A Class-Room Introduction to Logic

May 4, 2009

Unit-VI: Translating Sentences into Standard form Propositions

Filed under: Logic,Translation to Standars form proposition — Dr. Desh Raj Sirswal @ 8:44 am

For reasoning in everyday life, as you know, people do not talk in standard categorical form. Categorical form is much too stilted for writing effective discourse. There is a need to develop skills of logical translation to standard form categorical propositions in order to minimize errors in evaluating syllogistic arguments. Very often translation into standard form reveals fallacies of equivocation and fallacies of amphiboly in the original text.
1.Translation Rules of Thumb:

The subject and predicate terms must be the names of classes. If the predicate term is a descriptive phrase, make it a substantive (i.e., noun phrase).

Translation must not (significantly) alter the original meaning of the sentence. Categorical propositions must have a form of the verb “to be” as the copula in the present tense.

The quality and quantity indicators are set up from the meaning of the sentences.

Quantity indicators: “All,” “No,” “Some.”

Quality indicators: “No,” “are,” “are not.”

The word order is rearranged according to the sense of the sentence.

This rule requires special care—in some instances, it may well be the most difficult rule to follow. On occasion, we need to divide one sentence into two or more propositions

Before we take up some special cases, let’s look at some typical examples:
The following translations are relatively straightforward.

“Ships are beautiful” translates to
“All ships are beautiful things.”
“The whale is a mammal” translates to
“All whales are mammals.”

“Whoever is a child is silly” translates to
“All children are silly creatures.”
“Snakes coil” translates to
“All snakes are coiling things.”

“All swans are not white” translates to
“Some swans are not white.”

“Nothing ventured, nothing gained” translates to
“No non-ventured things are gained things.”
Or the obverse…
“All non-ventured things are non-gained things.”

2. Singular propositions are to be treated as (but not usually translated into) a universal proposition (i.e., an A or an E).
E.g., “Socrates is a man” is an A proposition, but
“Socrates is not a god” is an E proposition.
3. Exclusive propositions have the cue words “only” or “none but.” The order of the subject and predicate terms must be reversed.
E.g., “None but A is B” translates to “All B is A.”
“Only A is B” translates to “All B is A.”
“None but red trucks are fire engines” translates to
“All fire engines are red things.”

4. Exceptive propositions are compound propositions.
E.g., “All except A is B” translates to “All non-A is B and “No A is B.
E.g., “All except human beings are nonsymbolic animals” translates to …”
“All nonhuman beings are nonsymbolic animals” and
“No human beings are nonsymbolic animals”
(or, of course the obverse, “All human beings are symbolic animals.”)

5. A Compound statement asserts two propositions.
E.g., “There is a time to sow and a time to reap” translates to
“Some occasions are times to sow” and
“Some occasions are times to reap.”

6. Abstract: An inductive strategy for mechanizing translation is illustrated.

We have at this time a kind of “took kit” to work on syllogisms. Our tools include:
obversion, conversion, and contraposition
Venn diagrams
logical analogies
rules and fallacies
various techniques for reducing the number of terms
translation strategies

The following inductive technique can be used for mechanizing translation by isolating the steps for testing the validity of a syllogism. The steps can be itemized as follows:

Identify the conclusion and premises.

Put the syllogism into standard order as best you can.

Supply the suppressed statements, if any.

Reduce the number of terms to three per syllogism.

Translate the statements to standard form.

Unit-VII: Immediate Inference-Square of Opposition

Filed under: Logic,Square of Oppsition — Dr. Desh Raj Sirswal @ 8:36 am

Inference: Inference is the act or process of deriving a conclusion based solely on what one already knows. Inference has two types: Deductive Inference and Inductive Inference. They are deductive, when we move from the general to the particular and inductive where the conclusion is wider in extent than the premises. In intelligence testing, mostly deductive inference ability is judged. Inference is studied within several different fields.

  • Human inference (i.e. how humans draw conclusions)is traditionally studied within the field of cognitive psychology.
  • Logic studies the laws of valid inference.
  • Statisticians have developed formal rules for inference (statistical inference) from quantitative data.
  • Artificial intelligence researchers develop automated inference systems.

IMMEDIATE INFERENCE:

Deductive inference may be further classified as (i) Immediate Inference (ii) Mediate Inference. In immediate inference there is one and only one premise and from this sole premise conclusion is drawn. Immediate inference has two types mentioned below: Square of Opposition , Eduction. Here we will study about Square of Opposition.

