argument and strive to prove each one is right. Arguments are also used by lawyers to convince the jury of the defendant s guilt or innocence. Diplom ats will approach a negotiating table with a certain argument in mind. Entrepreneurs will present po tential backers with an argument in support of their business model. Explanations are used all the time in the classroom to put across new items to st udents. Giving directions is a form of explanation. You will also find explanations included with most new pu rchases, especially those with some assembly required. When the aforementioned entrepreneur is presenting an argument about his business model, he may be asked to explain how it all works.

Summary: 1.Arguments and explanations are both used to get the point across when speaking or writing. 2.Arguments are persuasive and seek to make people understand that something is true, whereas explanations start with the assumption of truthfulness and tell why or how the s tatement has come into being. 3.Both arguments and explanations have wide application in education and busines s, but arguments are used

for persuasion and explanations are used for clarification. Explanation: An explanation is a set of statements constructed to describe a set of facts whi ch clarifies the causes, context, and consequences of those facts.

In scientific research, explanation is one of the purposes of research, e.g., ex ploration and description. Explanation is a way to uncover new knowledge, and to report relationships among different aspects of studied phenomena. 4 Types of Explanation:

1. A thing's material cause is the material of which it consists. (For a table, that might be wood; for a statue, that might be bronze or marble.) 2. A thing's formal cause is its form, i.e. the arrangement of that matter. 3. A thing's efficient or moving cause[4] is "the primary source of the change o r rest." An efficient cause of x can be present even if x is never actually produced and so should not be confused with a sufficient cause.[5] (Aristotle argues that, for a table, this would be t he art of table-making, which is the principle guiding its creation.)[2] 4. A thing's final cause is its aim or purpose. That for the sake of which a thi ng is what it is. (For a seed, it might be an adult plant. For a sailboat, it might be sailing. For a bal l at the top of a ramp, it might be coming to rest at the bottom.)

III. LANGUAGE

A. Uses of Language

1. Three Basic Functions of Language 2. Discourse Serving Multiple Functions 3. Forms of Discourse 4. Emotive Words 5. Kinds of Agreement and Disagreement 6. Emotively Neutral Language

1. Three Basic Functions of Language

1. Informative language function: essentially, the communication of information.

a. The informative function affirms or denies propositions, as in science or the statement of a fact. b. It is used to describe the world or reason about it (e.g.., whether a state o f affairs has occurred or not or what might have led to it). c. These sentences have a truth value; that is, the sentences are either true or false (recognizing, of course, that we might not know what that truth value is). Hence , they are important for logic.

2. Expressive language function: reports feelings or attitudes of the writer (or speaker), or of the subject, or evokes feelings in the reader (or listener).

a. Poetry and literature are among the best examples, but much of, perhaps most of, ordinary language discourse is the expression of emotions, feelings or attitudes . b. Two main aspects of this function are generally noted: (1) evoking certain fe elings and (2) expressing feelings. c. Expressive discourse, qua expressive discourse, is best regarded as neither t rue or false. E.g., Shakespeare's King Lear's lament, "Ripeness is all!" or Dickens' "I t was the best of

times, it was the worst of times; it was the age of wisdom; it was the age of fo olishness " Even so, the "logic" of "fictional statements" is an interesting area of inquiry .

3. Directive language function: language used for the purpose of causing (or pre venting) overt action. a. The directive function is most commonly found in commands and requests. b. Directive language is not normally considered true or false (although various logics of commands have been developed). c. Example of this function: "Close the windows." The sentence "You're smoking i n a nonsmoking area, although declarative, can be used to mean "Do not smoke in this area."

2. Discourse Serving Multiple Functions

Almost any ordinary communication will probably exhibit all three uses of langua ge. Thus a poem, which may be primarily expressive, also may have a moral and thus also be directive. And, of course, a poem may contain a certain amount of information as well. Effective communication often demands t hat language serve multiple functions.

3. Forms of Discourse

. Sentences are commonly divided into four grammatical forms: a. declarative, b. interrogative, c. imperative, and d. exclamatory.

. Much discourse is intended to serve two or possibly all three functions of lan guage informative, expressive, directive at once. In such cases each aspect or function of a given pa ssage is subject to its own proper criteria.

. Logicians are most concerned with truth and falsehood and the related notions of the correctness and incorrectness of arguments. Thus, to study logic we must be able to differen tiate discourse that functions informatively from discourse that does not.

. argumentation/ persuasion one of the four forms of discourse which uses logic, ethics, and emotional appeals (logos, ethos, pathos) to develop an effective means to convin ce the reader to think or act in a certain way

. persuasion relies more on emotional appeals than on facts

. argument form of persuasion that appeals to reason instead of emotion to convi nce an audience to think or act in a certain way

. description a form of discourse that uses language to create a mood or emotion exposition one of the four major forms of discourse, in which something is explained or "set forth " narrative the form of discourse that tells about a series of events.

4. Emotive Words

. Emotive words are words that carry emotional overtones. These words are said t o have emotive significance or emotive meaning or emotional impact.

1. Two different words or phrases can have literal (or denotative) meanings whic h are similar, but differ significantly in their emotive significance. 2. Often, we speak of "slanting" as emotive significance; i.e., a word or phrase can be positively slanted, neutral, or negatively slanted.

. The informative function derives from the literal meaning of the words in the sentence the objects, events, or attributes they refer to and the relationship among them asserted by th e sentence. The expressive content emerges because some of the words in the sentence may also ha ve emotional suggestiveness or impact. Words, then, can have both a literal meaning and an em otive meaning. The literal meanings and the emotive meanings of a word are largely independent of one another. . Language has a life of its own, independent of the facts it is used to describ e.

. The game confirms what common experience teaches: One and the same thing can b e referred to by words that have very different emotive impacts.

5. Kinds of Agreement and Disagreement

. Aexcessive reliance on emotively charged language can create the appearance of disagreement between parties who do not differ on the facts at all, and it can just as easily disguise substantive disputes under a veneer of emotive agreement. Since the degrees of agreement in belief and attitude are independent of each other, there are four possible combinations at work here: 1. Agreement in Belief and Agreement in Attitude;

2. Agreement in Belief but Disagreement in Attitude; 3. Disagreement in Belief but Agreement in Attitude; 4. Disagreement in Belief and Agreement in Attitude.

. Agreement in belief and agreement in attitude: There aren't any problems in th is instance, since both parties hold the same positions and have the same feelings about them. . Ex. Mr. Recto: The sun is very far since it s 90 million miles away. . Ms. Grace: Yes that is very far, indeed.

. Agreement in belief but disagreement in attitude: This case, if unnoticed, may become the cause of endless (but pointless) shouting between people whose feelings differ sharply ab out some fact upon which they are in total agreement. . Ex.Mr. Abella: The sun is not so far; it s only 93 million miles away. . Ms. Tiffany: The sun is, indeed, very far since it s 93 million miles away.

. Disagreement in belief but agreement in attitude: In this situation, parties m ay never recognize, much less resolve, their fundamental difference of opinion, since they are lulle d by their shared feelings into supposing themselves allied. . Ex. Mr. Magarin: The sun is incredibly far from the earth; it s 60 million miles away. . Ms. Smith: Yes, the sun is extremely far from the earth, but it s 90 million mil es away.

