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- Understanding the structure of arguments: argument forms, structure of categorical propositions, Mood and Figure, Formal and Informal fallacies, Uses of language, Connotations and denotations of terms, Classical square of opposition.
- Evaluating and distinguishing deductive and inductive reasoning.
- Venn diagram: Simple and multiple use for establishing validity of arguments.
- Indian Logic: Means of knowledge.
- Pramanas: Pratyaksha (Perception), Anumana (Inference), Upamana (Comparison), Shabda (Verbal testimony), Arthapatti (Implication) and Anupalabddhi (Non-apprehension).
- Structure and kinds of Anumana (inference), Vyapti (invariable relation), Hetvabhasas (fallacies of inference).
UNDERSTANDING THE STRUCTURE OF ARGUMENTS
Mood and Figure
Figure and mood together determine the structure of syllogism. An appraisal of the significance of structure in deductive inference in general and syllogism in particular is made much easier when we deal with ‘figures and moods’ of syllogism.
An analysis of the structure of argument in deductive inference is a pre-requisite to the classification of arguments into good (valid) and bad (invalid) . Since the very function of logic is to distinguish arguments in the aforesaid manner, a study of figure and mood occupies an important position in our study of syllogism.
An appraisal of the significance of structure in deductive inference in general and syllogism in particular is made much easier when we deal with ‘figures and moods’ of syllogism.
Stating the arguments in what is called standard-form. Accordingly, the major premise is stated first followed by the minor premise and ending with the conclusion.
The following example illustrates what standard-form means:
- All humans are mortal.
- Joseph is a human.
∴ Joseph is mortal
The arguments can be paraphrased in standard-form by following process:
- Identify the conclusion which is to be placed in the final position.
- Whichever premise contains the predicate term of the conclusion automatically occupies the first place because the major premise should be stated first.
In the above arguments, ‘mortal’ is the predicate of the conclusion which appears in the first place in the argument followed by the minor premise. Therefore this type of arrangement subscribers to standard-form.
The Mood of a Syllogism:
There are four types of categorical proposition:
- universal affirmative (A)
- universal negative (E)
- particular affirmative (I)
- particular negative (O)
- The mood of a syllogism is simply a combination of categorical propositions (A, E, I, or O) which the argument comprises of.
- 64 moods can be enlisted.
Figures of Syllogism
The ‘figure’ of a syllogism is determined by the position of ‘middle term’. The ‘middle term’ appears both in the major and in the minor premises. Therefore its possible positions in premises result in four different configurations.
A schematic representation is preferable to verbal description. (S = Subject, P= Predicate, M= Middle Term)
|VERBAL DESCRIPTION||M− P
|EXAMPLES|| M P
All artists are poets. ( AAP)
| P M
All saints are pious. (SAP)
All great works are
worthy of study. (GAW)
| P M
No soldiers are traitors. (SET)
| S M
All musicians are artists. (MAA)
No criminals are
| M S
All great works are epics. (GAE)
| M S
All traitors are sinners. (TAS)
|Conclusion|| S P
All musicians are poets. (MAP)
No criminals are
Some epics are worthy of
Some sinners are not
Moods (only 6 moods are possible )
AAA, AAI, EAE, EAO, EIO and AII
AEE, AEO, EAE, EAO, EIO and AOO
AAI, AII, IAI, EAO, EIO and OAO
AAI, IAI, AEE, AEO, EAO, and EIO
|First letter stands for the major premise, second for the minor premise and third premise is conclusion.
Moods represented in 3 ways-
Strengthened moods: When the laws of syllogism permit two universal premises to yield logically only particular conclusion, then such moods are called strengthened moods.
Weakened moods: If particular conclusion is deduced from two universal premises, even when the laws of syllogism permit two universal premises to yield logically a universal conclusion, then such moods are called weakened moods.
Mnemonic verses : The process was give by Pope John XXI. It gives a technique to remember the method of reducing arguments from other figures o the first figure. Each mood, excluding weakened moods, was given a special name:
AII – DARII
EIO – FERISON
EIO – FRESISON
Incomplete syllogism and compound syllogism
- Enthymeme: Enthymeme is called an incomplete syllogism in which one or the other proposition is not stated explicitly.
- Sorites: If an argument consists of three or more than three premises, then such an argument is called sorites. It is also called polysyllogism. There are two kinds of sorties: Aristotelian sorites and Goclenian sorites. The primary rules which govern sorites are the rules of the categorical syllogism only.
Formal and Informal fallacies
Literal meaning of ‘fallacy’ is ‘ a mistaken belief and in the field of logic means ‘ a failure in reasoning which renders an argument valid’ or ‘faulty reasoning’.