Square of Opposition:

Any logical relation among the kinds of categorical propositions (A, E,I and O) exhibited on the Square of Opposition. There are four ways in which propositions may be “observed” –as Contradictories, Contraries, Sub-contraries and sub-alternation. These are representing with an important and widely used diagram called the Square of Opposition. This is given below:

square

The four corners of this diagram represent the four basic forms of propositions recognized in classical logic:

A propositions, or universal affirmatives take the form: All S are P.
E propositions, or universal negations take the form: No S are P.
I  propositions, or particular affirmatives take the form: Some S are P.
O propositions, or particular negations take the form: Some S are not P.

Given the assumption made within classical (Aristotelian) categorical logic, that every category contains at least one member, the following relationships, depicted on the square, hold:

Contradictory:

Propositions are contradictory when the truth of one implies the falsity of the other, and conversely. A and O propositions are contradictory, as are E and I propositions. Here we see that the truth of a proposition of the form All S are P implies the falsity of the corresponding proposition of the form Some S are not P. For example, if the proposition “all industrialists are capitalists” (A) is true, then the proposition “some industrialists are not capitalists” (O) must be false. Similarly, if “no mammals are aquatic” (E) is false, then the proposition “some mammals are aquatic” must be true.

Contrary:

Propositions are contrary when they cannot both be true; if one is true, then other must be false. They can both be false. A and E propositions are contrary. An A proposition, e.g., “all giraffes have long necks” cannot be true at the same time as the corresponding E proposition: “no giraffes have long necks.” Note, however, that corresponding A and E propositions, while contrary, are not contradictory. While they cannot both be true, they can both be false, as with the examples of “all planets are gas giants” and “no planets are gas giants.”

Subcontrary:

Propositions are subcontrary when it is impossible for both to be false; if one is false then other must be true. They can both be true. I and O propositions are subcontrary. Because “some lunches are free” is false, “some lunches are not free” must be true. Note, however, that it is possible for corresponding I and O propositions both to be true, as with “some nations are democracies,” and “some nations are not democracies.” Again, I and O propositions are subcontrary, but not contrary or contradictory.

Subalternation:

Two propositions are said to stand in the relation of Subalternation when the truth of the first (“the superaltern”) implies the truth of the second (“the subaltern”), but not conversely. A propositions stand in the Subalternation relation with the corresponding I propositions. The truth of the A proposition “all plastics are synthetic,” implies the truth of the proposition “some plastics are synthetic.” However, the truth of the O proposition “some cars are not American-made products” does not imply the truth of the E proposition “no cars are American-made products.” In traditional logic, the truth of an A or E proposition implies the truth of the corresponding I or O proposition, respectively. Consequently, the falsity of an I or O proposition implies the falsity of the corresponding A or E proposition, respectively. However, the truth of a particular proposition does not imply the truth of the corresponding universal proposition, nor does the falsity of a universal proposition carry downwards to the respective particular propositions.

Inferences from Square of Opposition:

A number of very useful immediate inferences may be readily drawn from the information embedded in the traditional square of opposition. Given in the truth, or the falsehood, of any one of anyone of the four standards form categorical proposition, it will be seen that the truth or falsehood of some or all of the others can be inferred immediately.

A being given as True: E is false; I is true; O is false.

E being given as True: A is false; I is false; O is true.

I being given as True: E is false; A and O are undetermined.

O being given as True: A is false; E and I are undetermined.

A being given as  False : O is true , E and I are undetermined.

E being given as False: I is true; A and O are undetermined.

I  being given as False: A is false; E is true; O is true.

O being given as False: A is true; E is false; I is true.

For example:

What can you infer about the truth or falsity of the following if you assume “ Some reptiles are not poisonous” is false?

(1)    All reptiles are poisonous. – True

(2)   No reptiles are poisonous. –  False

(3)   Some reptiles are poisonous.- True

What is the name of the opposition relation in which the categorical statements differ:

  1. In quantity only?
  2. In both quality and quantity
  3. Between A and I.
  4. Between I and O.
  5. Between E and I.

Unit-VIII: Immediate Inference-Eduction

Filed under: Eduction: Immediate Inference,Logic — Dr. Desh Raj Sirswal @ 8:25 am


Eduction: The second form of Immediate Inference is Eduction. It has three types –Conversion, Obversion and Contraposition. These are not part of the square of opposition. They involve certain changes in their subject and predicate terms. The main concern is to converse logical equivalence.

Details are given below:

Conversion

An inference formed by interchanging the subject and predicate terms of a categorical proposition. Not all conversions are valid.

Conversion grounds an immediate inference for both E and I propositions That is, the converse of any E or I proposition is true if and only if the original proposition was true. Thus, in each of the pairs noted as examples either both propositions are true or both are false.

Steps for Conversion: Reversing the subject and the predicate terms in the premise.