. Disagreement in belief and disagreement in attitude: Here the parties have so little in common that communication between them often breaks down entirely. . Ex. Mr. Cade: The sun is really very close to earth, only 60 million miles. . Ms. Cade: No, the sun is incredibly far away; it s over 93 million miles from ea rth.

. It is often valuable, then, to recognize the levels of agreement or disagreeme nt at work in any exchange of views. That won't always resolve the dispute between two parties, of course, but it will ensure that they don't waste their time on an inappropriate method of argument o r persuasion.

5. Emotively Neutral Language

. Neutral language is to be preferred when factual truth is our objective. When we are trying to learn what really is the case, or trying to follow an argument, distractions will be f rustrating; and emotion is a powerful distraction. Therefore, when we are trying to reason about facts, referring to them in emotive language is a hindrance.

. Language that is altogether neutral may not be available when we deal with som e very controversial matters. Language that is heavily charged with emotional meaning is unlikely to advance the quest for truth.

. If our aim is to communicate information, and if we wish to avoid being misund erstood, we should use language with the least possible emotive impact.

B. Definition

1. Disputes, Verbal Disputes and Definitions 2. Kinds of Definition and the Resolution of Disputes 3. Denotation (Extension) and Connotation (Intension) 4. Extension, and Denotative Definitions 5. Intension, and Connotative Definition 6. Rules for Definition by Genus and Difference

1. Disputes, Verbal Disputes and Definitions

Disputes: Disputes have their origins in disagreements between individuals.

The disagreement only becomes a dispute when one or other party cannot live with the consequences of the disagreement, and insists on having it resolved.

Disputes mostly arise either from a genuine difference of opinion or from dising enuous self-interest.

2. Kinds of Definition and the Resolution of Disputes

Dispute Resolution: Generally refers to one of several different processes used to resolve disputes between parties, including negotiation, mediation, arbitration, collaborative law, and litigation. Dispute resolution is the process of

resolving a dispute or a conflict by meeting at least some of each side s needs an d addressing their interests. Dispute resolution, or conflict resolution to use another common term, is a rela tively new field, emerging after World War II. Scholars from the Program on Negotiation were leaders in est ablishing the field.

. Adjudicative processes, such as litigation or arbitration, in which a judge, j ury or arbitrator determines the outcome. . Consensual processes, such as collaborative law, mediation, conciliation, or n egotiation, in which the parties attempt to reach agreement. . Not all disputes, even those in which skilled intervention occurs, end in reso lution. Such intractable disputes form a special area in dispute resolution studies.

Theoretical definitions: Are special cases of stipulative or precising definition, distinguished by their attempt to establish the use of this term within the context of a broader intellectual framework. Since the adop tion of any theoretical definition commits us to the acceptance of the theory of which it is an integral part, we are rightly cautious in agreeing to it

Persuasive definition: Is an attempt to attach emotive meaning to the use of a term. Since this can onl y serve to confuse the literal meaning of the term, persuasive definitions have no legitimate use.

The most common way of preventing or eliminating differences in the use of langu ages is by agreeing on the definition of our terms. Since these explicit accounts of the meaning of a word or phrase can be offered in distinct contexts and employed in the service of different goals, it's useful to distinguish definitions of several kinds:

Kinds of Definition: 1. Stipulative Definition 2. Lexical Definition

Stipulative Definition: At the other extreme, a stipulative definition freely assigns meaning to a compl etely new term, creating a usage that had never previously existed.

Lexical Definition: A lexical definition simply reports the way in which a term is already used with in a language community.

Kinds of Resolutions of Disputes: 1. 2. 3. 4. Judicial Dispute Resolution Extrajudicial Dispute Resolution Online Dispute Resolution Genuine and Verbal Resolution

Judicial Dispute Resolution:

. The legal system provides a necessary structure for the resolution of many dis putes. However, some disputants will not reach agreement through a collaborative processes. Some disp utes need the

coercive power of the state to enforce a resolution. Perhaps more importantly, m any people want a professional advocate when they become involved in a dispute, particularly if th e dispute involves perceived legal rights, legal wrongdoing, or threat of legal action against them .

. The most common form of judicial dispute resolution is litigation. programs an nexed to the courts, to facilitate settlement of lawsuits.

Extrajudicial Dispute Resolution

. Some use the term dispute resolution to refer only to alternative dispute reso lution (ADR), that is, extrajudicial processes such as arbitration, collaborative law, and mediation us ed to resolve conflict and potential conflict between and among individuals, business entities, governm ental agencies, and (in the public international law context) states.

Online Dispute Resolution

. Dispute resolution can also take place on-line or by using technology in certa in cases. Online dispute resolution, a growing field of dispute resolution, uses new technologies to solv e disputes. Online Dispute Resolution is also called "ODR". Online Dispute Resolution or ODR also i nvolves the application of traditional dispute resolution methods to disputes which arise on line

Genuine and Verbal Resolution

. Genuine disputes involve disagreement about whether or not some specific propo sition is true. Since the people engaged in a genuine dispute agree on the meaning of the words by mea ns of which they convey their respective positions, each of them can propose and assess logical a

rguments that might eventually lead to a resolution of their differences.

. Merely verbal disputes, on the other hand, arise entirely from ambiguities in the language used to express the positions of the disputants. A verbal dispute disappears entirely on ce the people involved arrive at an agreement on the meaning of their terms, since doing so re veals their underlying agreement in belief.

3. Denotation (Extension) and Connotation (Intension)

4. Extension, and Denotative Definitions

A denotative definition tries to identify the extension of the term in question. Thus, we could provide a denotative definition of the phrase "this logic class" simply by listing all of our names.

The extension of a general term is just the collection of individual things to w hich it is correctly applied. Thus, the extension of the word "chair" includes every chair that is (or ever has been or ever will be) in the world.

5. Intension, and Connotative Definition

The intension of a general term, on the other hand, is the set of features which are shared by everything to which it applies. Thus, the intension of the word "chair" is (something like) "a piece of furniture designed to be sat upon by one person at a time."

A connotative definition tries to identify the intension of a term by providing a synonymous linguistic expression or an operational procedure for determining the applicability of the term.

6. Rules for Definition by Genus and Difference

Classical logicians developed an especially effective method of constructing con notative definitions for general terms, by stating their genus and differentia.

Five rules by means of which to evaluate the success of connotative definitions by genus and differentia:

1. Focus on essential features. Although the things to which a term applies may share many distinctive properties, not all of them equally indicate its true nature.

2. Avoid circularity. Since a circular definition uses the term being defined as part of its own definition, it can't provide any useful information; either the audience already understands the meaning of the term, or it cannot understand the explanation that includes that term. Thus, for example, there isn't much point in defining "cordless 'phone" as "a telephone th at has no cord."

3. Capture the correct extension. A good definition will apply to exactly the sa me things as the term being defined, no more and no less. There are several ways to go wrong. Con sider

alternative definitions of "bird": "warm-blooded animal" is too broad, since that would include horses, dogs, and a ardvarks along with birds. "feathered egg-laying animal" is too narrow, since it excludes those birds who h appen to be male, and "small flying animal" is both too broad and too narrow, since it includes bats ( which aren't birds) and excludes ostriches (which are). Successful intensional definitions must be satisfied by all and only those thing s that are included in the extension of the term they define.