Logical fallacy is the reasoning that is evaluated as logically incorrect and that undermines the logical validity of the argument and permits its recognition as unsound. Logical fallacy can occur as accidental or can be deliberately used as an instrument of manipulation. Basically, errors in reasoning are called fallacies. The cause of invalidity of any argument is also the cause of fallacies. Fallacies may occur either due to the violation of rules of inference or it may occur due to deduction of false conclusion from true premise or premises. These two possibilities may or may not overlap.
The different kind of fallacies are as follows:
Formal Fallacy: A formal fallacy is an error in logic that can be seen in the argument´s form. Any hypothetical proposition consists of antecedent and consequent and its relevance lies in its application as a part of argument or pseudo-argument. According to the rule, in the second premise either antecedent must be affirmed or consequent must be denied.
- Appeal to probability: is a statement that takes something for granted because it would probably be the case (or might be the case).
- Argument from fallacy: is the assumption that if an argument for some conclusion is fallacious, then the conclusion is false.
- Base rate fallacy: is making a probability judgment based on conditional probabilities, without taking into account the effect of prior probabilities.
- Conjunction fallacy: is the assumption that an outcome simultaneously satisfying multiple conditions is more probable than an outcome satisfying a single one of them.
- Masked-man fallacy: (illicit substitution of identical) is when the substitution of identical designators in a true statement can lead to a false one.
- A propositional fallacy: is an error in logic that concerns compound propositions. For a compound proposition to be true, the truth values of its constituent parts must satisfy the relevant logical connectives that occur in it.
- Affirming a disjunct : is conclusion that one disjunct of a logical disjunction must be false because the other disjunct is true.
- Affirming the consequent: is when the antecedent in an indicative conditional is claimed to be true because the consequent is true.
- Denying the antecedent: is when the consequent in an indicative conditional is claimed to be false because the antecedent is false.
- A quantification fallacy: is an error in logic where the quantifiers of the premises are in contradiction to the quantifier of the conclusion.
- Existential fallacy: is an argument that has a universal premise and a particular conclusion.
- Formal syllogistic fallacies: are logical fallacies that occur in syllogisms.
- Affirmative conclusion from a negative premise :(illicit negative) is when a categorical syllogism has a positive conclusion, but at least one negative premise.
- Fallacy of exclusive premises: is a categorical syllogism that is invalid because both of its premises are negative.
- Fallacy of four terms: is a categorical syllogism that has four terms.
- Illicit major: is a categorical syllogism that is invalid because its major term is not distributed in the major premise but distributed in the conclusion.
- Illicit minor: is a categorical syllogism that is invalid because its minor term is not distributed in the minor premise but distributed in the conclusion.
- Negative conclusion: from affirmative premises (illicit affirmative) is when a categorical syllogism has a negative conclusion but affirmative premises.
- Fallacy of the undistributed: middle is when the middle term in a categorical syllogism is not distributed.
- Modal fallacy: is when confusing possibility with necessity.
- Modal scope fallacy: is when a degree of unwarranted necessity is placed in the conclusion.
Informal Fallacies : Informal fallacies are arguments that are fallacious for reasons other than structural (formal) flaws and that usually require examination of the argument´s content.
- Argument to moderation (false compromise, middle ground, fallacy of the mean, argumentum ad temperantiam) is when assuming that the compromise between two positions is always correct.
- Continuum fallacy:(fallacy of the beard, line-drawing fallacy, sorites, fallacy, fallacy of the heap, bald man fallacy) is when improperly rejecting a claim for being imprecise.
- Suppressed correlative: is when a correlative is redefined so that one alternative is made impossible.
- Definist fallacy: involves the confusion between two notions by defining one in terms of the other.
- Divine fallacy (argument from incredulity) is when arguing that, because something is so incredible or amazing, it must be the result of superior, divine, alien or paranormal agency.
- Double counting is counting events or occurrences more than once in probabilistic reasoning, which leads to the sum of the probabilities of all cases exceeding unity.
- Equivocation is the misleading use of a term with more than one meaning.
- Ecological fallacy is when inferences about the nature of specific individuals are based solely upon aggregate statistics collected for the group to which those individuals belong.
- Etymological fallacy is the reasoning that the original or historical meaning of a word or phrase is necessarily similar to its actual present-day usage.
- Fallacy of composition is when assuming that something true of part of a whole must also be true of the whole.
- Fallacy of division is when assuming that something true of a thing must also be true of all or some of its parts.