Valid Conversions

Convertend                                       Converse

A: All S is P.                                        I: Some P is S (by limitation)

E: No S is P                                          E: No P is S

I : Some S is P                                     I : Some P is S

O: Some S is not P                              (conversion not valid)

Example:

All bags are mangoes.-A

Some mangoes are bags.-I

No men are intelligent.-E

No intelligent are men.-E

Some cows are tables.-I

Some tables are cows.-I

Some students are not cats.

(not valid)

Obversion

An inference formed by changing the quality of a proposition and replacing the predicate term by its complement. Obversion is valid for any standard form Categorical proposition.

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

Steps for Obversion:

  1. Replace the quality of the given statements. That is, if affirmative, change it into negative, and if negative, change it into affirmative.
  2. Replace the predicate term by its complementary term.

 Valid Obversions

Obverted                                             Obverse

A: All S is P.                                       E: No S is non-P.

E: No S is P                                        A: All S is non-P.

I : Some S is P                                    O : Some S is not non-P.

O: Some S is not P                              I: Some S is non-P.

Example:

All females are perfect beings.-A

No females are non-perfect beings.-E

No female are perfect beings.-E

All female are non-perfect beings.-A

Some female are perfect beings.-I

Some females are not non-perfect beings.-O

Some female are not perfect beings.-O

Some female are non-perfect beings.-I

Contraposition

An inference formed by replacing the subject term of a proposition with the complement of its predicate term, and replacing the predicate term by the complement of its subject term. Not all contrapositions are valid.

Contraposition is a reliable immediate inference for both A and O propositions; that is, the contrapositive of any A or O proposition is true if and only if the original proposition was true. Thus, in each of the pairs, both propositions have exactly the same truth-value.

Note: In contraposition the subject of the conclusion is contradictory of the predicate of the premise and predicate of the conclusion is contradictory of the subject of the premise.

Steps for Contraposition:

a. Convert the statement: reverse the subject and the predicate terms.

b. Replace both terms by their complementary terms.

Valid Contrapositions

Premises                                               Contrapositive

A: All S is P.                                      A: All non-P is non-S.

E: No S is P                                       O: Some non-P is not non-S.

(By limitation)

I : Some S is P                                    (Contraposition not valid)

O: Some S is not P                            O: Some non-P is not non-S.

Example:

All citizens are voter.-A

All non-voters are non-citizens.-A

No politicians are honest.-E

Some-non-honest are not non-politicians.-O

(by limitation)

Some applicants are graduate. -I

(cannot be contraposited)

Some students are not scholarship holders.-O

Some non-scholarship holders are not non-students.-O

Existential Import

It is time to express more explicitly an important qualification regarding the logical relationships among categorical propositions. There must be some things a certain kind. This special assumption, that the class designated by the subject term of a universal proposition has at least one member, is called existential import . Classical logicians typically presupposed that universal propositions do have an existential import.

Exercises:

Make the conversion, obversion and contraposition of the following:

  1. Some monks are not vegetarians.
  2. All journalists are pessimists.
  3. No reptiles are warm blooded animals.
  4. Some men are happy.
  5. No businessmen are philosophers.
  6. Artists are professionals.

Unit-IX: Mediate Inference –Categorical Syllogism

Filed under: Logic,Syllogism in Logic — Dr. Desh Raj Sirswal @ 8:17 am

In mediate inference conclusion draw from two and more than two premises. Both premises jointly imply the conclusion.

Syllogism: A syllogism is a form of mediate deductive inference, in which the conclusion is drawn from two premises take jointly. There are three major types of syllogism:

  • Conditional syllogism
  • Categorical syllogism
  • Disjunctive syllogism

Categorical Syllogism:

A categorical syllogism is a deductive argument consisting of exactly three categorical propositions (two premises and a conclusion) in which there appear a total of exactly three categorical terms, each of which is used exactly twice.

In a standard form categorical syllogism, major premise comes first, then the minor premise occurs and conclusion comes in the end. Standard form order of  a syllogism is the following format:

Major premise: A general statement.
Minor premise: A specific statement.
Conclusion: based on the two premises.

Consider, for example, the categorical syllogism:

No geese are felines.

Some birds are geese.

Therefore, Some birds are not felines.

 

Terms Used in Categorical Syllogism

A syllogism contains exactly three terms or class names:

Major Term/Major Premise: The major term is the term that occurs  as the predicate of the conclusion in a standard-form syllogism.

The major premise is the premise that contains the major term.

Minor Term/Miner Premise: The minor term is the term that occurs as the subject of the conclusion in a standard form syllogism. 

The minor premise is the premise that contains the minor term.

Middle term: The term that occurs in both premises, but not in the conclusion, of a standard form syllogism.