4. Avoid figurative or obscure language. Since the point of a definition is to e xplain the meaning of a term to someone who is unfamiliar with its proper application, the use of lang uage that doesn't help such a person learn how to apply the term is pointless. Thus, "happ iness is a warm puppy" may be a lovely thought, but it is a lousy definition.

5. Be affirmative rather than negative. It is always possible in principle to ex plain the application of a term by identifying literally everything to which it does not apply. In a f ew instances, this may be the only way to go: a proper definition of the mathematical term "infinit e" might well be negative, for example. But in ordinary circumstances, a good definition uses pos itive designations whenever it is possible to do so. Defining "honest person" as "some one who rarely lies" is a poor definition.

IV. DEDUCTIVE REASONING

A. Categorical Propositions

1. Categorical Propositions and Classes 2. Quality, Quantity and Distribution 3.The Traditional Square of Opposition 4. Further Immediate Inferences 5 Existential Import 6. Symbolism and Diagrams for Categorical Propositions

1. Categorical Propositions and Classes:

. Propositions are Statements or sentences where the content or meaning of a mea ningful declarative sentence. Statements or sentences that posses a quality or property of being either TRUE or FALSE. External manifestation of the mental product of Judgement

. Categorical Propositions are IS THE KIND OF PROPOSITION WHEREIN THE JUDGEMENT IS DONE IN ABSOLUTE MANNER, i.e., The agreement or disagreement between the subject and the predicate, is done in an absolute manner and MAKES A DIRECT ASSERTION OF AGREEMENT BETWEEN THE SUBJECT AND THE PREDICATE.

. Examples: a) HONEST FILIPINOS AVOID CHEATERS. (asserts that the entire class of honest Filipinos is included in the class of p eople who avoid liars.) b) PEDICABS DO NOT BELONG TO EXPRESSWAYS.

(asserts that the entire class of Pedicabs is excluded from the class of vehicle s that beling to expressways.) c) MANY STUDENTS HAVE CELLPHONES. d) NOT ALL LONG DISTANCE RELATIONSHIPS HAVE HAPPY ENDINGS. e) ANN CURTIS IS A FAMOUS SHOWBIZ PERSONALITY.

Classes of Categorical Propositions: 1. THOSE THAT ASSERT THAT THE WHOLE SUBJECT CLASS IS INCLUDED IN THE PREDICATE CLASS.

2. THOSE THAT ASSERT THAT PART OF THE SUBJECT CLASS IS INCLUDED IN THE PREDICATE CLASS.

3. THOSE THAT ASSERT THAT THE WHOLE SUBJECT CLASS IS EXCLUDED FROM THE PREDICATE CLASS

4. THOSE THAT ASSERT THAT PART OF THE SUBJECT CLASS IS EXCLUDED FROM THE PREDICATE CLASS.

3 Elements of a Proposition: 1. Subject This man.. 2. Coupla is...

3. Predicate

a doctor.

2. Quantity, Quality and Distribution

Quantity of a Proposition: Is equivalent of a quantity of a subject.

1. SINGULAR - Stands for a single definite individual group - Example: Aristotle is the father of logic.

2. PARTICULAR - The subject designates an indefinite part of its total extension. - Example: Some philosophers are atheists.

3. UNIVERSAL - subject can apply to every portion of the term being indicated. - Example: Love is not selfish.

Quantity of a Predicate: 1. Singular - The predicate indicates any signs of singularity - Ex. The new Dean of the College of Law of LDCU is Atty.Adel Tamano.

2. Particular - The predicate of a categorical is not singular and the copula is affirmative. - A truly happy life is a life of goodness

3. Universal - The predicate is not singular, and the copula is negative, then the predicate is universal. - Most politicians are not moral persons.

A Proposition E Proposition

UNIVERSAL / SINGULAR QUANTITY

UNIVERSAL / SINGULAR QUANTITY

AFFIRMATIVE QUALITY NEGATIVE QUALITY PARTICULAR QUANTITY PARTIcULAR QUANTITY AFFIRMATIVE QUALITY NEGATIVE QUALITY

I Proposition O Proposition

A Proposition: EXAMPLES: 1. ALL SOLDIERS ARE PATRIOTIC. 2. EVERY PHILOSOPHER IS A LOVER OF WISDOM. 3. AN ORANGUTAN IS AN APE.

*Quantity is UNIVERSAL/SINGULAR Quality is AFFIRMATIVE

E Proposition: EXAMPLES: 1. NO SINNERS ARE SAINTS. 2. A GREEN MANGO IS NOT SWEET. 3. NEITHER DIAMONDS NOR GOLD IS EXPENSIVE.

*Quantity is UNIVERSAL/SINGULAR Quality is NEGATIVE

I Proposition: EXAMPLES: 1. SOME PHILOSOPHERS ARE ATHEISTS. 2.FILIPINOS ARE NATURE-LOVERS. 3. ALMOST ALL PEOPLE ARE GOD-FEARERS.

* Quantity

PARTICULAR Quality -AFFIRMATIVE

images.jpg O Proposition: EXAMPLES: 1. NOT ALL SENATORS ARE HONEST POLITICIANS. 2.NOT EVERYONE WHO CALLS TO ME LORD, LORD ARE PERSONS TO BE SAVED.

3. SOME FILIPINOS ARE NOT PATRIOTIC.

*Quantity

PARTICULAR Quality - NEGATIVE

Quality of a Proposition: 1. AFFIRMATIVE affirms class membership. - connotes that all, or some, of the members indicated by the subject are contai ned in the class indicated by the predicate.

2. NEGATIVE denies class membership. If all, or some of the members indicated by the subject are not contained in the class indicated by the predicate

3.The Traditional Square of Opposition . . . . CONTRARIES CANNOT BOTH BE TRUE, BUT BOTH CAN BE FALSE. SUBCONTRARIES CANNOT BOTH BE FALSE, BUT BOTH CAN BE TRUE. SUBALTERN PAIRS CAN BOTH BE TRUE OR BOTH BE FALSE. CONTRADICTORIES CANNOT BE TRU AND CANNOT BE FALSE.

4. Further Immediate Inferences 1. Conversion - is the new categorical proposition that results from putting the PREDICATE term of the original proposition in the SUBJECT place of the NEW PROPOSITION and the SUBJECT term of the original in the PREDICATE place of the new.

. Example. No cats are canines. Some snakes are poisonous animals No canines are cats. Some poisonous Animals are snakes

- The converse of any E or I propositions are true only if the original proposit ion was true. From either pair of examples above, both propositions are true or both are false. That is why conversion grounds an immediate inference for both E and I propositi ons.

2. Subalternation - if we first perform a subalternation and then convert our result, then the tru th of an A proposition may be said, in "conversion by limitation," to entail the truth of a n I proposition with subject and predicate terms reversed

. Example: All dogs are mammals" to be true while "All mammals are dogs" is fals e

"Some females are not mothers" to be true while "Some mothers are not females" i s false. Thus, conversion does not warrant a reliable immediate inference with res pect to A and O propositions.

3. Obversion - In order to form the obverse of a categorical proposition, we replace the pred icate term of the proposition with its complement and reverse the quality of the propo sition, either from affirmative to negative or from negative to affirmative

. Example: "All ants are insects" is "No ants are non-insects" "No fish are mammals" is "All fish are non-mammals"

- Obversion is the only immediate inference that is valid for categorical propos itions of every form. In each of the instances cited above, the original proposition and i ts obverse must have exactly the same truth-value, whether it turns out to be true or false .