- False attribution is when an advocate appeals to an irrelevant, unqualified, unidentified, biased or fabricated source in support of an argument. Example: Fabricated scientific reports.
- Fallacy of quoting out of context refers to the selective excerpting of words from their original context in a way that distorts the source´s intended meaning.
- False authority (single authority) is when using an expert of dubious credentials or using only one opinion to sell a product or idea.
- False dilemma (false dichotomy, fallacy of bifurcation, black-or-white fallacy) is when two alternative statements are held to be the only possible options when in reality there are more.
- False equivalence is when describing a situation of logical and apparent equivalence, when in fact there is none.
- Incomplete comparison is when insufficient information is provided to make a complete comparison.
- Mind projection fallacy is when subjective judgments are projected to be inherent properties of an object, rather than being related to personal perceptions of that object.
- Nirvana fallacy (perfect-solution fallacy) is when solutions to problems are rejected because they are not perfect.
- Psychologist´s fallacy is when an observer presupposes the objectivity of their own perspective when analyzing a behavioral event.
- Reification (hypostatization, fallacy of misplaced concreteness) is a fallacy of ambiguity, when an abstraction (abstract belief or hypothetical construct) is treated as if it were a concrete, real event or physical entity. It is the error of treating as a real thing something that is not a real thing, but merely an idea.
- Slippery slope (thin edge of the wedge,camel´s nose) is when asserting that a relatively small first step inevitably leads to a chain of related events culminating in some significant impact/event that should not happen, thus the first step should not happen.
- Begging the question is when providing what is essentially the conclusion of the argument as a premise.
- Circular reasoning is when the reasoned begins with what he or she is trying to end up with.
- Fallacy of many questions is when someone asks a question that presupposes something that has not been proven or accepted by all the people involved.
- Faulty generalization means to reach a conclusion from weak premises.
- Accident is when an exception to a generalization is ignored.
- Cherry picking is the act of pointing at individual cases or data that seem to confirm a particular position, while ignoring a significant portion of related cases or data that may contradict that position.
- False analogy is an argument by analogy in which the analogy is poorly suited.
- Hasty generalization is when basing a broad conclusion on a small sample or the making of a determination without all of the information required to do so.
- Misleading vividness involves describing an occurrence in vivid detail, even if it is an exceptional occurrence, to convince someone that it is a problem.
- Questionable cause is a general type of error with many variants. Its primary basis is the confusion of association with causation, either by inappropriately deducing (or rejecting) causation or a broader failure to properly investigate the cause of an observed effect.
- Cum hoc ergo propter hoc is a faulty assumption that, because there is a correlation between two variables, one caused the other.
- Fallacy of the single cause is when it is assumed that there is one, simple cause of an outcome when in reality it may have been caused by a number of only jointly sufficient causes.
- Gambler´s fallacy is the incorrect belief that separate, independent events can affect the likelihood of another random event.
- Magical thinking is fallacious attribution of causal relationships between actions and events.
- Appeal to the stone is when dismissing a claim as absurd without demonstrating proof for its absurdity.
- Argument from ignorance is when assuming that a claim is true because it has not been or cannot be proven false, or vice versa.
- Argument from incredulity is when someone cannot imagine how something can be true. Therefore, it must be false.
- Argument from repetition is when repeating an argument until nobody cares to discuss it any more.
- Argument from silence is when assuming that a claim is true based on the absence of textual or spoken evidence from an authoritative source, or vice versa.
Red Herring Fallacies : A red herring fallacy, one of the main sub types of fallacies of relevance, is an error in logic where a proposition is, or is intended to be, misleading in order to make irrelevant or false inferences.
Types of Red Herring Fallacy:
- Ad hominem is when attacking the arguer instead of the argument.
- Appeal to authority is when an assertion is deemed true because of the position of authority of the person asserting it.
- Appeal to consequences is when the conclusion is supported by a premise that asserts positive or negative consequences from some course of action in an attempt to distract from the initial discussion.
- Appeal to emotion is when an argument is made due to the manipulation of emotions,rather than the use of valid reasoning. These fallacies are appeal to fear, appeal to flattery, appeal to pity, appeal to ridicule, appeal to spite, judgmental language, pooh-pooh and wishful thinking.
- Appeal to novelty is when a proposal is claimed to be superior or better solely because it is new or modern.
- Appeal to wealth is when supporting a conclusion because the arguer is wealthy (or refuting because the arguer is poor).
- Appeal to poverty is the opposite of appeal to wealth.