Syllogistic Moods:

Logicians also speak of syllogistic moods. Moods are defined as the arrangement of the premises according to quantity (universal or particular) and quality (affirmative or negative). In other words, we can say that mood is determined by the type of standard form categorical propositions of the syllogism contains. Example:

A-    All M is P.

A-    All S is M.

A-    All S is P.

So , AAA is the mood of this syllogism.

Now we will see what rules govern each figure and how these rules affect the validity of the single moods.

Figure in Syllogism:

The figure of a syllogism, determined by the positions of the middle term in its premises; there are four possible figures. When we use the term “syllogistic figure” we understand the disposition of the middle term (M) with respect to the major (P) and minor terms (S) in the premises of a syllogism.

The minor term (S) is always the subject and the major term (P) is always the predicate of the conclusion. Whatever variations that can take place in the relative position of the terms among themselves must occur in the premises.

In the major premise the middle term is compared with the major extreme. In the minor premise the middle term is compared with the minor extreme. This gives four different syllogistic figures:

Figure 1 Figure 2 Figure 3 Figure 4
M — P P — M M– P P — M
S — M S — M M– S M — S
S — P S — P S — P S — P

First Figure

M P All animals (M) are a nuisance (P).
S M All dogs (S) are animals (M).
S P Therefore, All dogs (S) are a nuisance (P).

The middle term is the subject of the major premise and the predicate of the minor premise.

Second Figure

P M No statesmen are good politicians.
S M Some journalists are good politicians.
S P Therefore, Some journalists are not statesmen.

The middle term is the predicate of both premises.

Third Figure

M P All writers are intelligent.
M S Some writers are American citizens.
S P Therefore, Some American citizens are intelligent.

The middle term is the subject of both premises.

Fourth Figure

P M All Americans are happy people.
M S All happy people are fun-loving
S P Therefore, Some fun-loving people are Americans.

The middle term is the predicate of the major premise and the subject of the minor premise.

The First Figure has been considered to be the perfect syllogism because it is the way we tend to make statements normally and naturally. The other three figures, however, are correct forms of syllogistic reasoning, even if they seem to be somewhat stilted and unnatural.

Rule of remind figures: SPIRIT OPPRESSED THE PSALMIST.

Meaning: SP- First figure, PP- Second figure, SS –Third figure and PS- Fourth figure.

Unit-X: Rules and Fallacies for Categorical Syllogisms

Filed under: Logic,Rules for validity of syllogism — Dr. Desh Raj Sirswal @ 8:09 am
Tags:


Aristotle and other traditional logicians provided certain rules which determine the validly/invalidity of syllogism. Here are some rules to check the validity of a syllogism.

Rule 1: Avoid Four Terms

Fallacy: Fallacy of four terms (A formal mistake in which a categorical syllogism contains more than three terms.)

Example:

             All men are rational animal.

All chalks are white.

Therefore, ————————.

Justification: A valid standard-from categorical syllogism must contain exactly three terms, each of which is used in the same sense throughout the argument. If there is more terms than, it cannot be in standard-form syllogism, we cannot call it syllogism.

Rule 2: The middle term must be distributed at least once.

Fallacy: Undistributed middle( A formal mistake in which a categorical syllogism contains a middle term that is not distribute in either premise.)

Example:

All sharks are fish.All salmon are fish.Therefore, All salmon are sharks

Justification: 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.

Rule 3: If a term is distributed in the conclusion, then it must be distributed in a premise.

Fallacy: Illicit major (A formal mistake in which the major term of a syllogism is undistributed in the major premise, but is distributed in the conclusion.)

Illicit minor (A formal mistake in which the minor term of a syllogism is undistributed in the minor premise, but is distributed in the conclusion.)

Examples:

And: All horses are animals.Some dogs are not horses.Therefore, Some dogs are not animals.

All tigers are mammals.

All mammals are animals.

Therefore, All animals are tigers.

Justification: 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. Remember that the minor premise says nothing about the P class. Therefore, the conclusion contains information that is not contained in the premises, making the argument invalid.

Rule 4: No conclusion drawn from two negative premises.

Fallacy: Exclusive premises (A formal mistake in which both premises of a syllogism are negative)

Example:

No fish are mammals.Some dogs are not fish.Therefore, Some dogs are not mammals.

Justification: 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.

Rule 5: A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise. (Alternate rendering: Any syllogism having exactly one negative statement is invalid.)

Fallacy: Drawing an affirmative conclusion from a negative premise, or drawing a negative conclusion from an affirmative premise. (A formal mistake in which one premise of a syllogism is negative but the conclusion is affirmative.)

Example:

All crows are birds.Some wolves are not crows.Therefore, Some wolves are birds.