4. Contrapositions - The contrapositive of any categorical proposition is the new categorical propo sition that results from putting the complement of the predicate term of the original proposition in the subject place of the new proposition and the complement of th e subject term of the original in the predicate place of the new

. Examples: "Some carnivores are not mammals" is "Some non-mammals are not noncarnivores."

- the contrapositive of any A or O proposition is true if and only if the origin al proposition was true. If we form the contrapositive of our result after performing subalter nation, then an E proposition, in "contraposition by limitation," entails the truth of a related O proposition

. If "No bandits are biologists" then "Some non-biologists are not non-bandits,"

(Provided that there is at least 1 member of the class designated by

bandits )

- contraposition is not valid for E and I propositions. . Example: "No birds are plants" and "No non-plants are non-birds" need not have the same truth-value

5. Existential Import . A term, whether subject or predicate, is said to have existential import if the term implies the actual existence of members of that category. Aristotelian logic inferred affirmative propositions such as "All cats are felines" include that there are such entities as cats. Aristotelian logic also inferred the existence of members of I propositions. "Some diesel engines are engines powered by coconut oil," also implied the existence of such diesel engines

6. Symbolism and Diagrams for Categorical Propositions

. The modern interepretation of categorical logic also permits a more convenient way of assessing the truth-conditions of categorical propositions, by drawing Venn diagrams, topological representations of the logical relationships among the classes designated by categorical terms. The basic idea is fairly straightforward:

Each categorical term is represented by a labelled circle. The area inside the circle represents the extension of the categorical term, and the area outside the circle its complement. Thus, members of the class designated by the categorical term would be located within the circle, and everything else in the world would be located outside it.

We indicate that there is at least one member of a specific class by placing an inside the circle; an outside the circle would indicate that there is at least one member of the complementary class.

To show that there are no members of a specific class, we shade the entire area inside the circle; shading everything outside the circle would indicate that there are no members of the complementary class.

In order to represent a categorical proposition, we must draw two overlapping ci rcles, creating four distinct areas corresponding to four kinds of things: 1. those that are members of the class designated by the subject term but not o f that designated by the predicate term; 2. those that are members of both classes; 3. those that are members of the class designated by the predicate term but not of that designated by the subject term; 4. and those that are not members of either class.

The universal negative (E) proposition asserts that nothing is a member of both classes designated by its terms, so its diagram shades the area in which the two circles overlap.

The particular affirmative (I) proposition asserts that there is at least one thing that is a member of both classes, so its diagram places an in the area where the two circles overlap.

two diagrams models the contradictory relationship between E and I propositions; one of them must be true and the other false, since either there is at least one member that the two classes have in common or there are none.

The particular negative (O) proposition asserts that there is at least one thing that is a member of the class designated by its subject term but not of the class designated by its predicate term, so its diagram places an in the area inside the circle that represents the subject term but outside the circle that represents the predicate term.

Finally, the universal affirmative (A) proposition asserts that every member of the subject class is also a member of the predicate class. Since this entails that there is nothing that is a member of the subject class that is not a member of the predicate class, an A proposition can be diagrammed by shading the area inside the subject circle but outside the predicate circle.

B. Categorical Syllogisms

1. Standard-Form Categorical Syllogisms 2. The Formal Nature of Syllogistic Argument 3. Venn Diagram: Technique for Testing Syllogisms 4. Six Rules of Categorical Syllogisms

1. Standard-Form Categorical Syllogism

- A categorical syllogism is a verbal expression of an inference. It is an oral or written discourse showing the agreement or disagreement between two terms on the basis of their re spective relation to a common third term

- Is any argumentation in which, from two prepositions called the premise, we co nclude a third proposition called the conclusion, which is so related to the premise taken join tly that if the premises are true, the conclusion must also be true

- The logical form is the structure of a categorical syllogism indicated by its figures and moods.

. Figure is the arrangement of the terms. major, minor, middle terms of the argument.

a. Major term is found in the major premise, used either as subject or predicate, or as predicate of the conclusion. Since the major term has the greatest extension, it has the greatest concept compared to the other terms.

b. Minor term is the term whose function is to mediate between the other two terms. In an affirmative syllogism. The middle term unites the major term and the minor term. In a negative syllogism, it separates the two. thus, the quality of the syllogism depends on the role of the middle term toward the other two terms. The middle term is never to be found in the conclusion. It is used either as subject or predicate in the premises.

. Mood is the arrangement of the preposition by quantity or quality.

2. The Formal Nature of Syllogistic Argument

A categorical syllogism must always have that SEQUENTIAL RELATION as a differentiating mark of a true and valid syllogism from what is not.

SEQUENTIAL RELATION refers to the interdependence of the premise upon one another. The sure sign of the sequential relation is the presence of a middle term in the premise. NOT A VALID SYLLOGISM. . No sequential relation between the first and second premise because there is no

middle term to connect or disconnect the two terms. . There is no relation between the two premise in which to derive a valid conclusion. . The above supposed conclusion is no conclusion at all.

EXAMPLE: Every man is biped. But every cow is quadruped. Therefore, every cow is not a man.

. VALID AND TRUE SYLLOGISM . There exist a sequential flow of thought from the first premise to the second premise, and then from the premise to the conclusion. . The propositions are connected to one another through the middle term creature which connects the major term mortal and the minor term person.

EXAMPLE: All creatures are mortal. But a person is a creature. Therefore, a person is mortal.

21.jpg minor.jpg 20.jpg

A syllogism is called categorical if the premise and the conclusion composing it are categorical propositions expressed in a declarative form like women are beautiful . not all Filipinos are poor. Or every creature is good .

3. Venn Diagram: Technique for Testing Syllogisms In order to test a categorical syllogism by the method of Venn diagrams, one mus t first represent both of its premises in one diagram. That will require drawing three overlapping circles, fo r the two premises of a standard-form syllogism contain three different terms-minor term, major term, an d middle term. How to draw Venn Diagram:

STEP 1:

First draw three overlapping circles and label them to represent the major, minor, and middle terms of the syllogism.

No M are P. Some M are S. Therefore, Some S are not P.

STEP 2:

Since the major premise is a universal proposition, we may begin with it. The diagram for "No M are P" must shade in the entire area in which the M and P circles overlap. (Notice that we ignore the S circle by

shading on both sides of it.)

22.jpg 23.jpg 24.jpg

Now we add the minor premise to our drawing. The diagram for "Some M are S" puts an inside the area where the M and S circles overlap. But part of that area (the portion also inside the P circle) has already been shaded, so our must be placed in the remaining portion.

Next, on this framework, draw the diagrams of both of the syllogism's premises. a. Always begin with a universal proposition, no matter whether it is the major or the minor premise. b. Remember that in each case you will be using only two of the circles in each case; ignore the third circle by making sure that your drawing (shading or ) straddles it.

STEP 3:

Ignoring the M circle entirely, we need only ask whether the drawing of the conclusion "Some S are not P" has already been drawn.

Finally, without drawing anything else, look for the drawing of the conclusion. If the syllogism is valid, then that drawing will already be done.

Since it perfectly models the relationships between classes that are at work in categorical logic, this procedure always provides a demonstration of the validity or invalidity of any c ategorical syllogism.

