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1. What is dilemma? How we can avoid dilemma?
Ans. A dilemma is a problem offering two possibilities, neither of which is unambiguously acceptable or preferable. The possibilities are termed the horns of the dilemma, a clichéd usage, but distinguishing the dilemma from other kinds of predicament as a matter of usage.
The term dilemma is attributed by Gabriel Nuchelmans to Lorenzo Valla in the 15th century, in later versions of his logic text traditionally called Dialectica. Valla claimed that it was the appropriate Latin equivalent of the Greek dilemmaton. Nuchelmans argued that his probable source was a logic text of c.1433 of George of Trebizond. He also concluded that Valla had reintroduced to the Latin West a type of argument that had fallen into disuse.
Valla’s neologism did not immediately take hold, preference being given to the established Latin term complexio, used by Cicero, with conversio applied to the upsetting of dilemmatic reasoning. With the support of Juan Luis Vives, however, dilemma was widely applied by the end of the 16th century.
In the form “you must accept either A, or B” — here A and B are propositions each leading to some further conclusion — and applied incorrectly, the dilemma constitutes a false dichotomy, that is, a fallacy. Traditional usage distinguished the dilemma as a “horned syllogism” from the sophism that attracted the Latin name cornutus. The original use of the word horns in English has been attributed to Nicholas Udall in his 1548 book Paraphrases, translating from the Latin term cornuta interrogatio.
The dilemma is sometimes used as a rhetorical device. Its isolation as textbook material has been attributed to Hermogenes of Tarsus in his work On Invention. C. S. Peirce gave a definition of dilemmatic argument as any argument relying on excluded middle.
In propositional logic, dilemma is applied to a group of rules of inference, which are in themselves valid rather than fallacious. They each have three premises, and include the constructive dilemma and destructive dilemma. Such arguments can be refuted by showing that the disjunctive premise — the “horns of the dilemma” — does not in fact hold, because it presents a false dichotomy. You are asked to accept “A or B”, but counter by showing that is not all. Successfully undermining that premise is called “escaping through the horns of the dilemma”.
Dilemmatic reasoning has been attributed to Melissus of Samos, a Presocratic philosopher whose works survive in fragmentary form, making the origins of the technique in philosophy imponderable. It was established with Diodorus Cronus (died c. 284 BCE). The paradoxes of Zeno of Elea were reported by Aristotle in dilemma form, but that may have been to conform with what Plato said about Zeno’s style.
In cases where two moral principles appear to be inconsistent, an actor confronts a dilemma in terms of which principle to follow. This kind of moral case study is attributed to Cicero, in book III of his De Officiis. In the Christian tradition of casuistry, an approach to abstract ranking of principles introduced by Bartolomé de Medina in the 16th century became tainted with the accusation of laxism, as did casuistry itself. Another approach, with legal roots, is to lay emphasis on particular features present in a given case: in other words, the exact framing of the dilemma.
2. What is conditional proof method? Write an essay on the significance and the advantage of
conditional proof method.
Ans. A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent.
The assumed antecedent of a conditional proof is called the conditional proof assumption (CPA). Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion necessarily follows. The validity of a conditional proof does not require that the CPA be true, only that if it were true it would lead to the consequent.
Conditional proofs are of great importance in mathematics. Conditional proofs exist linking several otherwise unproven conjectures, so that a proof of one conjecture may immediately imply the validity of several others. It can be much easier to show a proposition’s truth to follow from another proposition than to prove it independently.
A famous network of conditional proofs is the NP-complete class of complexity theory. There is a large number of interesting tasks, and while it is not known if a polynomial-time solution exists for any of them, it is known that if such a solution exists for some of them, one exists for all of them. Similarly, the Riemann hypothesis has many consequences already proven.
Sometimes you find yourself needing to derive a conditional. When this is the case, it is convenient to use a technique called a “conditional proof” (CP). CP allows you derive a conditional (hence the name) that you need in a proof, either as the conclusion or as an intermediate step. This technique allows one to assume a proposition, then derive something from it (and any other available propositions). The assumed proposition and the resulting proposition are then linked together in a conditional statement. It is usually possible to derive the resulting conditional could have been arrived at in some other way, but a CP is often more convenient.
The theory is simple. If premise (P) is assumed and from premise (P) together with given premises and the application of inference and equivalence rules, another proposition (Q) can be shown to be derivable from that Assumed Premise (P), then the conditional ‘if P then Q’ is demonstrated. The conditional proof must be bracketed from the assumed premise to the conclusion with the last line outside the bracket always a material implication. In a conditional proof only the final line beyond the conditional proof is proven. The final line must have the horse shoe as the dominant operator.
b) Write an essay on the square of opposition.