Justification: Two directions, here. Take a positive conclusion from one negative premise. The conclusion states that the S class is either wholly or partially contained in the P class. The only way that this can happen is if the S class is either partially or fully contained in the M class (remember, the middle term relates the two) and the M class fully contained in the P class. Negative statements cannot establish this relationship, so a valid conclusion cannot follow.

Take a negative conclusion. It asserts that the S class is separated in whole or in part from the P class. If both premises are affirmative, no separation can be established, only connections. Thus, a negative conclusion cannot follow from positive premises.

Note: These first four rules working together indicate that any syllogism with two particular premises is invalid.

Rule 6: If both premises are universal, the conclusion cannot be particular. And also there is no conclusion from two particular premises.

Fallacy: Existential fallacy (As a formal fallacy, the mistake of inferring a particular conclusion from two universal premises.)

Example:

All mammals are animals.All tigers are mammals.Therefore, Some tigers are animals.

Justification: On the Boolean model, Universal statements make no claims about existence while particular ones do. 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.

 

Unit-XI: Check the Validity of Syllogism through Venn-Diagrams

Filed under: Logic,Venn Diagrams — Dr. Desh Raj Sirswal @ 8:03 am

The modern interpretation offers a more efficient method of evaluating the validity of categorical syllogisms. By combining the drawings of individual propositions, we can use Venn diagrams to assess the validity of categorical syllogisms by following a simple three-step procedure:

  1. Draw three overlapping circles and labels them to represent the major, minor, and middle terms of the syllogism.
  2. Draw the diagrams of both of the syllogism’s premises. Two things always remember: (i) Always begin with a universal proposition, no matter whether it is the major or the minor premise. (ii)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.
  3. Without drawing anything else, look for the drawing of the conclusion.
  4. If  conclusion draws, then the syllogism valid.
  5. If No, then the syllogism invalid.

Here are the examples of several other syllogistic forms. In each 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.

 

Example of  Valid Syllogism -AAA

All M are P.

All S are M.

Therefore, All S are P.

Example of Invalid Syllogism- AAA 

All M are P.

All M are S.

Therefore, All S are P.

Exercises:

Some other examples to be verified by the Venn diagrams:

(i) All P are M.

No M is S.

Therefore, No S are P.

(ii)No M is P.

No S are M.

Therefore, No S are P.

(iii) All P are M.

Some M are S.

Therefore, Some S are P.

(iv)  All M are P.

All S are M.

Therefore, All S are P.

(v) No humans are cats, and all humans are mammals . Therefore, no cats are mammals.

(vi) Elephants are not tigers and no elephants are carnivorous, so no tigers are carnivorous.

Unit-XII: Arguments of Ordinary language

Filed under: Argument in Ordinary Language,Uncategorized — Dr. Desh Raj Sirswal @ 7:52 am

Argument:

An argument is a connected series of statements or propositions, some of which are intended to provide support, justification or evidence for the truth of another statement or proposition. Arguments consist of one or more premises and a conclusion. The premises are those statements that are taken to provide the support or evidence; the conclusion is that which the premises allegedly support.

Argument Form: In logic, the argument form or test form of an arguement results from replacing the different words, or sentences, that make up the argument with letters, along the lines of algebra; the letters represent logical variables. The sentence forms which classify argument forms of common important arguments are studied in logic.

Here is an example of an argument:

All humans are mortal. Socrates is human. Therefore, Socrates is mortal.

We can rewrite argument  by putting each sentence on its own line:

All humans are mortal.

Socrates is human.

Therefore, Socrates is mortal.

To demonstrate the important notion of the form of an argument, substitute letters for similar items :

All S are P.

a is S.

Therefore, a is P.

Thus arguments are structural pieces of articulated critical reasoning. Every argument must have a conclusion and a premise or some premises.

 

Deductive and Inductive Arguments

There are two types of arguments:

Deductive Argument: A deductive argument is an argument in which it is thought that the premises provide a guarantee of the truth of the conclusion. Here the premises are intended to provide support for the conclusion that is so strong that, if the premises are true, it would be impossible for the conclusion to be false.

Inductive Arguments: An inductive argument is an argument in which it is thought that the premises provide reasons supporting the probable truth of the conclusion. Here the premises are intended only to be so strong that, if they are true, then it is unlikely that the conclusion is false.

Difference between Deductive and Inductive Argument:

The difference between the two comes from the sort of relation the author or expositor of the argument takes there to be between the premises and the conclusion. If the author of the argument believes that the truth of the premises definitely establishes the truth of the conclusion due to definition, logical entailment or mathematical necessity, then the argument is deductive. If the author of the argument does not think that the truth of the premises definitely establishes the truth of the conclusion, but nonetheless believes that their truth provides good reason to believe the conclusion true, then the argument is inductive.