Here is a diagram of a syllogistic form. In such case, both of the premises have already been drawn in the appropriate way, so if the drawing of the conclusion is already drawn, the syllogism must be valid, and if it is not, the syllogism must be invalid.

AAA-1 (valid) All M are P. All S are M. Therefore, All S are P.

4. Six Rules of Categorical Syllogisms Rule 1: A syllogism must contain exactly three terms, each of which is used in t he same sense. - The use of exactly three categorical terms is part of the definition of a cate gorical syllogism, and we saw earlier that the use of an ambiguous term in more than one of its senses amounts to the use of two distinct terms. In categorical syllogisms, using more than three terms commits the fallac y of four terms.

This syllogism appears to have only three terms, but there are really four since one of them, the middle term power is used in different senses in the two premises. Example of INVALID SYLLOGISM: Power tends to corrupt. Knowledge is power. Knowledge tends to corrupt

Rule 2: In a valid categorical syllogism the middle term must be distributed in at least one of the premises. - In order to effectively establish the presence of a genuine connection between the major and minor terms, the premises of a syllogism must provide some information about the entire class designated by the middle term. If the middle term were undistributed in both premises, then the two porti ons of the designated class of which they speak might be completely unrelated to each other. Syllogisms that violate this rule are said to commit the fallacy of the undistributed middle. The middle term is what connects the major and the minor term. If the middle term is never distributed, then the major and minor terms might be related to different parts of the M class, thus giving no common ground to relate S and P.

Example of INVALID SYLLOGISM: All sharks are fish All salmon are fish

All salmon are sharks

Rule 3: In a valid categorical syllogism if a term is distributed in the conclus ion, it must be distributed in the premises. - A premise that refers only to some members of the class designated by the majo r or minor term of a syllogism cannot be used to support a conclusion that claims to tell us about ev ery member of that class. Depending which of the terms is misused in this way, syllogisms in violation com mit either the fallacy of the illicit major or the fallacy of the illicit minor. When a term is distributed in the conclusion, let s say that P is distributed, then that term is saying something about every member of the P class. If that same term is NOT distributed in the major premise, then the major premise is saying something about only some members of the P class.

Example of INVALID SYLLOGISM: All horses are animals Some dogs are not horses Some dogs are not animals

Rule 4: No syllogism can have two negative premises. - The purpose of the middle term in an argument is to tie the major and minor te rms together in such a way that an inference can be drawn, but negative propositions state that the terms o f the propositions are exclusive of one another. In an argument consisting of two negative propositions the middle term is excluded from both the major term and the minor term, and thus there is no connection bet ween the two and no inference can be drawn. A violation of this rule is called the fallacy of exclus ive premises.

If the premises are both negative, then the relationship between S and P is denied. The conclusion cannot, therefore, say anything in a positive fashion. That information goes beyond what is contained in the premises.

Example of INVALID SYLLOGISM: No fish are mammals Some dogs are not fish Some dogs are not mammals

Rule 5: If either premise of a valid categorical syllogism is negative, the conc lusion must be negative. - An affirmative proposition asserts that one class is included in some way in a nother class, but a negative proposition that asserts exclusion cannot imply anything about inclusion. For th is reason an argument with a negative proposition cannot have an affirmative conclusion. An argument that vio lates this rule is said to commit the fallacy of drawing an affirmative conclusion from a negative premise. Example of INVALID SYLLOGISM: All crows are birds Some wolves are not crows Some wolves are birds Rule 6. In valid categorical syllogisms particular propositions cannot be drawn properly from universal premises. - Because we do not assume the existential import of universal propositions, the y cannot be used as premises to establish the existential import that is part of any particular proposition. The existential fallacy violates this rule. Although it is possible to identify additional features shared by all vali d categorical syllogisms (none of them, for example, have two particular premises), these six rules are jointly su fficient to distinguish between valid and invalid syllogisms.

Thus, if the syllogism has universal premises, they necessarily say nothing about existence. Yet if the conclusion is particular, then it does say something about existence. In which case, the conclusion contains more information than the premises do, thereby making it invalid. Example of INVALID SYLLOGISM: All mammals are animals All tigers are mammals Some tigers are animals

C. Arguments in Ordinary Language

1. Reducing the Number of Terms in a Syllogistic Argument 2. Translating Categorical Propositions into Standard Form 3.Uniform Translation 4.Enthymemes 5 Sorites 6. Disjunctive and Hypothetical Syllogisms 7.The Dilemma

1. Reducing the Number of Terms in a Syllogistic Argument In slightly more complicated instances, an ordinary argument may deal with more than three terms, but it may still be possible to restate it as a categorical syllogism. Two kinds of too ls will be helpful in making such a transformation: STEP 1: First, it is always legitimate to replace one expression with another th at means the same thing. Of course, we need to be perfectly certain in each case that the expressions are ge nuinely synonymous. But in many contexts, this is possible: in ordinary language, "husbands" and "married m ales" almost always mean the same thing. STEP 2: Second, if two of the terms of the argument are complementary, then appr opriate application of the immediate inferences to one of the propositions in which they occur will enable us to reduce the two to a single term. Consider, for example, "No dogs are non-mammals, and some non-canines are not n on-pets, so some nonmammals are pets." Replacing the first proposition with its (logically equivalen t) obverse, substituting "dogs" for the synonymous "canines" and taking the contrapositive of the second, and ap plying first conversion and then obversion to the conclusion, we get the equivalent standard-form categorica l syllogism: All dogs are mammals. The invalidity of this syllogism is more readily apparent than that of the argument from which it was derived. Some pets are not dogs.

Therefore, Some pets are not mammals.

2. Translating Categorical Propositions into Standard Form We may need only to re-arrange the propositions of the argument in order to tran slate it into a standard-form categorical syllogism.

Thus, for example, "Some birds are geese, so some birds are not felines, since n o geese are felines" is just a categorical syllogism stated in the non-standard order minor premise, conclusion, major premise; all we need to do is put the propositions in the righ t order, and we have the standard-form syllogism: No geese are felines. Some birds are geese. Therefore, Some birds are not felines.

3.Uniform Translation

In order to achieve the uniform translation of all three propositions contained in a categorical syllogism, it is sometimes useful to modify each of the terms employed in an ordinary-language ar gument by stating it in terms of a general domain or parameter. The goal here, as always, is faithfully to represent the intended meaning of eac h of the offered propositions, while at the same time bringing it into conformity with the others, making it po ssible to restate the whole as a standard-form syllogism. Thus, for example, in the argument, "The attic must be on fire, since it's full of smoke, and where there's smoke, there's fire," the crucial parameter is location or place. If we suppose the terms of this argument to be "places where fire is," "places where smoke is," and "places that are the att ic," then by applying our other techniques of restatement and re-arrangement, we can arrive at the syllogism: This standard-form categorical syllogism of the form AAA-1 is clearly valid.

All places where smoke is are places where fire is. All places that are the attic are places where smoke is. Therefore, All places that are the attic are places where fire is.