- Argumentum ad populum (appeal to widespread belief, bandwagon argument, appeal to the majority, appeal to the people) is when a proposition is claimed to be true or good solely because a majority or many people believe it to be so. This fallacy is similar to appeal to popularity (something is true because it is popular).
Straw man fallacy is when an argument is based on misrepresentation of an opponent´s position, especially in a way to attack a weaker version of it rather than the argument actually
Connotations and denotations of terms
Logic makes a sharp distinction between ‘word’ and ‘term’. All words are not terms, but all terms are words. Terms refer to specific classes of objects or qualities whereas words refer to none of them. A term is a word or group of words which is either a subject or a predicate of a proposition.
A term is a group of words which has the following characteristics:
- It is either a subject or predicate of a proposition
- It must have a definite meaning of its own.
Positive and negative: Positive terms signify the presence of desirable qualities e.g., light,health, etc.; negative terms signify, generally, undesirable qualities or qualities not desired, rightly or wrongly.
Concrete and abstract: Concrete terms are those which refer to perceptible entities; abstract terms are those which do not; e.g. man, animal, tall etc., are concrete terms; mankind, animalism, etc., are abstract terms.
Relative and absolute: Relative terms are those which express a relation between two or more than two persons or things, e.g. father, son, etc. Absolute terms do not express such relation, e.g. nationality, cone, etc. Comparative terms are obviously relative: e.g., larger, prettier, etc.
Singular and General: Singular terms denote specific objects. It points to one object only. All proper names are singular terms, General terms are just class names. terms. ‘The author of Principia Mathematica, The farthest planet from the sun’, etc. are singular terms. Vegetable, criminal, politician, etc., are examples for general terms.
Univocal and Equivocal terms: Univocal terms carry only one meaning. Entropy is an example for univocal terms. Equivocal terms are burdened with at least two meanings. Gravity is equivocal; so is astronomical.
Two important aspects of a term are:
- Denotation and Connotation
Denotation and Connotation:
Denotation of a term consists of the thing or object to which it applies and its connotation consists of the attributes which it implies. The connotation of a term is the characteristics determine the object(s) to which term intends and which are therefore sufficient to distinguish that object(s) from other objects. The denotation of a term refers to the extension to which it can be referred. The connotation refers to the intention with a term intends. The extension of a term is given by specifying the class members what the term denotes (the objects, the extension of a term is given by specifying the class members what the term denotes (the objects, the things, it refers to).The intention of a term is specified by listing the characteristics, properties, that the term connotes. The characteristics or qualities referred by the term do not always refer to all qualities, essential or accidental. Connotation indicates only essential qualities. Essential characteristics are those bare minimum qualities without which the existence of a thing, person or an object is not possible. Only the general terms (which are classes) have both connotation and denotation. Connotation and Denotation are related to each other by inverse variable relation, meaning connotation increases denotation decreases.
Distribution of terms:
A term is said to be distributed if the proposition in which it occurs, either includes or excludes the said term completely A term is distributed if it refers to all the members of its class. For example, on the proposition ,” All men are mortal”, the term “man” is distributed for it refers to all the members of the class man.
Classical Square of opposition
Square of Opposition: A standardized arrangement of standard form categorical propositions that portrays the relationships between the types of propositions and aids in drawing inferences on the basis of their properties.
The doctrine of the square of opposition originated with Aristotle in the 4th Century BC and has occurred in logic texts.
The square of opposition is a group of assumptions embodied in a diagram. The diagram is not essential to the assumptions; it is just a useful way to keep them straight. The theory concern logical relations among 4 logical forms:
|A||Every S is P||Universal Affirmative|
|E||No S is P||Universal Negative|
|I||Some S is P||Particular Affirmative|
|O||Some S is not P||Particular Negative|
The diagram for the traditional square of opposition is:
This is a genuine square of opposition if and only if all the following hold.
(1) ‘S is P’ and ‘S is not-P’ cannot both be true.
(2) ‘S is P’ and ‘S is not P’ are contradictory.
(3) ‘S is not-P’ and ‘S is not not-P’ are contradictory.
(4) ‘S is not not-P’ and ‘S is not P’ cannot both be false.
(5)’S is P’ implies ‘S is not not-P’.
(6) ‘S is not-P’ implies ‘S is not P’.
All the logical relations among the four categorical and, indeed, among four can be guaranteed by just two simple Aristotelian rules.
- The Law of Contradiction: An affirmation and its negation cannot both be true.
- The Law of Excluded Middle: Either an affirmation or its negation must be true.
- The Law of Incompatibility (An affirmation and its logical contrary cannot both be true) is derivable from the Law of Contradiction.