Arguments have certain special characteristics:

  1. Arguments are not claims.
  2. Every set of claims is not an argument.
  3. There is no fixed number of premises in the argument.
  4. Format of an argument may not always be simple.
  5. There may be unstated premises.
  6. There can be missing premises.
  7. Arguments have a standard format:

Premises

Therefore, Conclusion

To put arguments in the standard format, one has to do the followings:

  1. Separate the premises from the conclusion.
  2. State, the premises first in a sequential order and, if necessary, number them.
  3. Then state the conclusion with a conclusion marker, such as the symbol “/” or any of the conclusion-indicator words.

Recognizing and Argument:

Premise- indicator words: Since-For-Because-Given that.

Conclusion- indicator words: Therefore- Hence- It follows that-So-Consequently-Thus.

Unit-XIII: Types of Arguments

Filed under: Logic,types of argument in ordinary language — Dr. Desh Raj Sirswal @ 7:43 am

1. Enthymeme

An enthymeme, in its modern sense, is an informally stated syllogism (a three-part deductive argument) with an unstated assumption that must be true for the premises to lead to the conclusion. In an enthymeme, part of the argument is missing because it is assumed. In a broader usage, the term “enthymeme” is sometimes used to describe an incomplete argument of forms other than the syllogism.

Enthymeme’s three parts

The following quotation is an example of an enthymeme (used for humorous effect).

“There is no law against composing music when one has no ideas whatsoever. The music of Wagner, therefore, is perfectly legal.” —Mark Twain.

The three parts:

There is no law against composing music when one has no ideas whatsoever. (Premise)

The music of Wagner, therefore, is perfectly legal. (Conclusion)

Wagner has no ideas. (Implicit premise)

Further examples:

Socrates is mortal because he’s human.

The complete syllogism would be the classic:

All humans are mortal. (Major premise – assumed)

Socrates is human. (Minor premise – stated)

Therefore, Socrates is mortal. (Conclusion – stated)

Hidden premises are often an effective way to obscure a questionable or fallacious premise in reasoning. Typically fallacies of presumption (fallacies based on mistaken assumptions, such as ad hominem or two wrongs make a right) are attracted to enthymeme.

2. Sorites

An argument in which a conclusion is inferred from any number of premises through a chain of syllogistic inferences.

Example:

All babies are illogical persons.

All illogical persons are despised person..

No persons who can imagine crocodiles are despised persons.

Therefore, No babies are persons who can imgine crocodiles.

Solution:

            This Sorites consists of two syllogisms, as follows:

All I is D                                  No M is D

All B is I                                  All B is D

Therefore, All B is D              Therefore, No B is M.

3. The Disjunctive Syllogism

This syllogism presents two alternatives in an “either . . . or” form; one of the alternatives is for formal reasons assumed to be necessarily true, so that to deny one leaves the other as the only possibility. The two possibilities, called disjuncts, are stated in the major premise; one is and must be denied in the minor premise; and the other is affirmed in the conclusion. This is the valid form, which can be shown as follows:

Either A or B
Not A; therefore B
(Deny first disjunct; affirm the second)

Either A or B
Not B; therefore A
(Deny second disjunct; affirm the first)

The opposite procedure of first affirming and then denying is, however, incorrect. Except where the members are explicitly contradictory so that both could not possibly be true, the affirmation of one disjunct (in the minor premise) does not deny the other. For example, to say, “Either the power is off or the bulb is burned out; the power is off so the bulb is not burned out,” would be a fallacy, because, while we assume that one of the disjuncts is definitely true, both might be true–we did not check the bulb and so cannot be sure of its condition. Since the second disjunct has not been investigated, it cannot be denied by default. (Where the members of the disjunct are contradictory, as in “The plant is either alive or dead,” the argument should, to avoid confusion, be changed into the conjunctive form of syllogism and worked from there–see below, section #3.)

The fallacy, then, of first affirming one disjunct and then denying the other looks like this:

Either A or B
And A; therefore not B

Either A or B
And B; therefore not A

Fallacy of Affirming a Disjunct (AD)

4. Hypothetical Syllogisms

Hypothetical syllogisms are different from standard syllogisms and thus have their own rules. In a hypothetical syllogism the first premise (or major proposition) presents an uncertain condition (“if A, then B”) or a problem (“either A or B”; “S and T cannot both be true”) which must then be properly resolved by the second premise so that a valid conclusion can follow. The resolution of the problem is always in the form of affirmation or denial. In this article, the three types of hypothetical syllogism we will cover are the conditional syllogism, the disjunctive syllogism, and the conjunctive syllogism.

If P is true then Q is true.

If Q is true then R is true.

Therefore, If P is true then R is true.