4.Enthymemes - Another special case occurs when one or more of the propositions in a categori cal syllogism is left unstated. Incomplete arguments of this sort, called enthymemes are said to be "first-," "s econd-," or "third-order," depending upon whether they are missing their major premise, minor premise, or c onclusion respectively. In order to show that an enthymeme corresponds to a valid categorical syllogism, we need only supply the missing premise in each case. Thus, for example, "Since some hawks have sharp beaks, some birds have sharp bea ks" is a second-order enthymeme, and once a plausible substitute is provided for its missing minor pre mise ("All hawks are birds"), it will become the valid IAI-3 syllogism: Some hawks are sharp-beaked animals. All hawks are birds. Therefore, Some birds are sharp-beaked animals.

5.Sorites Finally, the pattern of ordinary-language argumentation known as sorites involve s several categorical syllogisms linked together. The conclusion of one syllogism serves as one of the premises for another syllogism, whose conclusion may serve as one of the premises for another, and so on. In any such case, of course, the whole procedure will comprise a valid inference so long as each of t he connected syllogisms is itself valid.

6. Disjunctive and Hypothetical Syllogisms (KINDS OF COMPOUND PROPOSITION) DISJUNCTIVE:

The disjunctive holds that at least one of the two components are true, allowing for the possibility that both are true. If we have a disjunction as one premise, and a denial of one of the di sjuncts as a second premise, we can validly infer that the other disjunct component is true P1 Either that's a gun in your pocket, or you're happy to see me P2 You don't have a gun in your pocket (This is implied - implied premises are c alled "enthymemes") C: You must be happy to see me

This argument is valid, because it eliminates one of the disjuncts. Take a look at this argument: P1 Either that's a gun in your pocket, or you're happy to see me P2 You have a gun in your pocket C: Therefore you're not happy to see me

This argument is invalid. It bears a superficial similiarty to the above argumen t, but rather than eliminate one of the disjuncts, it merely affirms one of them. HYPOTHETICAL SYLLOGISMS: We can call these statements If/Then statements, where the "If" part is the ante cedant and the part following after "Then" is the consequent. A conditional that contains conditional statements exclusively is called a pure hypothetical syllogism: Example: P1: If you study (antecedent), then you will become a good student (con sequent). P2: If you become a good student, then you will go to college Therefore, If you study, then you will go to college.

Notice that the first premise and the conclusion have the same antecedent, and t he second premise and the conclusion have the same consequent.

There are two valid and two invalid forms of a mixed hypothetical syllogism: a) modus ponens (From the Latin "ponere", "to affirm") b) modus tollens (Latin: "To deny")

The next form, Affirming the consequent, is invalid: Affirming the consequent . If P is true then Q is true . Q is true . Therefore, P is true

MODUS PONENS: If P is true, then Q is true P is true Therefore, Q is true

MODUS TOLLENS: If P is true, then Q is true Here the syllogism denies the consequent of the conditional premise, and the conclusion denies the antecedant.

Q is not true Therefore, P is not true

7.The Dilemma It is a form of argument that is composed of a conjunction of two conditional h ypothetical statements as its major premise. KINDS: 1. Simple Dilemma . A form wherein the conclusion is a categorical proposition. Example: The security officer of a certain university was called into the office of the c hief security on the issue of allowing students to enter the school premises without wearing their ID s. The chief security said to the guard: Either you were playing favorites or you were not; if you were playing favorites, then you should be punished for being incompetent. If you were not playing favorites, you deserve p unishment for dereliction of duty. Therefore, in either case, you should be punished.

2. Complex Dilemma . It is a form of dilemma wherein the conlusion is a disjunctive proposition off ering alternatives. Example: If the next president of the Phil. Is a re-electionist, then he has nothing new to offer. If the next president of the Phil. Is not a re-electionist, then he is not yet equipped in t he country s political and economic matters. Therefore, in either case, the next president of the Philippin es has nothing new to offer or he is not yet equipped in the country s political and economic matters.

D. Symbolic Logic

1. The Value of Special Symbols 2. The Symbols for Conjunction, Negation, and Disjunction 3. Conditional Statements and Material Implication 4. Argument Forms and Arguments 5. Statement Forms, Material Equivalence, Logical Equivalence 6. The Paradoxes of Material Implication 7. The Three Laws of Thought

2. The Symbols for Conjunction, Negation, and Disjunction

Conjunction: In logic, a conjunction is a compound sentence formed by using the word and to j oin two simple sentences. The symbol for this is .. (whenever you see . read 'and') When two si mple sentences, p and q, are joined in a conjunction statement, the conjunction is expressed symbo llically as p . q.

Simple Sentences Compound Sentence : conjunction p: Joe eats fries. q: Maria drinks soda. p . q : Joe eats fries, and maria drinks soda.

Conjunction =

and

=

Conjunction is only true if both conjuncts are true

Negation:

Indicates the opposite, usually employing the word not. The symbol to indicate n egation is : ~ Original Statement Negation of statment Today is monday. Today is not monday. That was fun. That was not fun.

Negation = not

= ~

Negation of a statement is true if statement is false Negation of a statement is false if statement is true

Disjunction: . In logic, a disjunction is a compound sentence formed by using the word or to join two simple sentences. The symbol for this is .. (whenever you see . read 'or') When two simple sentences, p and q, are joined in a disjunction statement, the disjunction is ex pressed symbollically as p . q.

Pneumonic: the way to remember the symbol for disjuntion is that, this symbol . looks like the 'r' in or, the keyword of disjunction statements. Simple Sentences Compound Sentence : disjunction p: The clock is slow. q: The time is correct. p . q : The clock is slow, or the time is correct.

Warning and caveat: The only way for a disjunction to be a false statement is if BOTH halves are false. A disjunction is true if either statement is true or if both statements a re true! In other words, the statement 'The clock is slow or the time is correct' is a false statement on ly if both parts are false! Likewise, the statement 'Mr. G teaches Math or Mr. G teaches Science' is true if Mr. G is teaches science classes as well as math classes!

Disjunction =

or

= v

Disjunction is true if either disjunct is true

Examples:

1. It is not true that evil spirits exist.

First step: Make a dictionary (define statements) Second step: Look at the sentence, symbolize statements correctly (using v) (Third step: Determine truth values) , ~, or

Answer: It is not true that evil spirits exist. ~E E=Evil spirits exist. If evil spirits do exist (E is True), then ~E is false. If evil spirits do not exist (E is False), then ~E is true.

2. Determine whether the following is true: ~(A v C) v ~(X ~Y) Given: A, B, and C are True X, Y, and Z are False

~(A v C) v ~(X . . . .