Ans. Conflicting statements and arguments are the objects of research of logic as a science, and special mechanisms exist that are designed to determine connections among the individual components of various hypotheses and their veracity. One of such tools is the square of opposition that is a common technique for analyzing judgments and interpreting them. As the key components of this scheme, particular and general arguments are applied, and this method allows correlating basic data and drawing the right conclusions.
In contrast with formal logic, categorical one implies dividing judgments and hypotheses into separate segments, or categories. According to Jacquette, one of the examples of such interrelations is the square of opposition, a mechanism that is believed to have been developed by Aristotle. Its purpose is to demonstrate the scheme of the mutual relation of four-type claims. Jacquette notes the main features of this principle of statement interaction and argues that the edges of the square allow identifying the relationship between simple judgments, including both compatibility and contradiction. The sides and diagonals symbolize those logical connections that arise between the parts of assumptions. As a result, when placing the signs of quality and the number of judgments on the vertices of the square, one can note the principles of hypotheses intersection and their relationship.
The Components of the Square of Opposition
The vertices of the square indicate the type of judgment by the combined classification of A, E, O, and I. Murinová and Novák decode these components and give their meaning in the context of categorical logic. The upper corner on the left is marked with the letter A – a sign of universally affirmative statements. The upper corner on the right is marked with the letter E – a sign of universally negative statements.
The lower corner on the left is marked with the letter I – a sign of particular affirmatives, and the lower corner on the right is marked with the letter O – a sign of particular negative judgments. When placing the signs of quality and the number of statements on the vertices of the square, one can note that the sides of the square AI and EO represent the relations of submission.
c) Write a note on Boolean Algebra
Ans. Boolean algebra is a branch of mathematics that deals with operations on logical values with binary variables.
The Boolean variables are represented as binary numbers to represent truths: 1 = true and 0 = false.
Elementary algebra deals with numerical operations whereas Boolean algebra deals with logical operations.
Boolean algebra is different from elementary algebra as the latter deals with numerical operations and the former deals with logical operations. Elementary algebra is expressed using basic mathematical functions, such as addition, subtraction, multiplication, and division, whereas Boolean algebra deals with conjunction, disjunction, and negation.
The concept of Boolean algebra was first introduced by George Boole in his book “The Mathematical Analysis of Logic,” and further expanded upon in his book “An Investigation of the Laws of Thought.”
Since its concept has been detailed, Boolean algebra’s primary use has been in computer programming languages. Its mathematical purposes are used in set theory and statistics.
Boolean Algebra in Finance
Boolean algebra has applications in finance through mathematical modeling of market activities. For example, research into the pricing of stock options can be aided by the use of a binary tree to represent the range of possible outcomes in the underlying security. In this binomial options pricing model, where there are only two possible outcomes, the Boolean variable represents an increase or a decrease in the price of the security.
This type of modeling is necessary because, in American options, which can be exercised at any time, the path of a security’s price is just as important as its final price. The binomial options pricing model requires the path of a security’s price to be broken into a series of discrete time ranges.
a) Write a note on the laws of thought.
Ans. laws of thought, traditionally, the three fundamental laws of logic: (1) the law of contradiction, (2) the law of excluded middle (or third), and (3) the principle of identity.
Aristotle cited the laws of contradiction and of excluded middle as examples of axioms. He partly exempted future contingents, or statements about unsure future events, from the law of excluded middle, holding that it is not (now) either true or false that there will be a naval battle tomorrow but that the complex proposition that either there will be a naval battle tomorrow or that there will not is (now) true. In the epochal Principia Mathematica (1910–13) of Alfred North Whitehead and Bertrand Russell, this law occurs as a theorem rather than as an axiom.
That the laws of thought are a sufficient foundation for the whole of logic, or that all other principles of logic are mere elaborations of them, was a doctrine common among traditional logicians. The law of excluded middle and certain related laws were rejected by the Dutch mathematician L.E.J. Brouwer, the originator of mathematical intuitionism, and his school, who did not admit their use in mathematical proofs in which all members of an infinite class are involved.
b) State the differences between monadic and dyadic models.