If P is true then Q is true.

P is true.

Therefore, Q is true.

5. The Dilemma

A dilemma means double proposition, it is a problem offering at least two solutions or possibilities, of which none are practically acceptable; one in this position has been traditionally described as being impaled on the horns of a dilemma, neither horn being comfortable.

The dilemma is sometimes used as a rhetorical device, in the form “you must accept either A, or B”; here A and B would be propositions each leading to some further conclusion. Applied in this way, it may be a fallacy, a false dichotomy.

Horned dilemmas can present more than two choices. The number of choices of Horned dilemmas can be used in their alternative names, such as two-pronged (two-horned) or dilemma proper , or three-pronged (three-horned) or trilemma, and so on.
Constructive dilemmas–

1. (If X, then Y) and (If W, then Z).

2. X or W.

3. Therefore, Y or Z.

Destructive dilemmas–

1. (If X, then Y) and (If W, then Z).

2. Not Y or not Z.

3. Therefore, not X or not W.

Unit-XIV: Fallacies (Formal & Informal)

Filed under: Logic,Uncategorized — Dr. Desh Raj Sirswal @ 7:37 am

From a psychological point of view, a fallacy is often defined as a mistake in reasoning used for deceptive purposes; however, many fallacies are, in fact, not deceptive to most persons. Even so, many of the informal fallacies are often used in the manipulation of opinion. Many of these mistakes in reasoning occur so often they deserve special study. This section investigates informal fallacies—those dependent upon language. An informal fallacy is one that arises from the content of an argument (the meaning what is said, not the grammar in terms of how the argument is expressed).

Our account of fallacies is in the tradition of I. M. Copi’s presentation: he reveals that some mistakes in reasoning arise from appeals to irrelevant factors and others from unsupported assumptions.

Nevertheless, as Joseph says in his Introduction to Logic (569): “Truth may have its norm, but error is infinite in its aberrations, and they cannot be digested in any classification.”

Not all irrelevant appeals and unsupported assumptions are fallacies. Fallacies occur in argumentative discourse. Thus, if no argument is offered, no fallacy is present

Fallacy:

A fallacy is a type of mistake in argumentation that might appear to be correct, but which proves upon examination not to be so.

Let us classify two basic types: 1. Informal Fallacy : those dependent upon language– i.e., a fallacy that arises from the content of an argument (the what is said, not the how it is said). 2. Formal Fallacy: those outside the content of language–i.e., a fallacy that arises from an error in the form of an argument; it is (usually) independent of content.

Fallacies of No Evidence

Argument Against the Person (argumentum ad hominem) This fallacy is committed when you attack a person s character or personal circumstances in order to oppose or discredit their argument or viewpoint. Also:

Tu Quoque Fallacy (you re one, too ) A type of abusive ad hominem that attempts to discredit a person s viewpoint or position by charging the person with hypocrisy or inconsistency. Essentially, the charge is, We don t need to take his argument seriously because he doesn’t practice what he preaches.

Guilt by association Fallacy A type of abusive ad hominem in which one person attacks a second person s associates in order to discredit the person and thereby his view or argument.

Appeal to Force (argumentum ad baculum, literally argument from the stick ) A fallacy committed when an arguer appeals to force or to the threat of force to make someone accept a conclusion.

Appeal to Pity (argumentum ad misericordiam) A fallacy committed when the arguer attempts to evoke pity from the audience and tries to use that pity to make the audience accept a conclusion.

Appeal to the People (argumentum ad populum) A fallacy committed when an arguer attempts to arouse and use the emotions of a group to win acceptance for a conclusion.

Snob Appeal Fallacy This is committed when the arguer claims that if you will adopt a particular conclusion, this will place you in a special, elite group or will make you better than everyone else.

Fallacy of Irrelevant Conclusion (ignoratio elenchi, meaning ignorance of the proof ) A fallacy in which someone puts forward premises in support of a stated conclusion, but the premises actually support a different conclusion.

Begging the Question Fallacy (petitio principii, meaning postulation of the beginning ) This is committed when someone employs the conclusion (usually in some disguised form) as a premise in support of that same conclusion.

Appeal to Ignorance (argumentum ad ignorantium) In this fallacy, someone argues that a proposition is true simply on the grounds that it has not been proven false (or that a proposition must be false because it has not been proven true).

Red Herring Fallacy A fallacy committed when the arguer tries to divert attention from his opponent s argument by changing the subject and drawing a conclusion about the new subject.

Genetic Fallacy A fallacy committed when someone attacks a view by disparaging the view s origin or the manner in which the view was acquired.

Poisoning the Well The use of emotionally charged language to discredit an argument or position before arguing against it.