~Y)

The main connective = the middle wedge (v) (disjunction) Therefore we have two disjuncts Left disjunct= ~(A v C) Right disjunct = ~(X ~Y)

. Strategy: determine truth values of each disjunct, then we know if at least on e disjunct is true, this will make the whole statement true . ~(A v C) v ~(X ~Y) . Left disjunct: ~(A v C) . Both A and C are true. This makes (A v C) true. . But (A v C) is negated, so ~(A v C) is false. . Right disjunct: ~(X ~Y) . X is false. . Y is false, so this means ~Y is true. . This makes the inner conjunction false (to be true, both conjuncts (X and ~Y) must both be true) . Because the whole statement (X ~Y) is false, this makes its negated form ~(X Y) true . Since the left disjunct is false, and the right disjunct is true, this means ~ (A v C) v ~(X ~Y) is true (since at least one disjunct is true)

~

3. Conditional Statements and Material Implication

Conditional statement: when two statements are combined by placing the word ore the first and then before the second . If then . We use the arrow . or horseshoe to represent the if-then phrase . Also called a hypothetical, an implication, or an implicative statement

if bef

The component statement that follows the if is called the antecedent The component statement that follows the then is called the consequent If (antecedent), then (consequent) A conditional statement asserts that if its antecedent is true, then its consequ ent is also true But as in disjunction, there are a few different senses in which a conditional c an be interpreted

4 Types of Implications: Logical Implication: the consequent follows logically from its antecedent Example: If all humans are mortal and Socrates is a human, then Socrates is mort al. Definitional Implication: the consequent follows the antecedent by definition Example: If Leslie is a bachelor, then Leslie is unmarried. Causal Implication: The connection between antecedent and consequent is discover ed empirically Example: If I put X in acid, then X will turn red. Decisional Implication: no logical connection nor one by definition between the consequent and antecedent. This is a decision of the speaker to behave in the specified way under the speci fied circumstances Example: Is we lose the game, then I ll eat my hat. Understanding the Implication: No matter what type of implication is asserted by a conditional statement, part of its meaning is the negation of the conjunction of its antecedent with the negation of its consequent For a conditional to be true (e.g. If p then q ), ~(p ~q) must be true:

Think p= A piece of blue litmus paper is placed in that solution. q= The piece of blue litmus paper will turn red. If p then q = false if paper is placed in solution, but doesn t turn red th

The horseshoe symbol does not stand, therefore, for all the meanings of if-then ere are several meanings. p q abbreviates ~(p ~q), whose meaning is included in the meanings of each kind of implication

Material Implication: . represents the material implication. A fifth type of implication E.g. If Hitler was a military genius, then I m a monkey s uncle.

No real connection between antecedent and consequent . This kind of relationship is what is meant by material implication . It just asserts that it is not the case that the antecedent is true when the consequent is false.

. Many arguments contain conditional statements of various kinds of implication, but the validity of all valid arguments (of the general type with which we will be concerned) is preserv ed, even if the additional meanings of their conditional statements are ignored.

If . . . .

can be replaced by such phrases as: in case provided that given that on condition that

Some indicator words for . implies... . entails

then

include:

4. Argument Forms and Arguments

Substitution-Instances Since the statements of the propositional calculus are propositions, they can be combined to form logicalarguments, complete with one or more premises and a single conclusion tha t may follow validly from them. Thus, for example, each of the following is an argument expressed in the l anguage of symbolic logic: A . B (D B) . ~E (A . E) . (D = B)

A D

B A . E

_______ _______________ ________________________

B ~E D = B What is more, notice that all three of these arguments share a common structure: the first premise of each is a . statement; the second premise is the antecedent of that statement; and the c onclusion is its consequent. We can exhibit this common structure more clearly by using statement variables t o express the argument forminvolved: p . q

p ________

q Each of the three arguments above is a substitution instance of this argument fo rm, since each of them results from the substitution of an appropriate (simple or compound) statement f or each of the statement variables in the argument form. Notice that these substitutions must be consiste nt in each application; once we've put D B in the place of p in the first premise of the second argument, for example, we must also put it in the place of p in the second premise. In the same way, the first and third arguments above along with indefinitely many others can be shown to be substitution-instances of the same arg ument form. Most arguments are substitution-instances of several distinct argument forms, each of which can be no more complex in structure than the argument itself. Testing for Validity

Recognizing individual arguments as substitution-instances of more general argum ent forms is an important skill because, as we've already seen, the validity of any argument dep ends solely upon its logical form. An argument in the propositional calculus is valid whenever it is a substi tution-instance of an argument form in which it is impossible for the premises to be true and the conclusion fa lse. Since the argument form reliably leads from premises of a certain general structure to a conclusion of a different structure, every substitution-instance of that argument form must express a valid argument. Thus, the same truth-tables we used to define the statement connectives provide an effective decision procedure for determining the validity of arguments in the propositional calculu s. We simply chart the truthvalues of each premise and the conclusion of an argument form for every possible combination of truthvalues for the statement variables involved, and look to see what happens on tho se lines of the truth-table in which all of the premises are true. If the conclusion is also true on each of th ese lines, then the inference captured by the argument form is a valid one, and arguments of this form must al l be valid. If, however, there is even a single line on which all of the premises are true but the conclusion i s false, then the inference is

invalid, and we cannot be sure whether arguments of this form are valid or inval id. (They certainly are not valid because of this form, but of course some of them may happen to be substitu tion-instances of other argument forms whose inferences are valid.) Modus Ponens Consider, for example, what happens when we construct a truth-table that lists e ach of the four combinations of truth-values that the component statements could exhibit in the simple argument form that we identified at the top of this page.

1st Premise 2nd Premise Conclusion p q p . q p q T T T T T T F F T F F T T

F T F F T F F

p . q

p _______

q This truth-table shows that (no matter what statements we substitute for p and q ) both of the premises of the argument will be true only on the first line (when both component statements are true). But on that line, the conclusion is also true, so the inference is valid. Whenever we come across an argument that shares this basic structure, we can be perfectly certain of its logical validity. In fact, arguments of this form are so common that the form itself has a name, Modus Ponens, which we will usually a bbreviate as M.P. On the other hand, consider what happens when we construct a truth-table for tes ting the validity of a distinct, though superficially similar, argument form:

1st Premise 2nd Premise Conclusion p q p . q

q p T T T T T T F F F T F T T T F F F T F F

p . q

q _______

p In arguments of this form, both premises are true on the first and on the third lines of the truth-table. While the conclusion is

true on the first line, on the third line it is false. Since it is therefore possible for the premises to be true while the conclusion is false, the inference is invalid. This unreliable argument form is called the fallacy of affirming the consequent. Although it might be mistaken for M.P. at a casual glance, the fallacy unlike its valid cousin does not guarantee the truth of its conclusion.

Modus Tollens Another common argument form with a valid inference is Modus Tollens (abbreviate d as M.T.), which has the form:

1st Premise 2nd Premise Conclusion p q p . q ~ q ~ p T T T F F T F F T F F T T F T F F

T T T

p . q

~ q _______

~ p As the truth-table shows, the premises are true only when both of the component statements are false, in which case the conclusion is also true. There is no line on which both premises are true and the conclusion false, so the inference is valid, as are all substitution-instances of this argument form. As with M.P., there is an argument form superficially similar to M.T. that yields entirely different results.

1st Premise 2nd Premise Conclusion p q p . q ~ p ~ q T T T F FT F F F T F T T T F F F T T T

p . q

~ p _______

~ q This is the fallacy of denying the antecedent. As the truth-table to the right clearly shows, it is an unreliable inference, since it is possible (on the third line) for both of its premises to be true while its conclusion is false. Substitution-instances of this argument form may not be valid. Hypothetical Syllogism

1st Premise 2nd Premise Conclusion p q r p . q q . r p . r T T T T T T T T F T F F

T F T F T T T F F F T F F T T T T T F T F T F T F F T T T T

F F F T T T

A larger truth-table is required to demonstrate the validity of the argument form called Hypothetical Syllogism (H.S.), since it involves three statement variables instead of two, and we must consider all eight of the possible combinations of their truth-values: p . q

q . r _______

p . r Despite its greater size, this truth-table establishes validity in exactly the same way as its more compact predecessors: both premises are true only on the first, fifth, seventh, and eighth lines, and the conclusion is also true on each of these lines. It follows that all arguments sharing in thisgeneral form m ust be valid. Disjunctive Syllogism Finally, consider the argument form known as Disjunctive Syllogism or D.S.