Fallacies of Little Evidence

Fallacy of Accident A fallacy committed when a general rule is applied to a specific case, but because of extenuating circumstances, the case is an exception to the general rule and the general rule should not be applied to the case.

Straw Man Fallacy A fallacy committed when an arguer (a) summarizes his opponent’s argument but the summary is an exaggerated, ridiculous, or oversimplified representation of the opponent s argument that makes the opposing argument appear illogical or weak;

(b) the arguer refutes the weakened, summarized argument; and (c) the arguer concludes that the opponent s actual argument has been refuted.

Appeal to Questionable Authority Fallacy (argumentum ad verecundiam) When someone attempts to support a claim by appealing to an authority that is untrustworthy, or when the authority is unqualified, or prejudiced, or has a motive to lie.

Fallacy of Hasty Generalization A fallacy committed when someone draws a generalization about a group on the basis of observing an unrepresentative sample of the group.

False Cause Fallacy A fallacy involving faulty reasoning about causality. Also:

In a Post Hoc Ergo Propter Hoc fallacy ( after this, therefore, because of this ) someone concludes that A is the cause of B simply on the grounds that A preceded B in time.

In a Non Causa Pro Causa fallacy ( not the cause for the cause ) someone claims that A is the cause of B, when in fact (1) A is not the cause of B, but (2) the mistake is not based merely on one thing coming after another thing. One version of this fallacy is the fallacy of accidental correlation: the arguer concludes that one thing is the cause of another thing from the mere fact that the two phenomena are correlated.

Slippery Slope Fallacy (or domino argument ) In this fallacy, someone objects to a position P on the grounds that P will set off a chain reaction leading to trouble; but no reason is given for supposing the chain will actually occur. Metaphorically, if we adopt a certain position, we will start sliding down a slippery slope and we won t be able to stop until we slide all the way to the bottom (where some bad result lies in wait).

Fallacy of Weak Analogy A fallacy committed when an analogical argument is presented but the analogy is too weak to support the conclusion.

Fallacy of False Dilemma A fallacy committed when someone assumes there are only two alternatives, eliminates one of these two, and concludes in favor of the second, when more than the two stated alternatives exist, but have not been considered.

Fallacy of Suppressed Evidence In this fallacy, evidence that would count heavily against the conclusion is left out of the argument or is covered up.

Fallacy of Special Pleading In this fallacy, the arguer applies a principle to someone else s case but makes a special exception to the principle in his own case.

Fallacies of Language

Fallacy of Equivocation In this fallacy, a particular word or phrase is used with one meaning in one place, that word or phrase is used with another meaning in another place, and what has been established on the basis of the one meaning is regarded as established with respect to the other meaning. As a result, the conclusion depends on a word (or phrase) being used in two different senses in the argument. The premises are true on one interpretation of the word, but the conclusion follows only from a different interpretation.

Fallacy of Amphiboly A fallacy containing a statement that is ambiguous because of its grammatical construction. One interpretation makes the statement true, the other makes it false. If the ambiguous statement is interpreted one way, the premise is true but the conclusion is false; but if the ambiguous statement is interpreted the other way, the premise is false. The meaning must shift if the argument is going to go from a true premise to a true conclusion. If the meaning is not allowed to shift during the argument, either the argument has false a premise or it is invalid.

Fallacy of Composition A fallacy in which someone uncritically assumes that what is true of a part of a whole is also true of the whole.

Fallacy of Division A fallacy in which someone uncritically assumes that what is true of the whole must be true of the parts.

Section-6 : Symbolic Logic

Filed under: Symbolic Logic — Dr. Desh Raj Sirswal @ 7:33 am

In the history of Western logic, Symbolic logic is a relatively recent development. What set symbolic logic apart from traditional logic is its leanings towards mathematics and symbolization. A general theory of deduction aims to explain the relations between premises and conclusion in deductive arguments and to provide techniques   for discriminating between valid and invalid deduction. Two great bodies of logical theory have sought to achieve these ends. The first, called “classical” (or Aristotelian) logic. The second, called “modern logic or symbolic logic.”  Symbolic logic of today owes its origin primarily to Frege and Russell, and then to Peano and many others. If they had helped the gensis of symbolic logic, other 20th CE mathematicians and philosophers such as Brouwer, Godel, Cantor, Hilbert, Wittgenstein, Tarski, Zermelo, Gentzen must be acknowledged  for joing in their efforts and for their worth contributions towards its steady growth. Here we  will only study Propositional Logic means Truth-functional Logic.                                                      

Unit-XV: Truth-functional Logic

Unit-XVI: Truth-functional Compound Statements

Unit-XVII: Validity and Invalidity by Truth-table Method

Unit-XVIII: Statement Forms

 


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