1st Premise 2nd Premise Conclusion p q

p . q ~ p q T T T F T T F T F F F T T T T F F F T F

p . q

~ p _____

q

The truth-table demonstration of its validity should look familiar by now. Whenever the premises are true (on the third line of the truth table), so is the conclusion. Once again, however, there is a similar form that embodies an invalid inference, the fallacy of affirming the alternative:

1st Premise 2nd Premise Conclusion p q p . q p ~ q T T T T F

T F T T T F T T F F F F F F T

p . q

p _____

~ q In this case, the first line of the truth-table shows that (with our inclusive sense of the . ) it is possible for the premises to be true and the co nclusion false. . 5. Statement Forms, Material Equivalence, Logical Equivalence Statement Forms: Statement Forms In exactly the same sense that individual arguments may be substitution-instance s of general argument forms, individual compound statements can be substitution-instances of general s tatement forms. In

addition, just as we employ truth-tables to test the validity of those arguments , we can use truth-tables to exhibit interesting logical features of some statement forms. Assessing Statement Forms Because all five of our statement connectives are truth-functional, the status o f every statement-form is determined by its internal structure. In order to determine whether a statement form is tautologous, selfcontradictory, or contingent, we simply construct a truth-table and inspect the appropriate column. Consider, for example, the statement form: p q (p . ~q) . ~(p T T T F F T F T T T F T F T T F F T q)

T T

(p . ~q) . ~(p

q)

Since the truth-table shows that statements of this form can be either true or false, depending upon the truth-values of their components, the statement form is contingent.

Tautology A statement form whose column in a truth-table contains nothing but Ts is said t o be tautologous. Consider, for example, the statement form: p ~ p p . ~ p

p \equiv q p \Leftrightarrow q T F T F T T

p . ~p Notice that whether the component statement p is true or false makes no differen ce to the truth-value of the statement form; it yields a true statement in either case . But it follows that any compound statement which is a substitution-instance of this for m no matter what its content can be used only to make true assertions. Contradiction A statement form whose column contains nothing but Fs, on the other hand, is sai d to be selfcontradictory. For example: p ~ p p T F F F T F ~ p

p

~p

Again, the truth-value of the component statement doesn't matter; the result is always false. Compound statements that are substitution-instances of this statement for m can

never be used to make true assertions. Contingency Of course, most statement forms are neither tautologous nor self-contradictory; their truth-tables contain both Ts and Fs. Thus: p q p . ~ q T T F T F T F T T F F T

p . ~q Since the column underneath it in the truth-table has at least one T and at leas t one F, this statement form is contingent. Statements that are substitution-instances of this statement form may be either true or false, depending upon the truth-value of their compon ent statements. Logical Equivalence: In logic, statements p and q are logically equivalent if they have the same logi cal content. This is a semantic concept; two statements are equivalent if they have the same truth va lue in everymodel (Mendelson 1979:56). The logical equivalence of p and q is sometimes expressed as , Epq, or . However, these symbols are also used for material equivalence; the pro

per interpretation depends on the context. Logical equivalence is different from material equivalen ce, although the two concepts are closely related. Example: The following statements are logically equivalent:

f \rightarrow e \neg e \rightarrow \neg f 1. If Lisa is in France, then she is in Europe. (In symbols, .) 2. If Lisa is not in Europe, then she is not in France. (In symbols, .) Syntactically, (1) and (2) are derivable from each other via the rules of contra position and double negation. Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); namely, those in which either Lisa is in France is false or Lisa is in Europe is true. (Note that in this example classical logic is assumed. Some non-classical logics do not deem (1) and (2) logically equivalent.) Difference between Material Equivalence: Logical equivalence is different from material equivalence. The material equival ence of p and q (often written p.q) is itself another statement, call it r, in same object language as p and q. rexpresses the idea "p if and only if q". In particular, the truth value of p.q can change from one model to another. The claim that two formulas are logically equivalent is a statement in the metal anguage, expressing a relationship between two statements p and q. The claim that p and q are semantic ally equivalent does not depend on any particular model; it says that in every possible model, p will hav e the same truth value as q. The claim that p and q are syntactically equivalent does not depend on models at all; it states that there is a deduction of q from p and a deduction of p from q. There is a close relationship between material equivalence and logical equivalen ce. Formulas p and q are syntactically equivalent if and only if p.q is a theorem, while p and q are sema ntically equivalent if and only if p.q is true in every model (that is, p.q is logically valid). Material Equivalence: Material Equivalence In similar fashion, Material Equivalence (Equiv.) provides alternative definitions of the = connective. p q p=q = (p.q) (q.p) T

T T T T T F F T F F T F T F F F T T T

Its first form defines = in terms of . , justifying the use of the term "biconditional:" *p=q+=*(p.q) (q.p)]

p q p=q

= (p q).(~p ~q) T T T T T

(\neg p \land p) \to q p \to (q \to p) \neg p \to (p \to q) p \to (q \lor \neg q) (p \to q) \lor (q \to r) \neg(p \to q) \to (p \land \neg q) T F F T F F T F T F F F T T T

Its second form defines = by pointing out its basic truth-conditions: *p=q+=*(p q).(~p ~q)+ Again, the logical equivalence of these three expressions provides us with a convenient way to comprehend and employ what is asserted in any statement of mat erial equivalence. 6. The Paradoxes of Material Implication

The paradoxes of material implication are a group of formulas which are truths o f classical logic, but which are intuitively problematic. One of these paradoxes is the paradox of entailment . The root of the paradoxes lies in a mismatch between the interpretation of the v alidity of

logical implication in natural language, and its formal interpretation in classi cal logic, dating back to George Boole's algebraic logic. In classical logic, implication describes conditional i f-then statements using a truthfunctional interpretation, i.e. "p implies q" is defined to be "it is not the ca se that p is true and q false". Also, "p implies q" is equivalent to "p is false or q is true". For example, "if it is raining, then I will bring an umbrella", is equivalent to "it is not raining, or I will bring an umbrella, or both". This truth-functional interpretation of implication is called material implication or material conditi onal. The paradoxes are logical statements which are true but whose truth is intuitive ly surprising to people who are not familiar with them. If the terms 'p', 'q' and 'r' stand for arbitrary pr opositions then the main paradoxes are given formally as follows: 1. , p and its negation imply q. This is the paradox of entailment. 2. , if p is true then it is implied by every q. 3. , if p is false then it implies every q. This is referred to as 'explosion'. 4. , either q or its negation is true, so their disjunction is implied by every p. 5. , if p, q and r are three arbitrary propositions, then either p implies q or q implies r. This is because if q is true then p implies it, and if it is false th en q implies any other statement. Since r can be p, it follows that given two arbitrary propositions, o ne must imply the other, even if they are mutually contradictory. For instance, "Nadia is in Barce lona implies Nadia is in Madrid or Nadia is in Madrid implies Nadia is in Barcelona." This truism sounds like nonsense in ordinary discourse. 6. , if p does not imply q then p is true and q is false. NB if p were false then it would imply q, so p is true. If q were also true then p would imply q, h ence q is false. This paradox is particularly surprising beca