- Monadic, a relation or function having an arity of one in logic, mathematics, and computer science
- Monadic, an adjunction if and only if it is equivalent to the adjunction given by the Eilenberg–Moore algebras of its associated monad, in category theory
- Monadic, in computer programming, a feature, type, or function related to a monad (functional programming)
- Monadic or univalent, a chemical valence
Two other dyadic models exist: the common-fate model and the mutual feedback model. 3 In the common-fate model, dyad members do not influence each other and the interdependency in the outcome errors is due to an outside factor that influences both dyad members.
For example, in the illustrative example, correlation in depressive symptomology could be due to a traumatic event that affected both dyad members. The mutual feedback model assumes that an individual’s outcome affects his/her partner’s outcome and vice-versa. For example, depressive symptomology in one dyad member can have an effect in their partner’s depressive symptomology and this could be the cause of the interdependency in the outcome errors. Multilevel methods have not been developed to estimate the common-fate or mutual feedback models.
c) Write a note on the problems of induction.
Ans. According to Popper, the problem of induction as usually conceived is asking the wrong question: it is asking how to justify theories given they cannot be justified by induction. Popper argued that justification is not needed at all, and seeking justification “begs for an authoritarian answer”. Instead, Popper said, what should be done is to look to find and correct errors. Popper regarded theories that have survived criticism as better corroborated in proportion to the amount and stringency of the criticism, but, in sharp contrast to the inductivist theories of knowledge, emphatically as less likely to be true. Popper held that seeking for theories with a high probability of being true was a false goal that is in conflict with the search for knowledge. Science should seek for theories that are most probably false on the one hand (which is the same as saying that they are highly falsifiable and so there are many ways that they could turn out to be wrong), but still all actual attempts to falsify them have failed so far (that they are highly corroborated).
Wesley C. Salmon criticizes Popper on the grounds that predictions need to be made both for practical purposes and in order to test theories. That means Popperians need to make a selection from the number of unfalsified theories available to them, which is generally more than one. Popperians would wish to choose well-corroborated theories, in their sense of corroboration, but face a dilemma: either they are making the essentially inductive claim that a theory’s having survived criticism in the past means it will be a reliable predictor in the future; or Popperian corroboration is no indicator of predictive power at all, so there is no rational motivation for their preferred selection principle.
d) Write an essay on the fallacy of presumption.
Ans. Fallacies of presumption are arguments that depend on some assumption that is typically unstated and unsupported. Identifying the implicit assumption often exposes the fallacy.
A complex question, or plurium interrogationum is a question that has a presupposition—an implicit assumption assumed to be true—that is complex because it contains several subparts. A question is complex when it is asked in such a way as to presuppose the truth of some conclusion buried in the question. A complex question is a loaded question if affirming the implicit assumption is detrimental to the person answering the question.
This fallacy has the form of a question with a false, disputed, or circular (begging the question) presumption.
This fallacy is often illustrated by the question “Have you stopped beating your wife?” The question presupposes that you have beaten your wife prior to its asking, as well as presupposing that you have a wife. If you have no wife, or have never beaten your wife, then the question is loaded.
Since this example is a yes/no question, there are only the following two direct answers:
- “Yes, I have stopped beating my wife”, which entails “I was beating my wife.”
- “No, I haven’t stopped beating my wife”, which entails “I am still beating my wife.”
Thus, either direct answer entails that you have beaten your wife, which is, therefore, a presupposition of the question. So, a loaded question is one which you cannot answer directly without implying a falsehood or a statement that you deny. For this reason, the proper response to such a question is not to answer it directly, but to either refuse to answer or to identify the fallacy and reject the question.
5. Write short notes on any five in about 100 words.
Ans. Denotation is the objective meaning of a word. The term comes from the Latin word “denotationem,” meaning “indication.” The denotation of a word is its literal definition—its dictionary definition—and contains no emotion.
While denotation is a relatively straightforward concept, there are a number of things about it worth keeping in mind.
- Every word has a denotation. No matter the language or part of speech, every word has a dictionary definition.
- Denotation is objective. Connotative meaning can change, but the denotative meaning does not. You understand the connotation of a word depending on your background. For example, two people may ascribe different connotations to the word “mother” depending on their life experiences. The dictionary definition of a word will always be the same for everybody, no matter what.
- Multiple words can denote the same thing. Sometimes, similar words have the same dictionary definition. For example, the words “trash” and “garbage” denote the same discarded rubbish.
- Denotation isn’t always neutral. Connotation can add a positive or negative spin to a word’s denotation, but the dictionary definition of a word can be positive or negative on its own. For example, the definition of the word “smirk” is a smug and offensive smile, which is inherently negative.
Ans. A disjunction is a compound statement formed by joining two statements with the connector OR. The disjunction “p or q” is symbolized by pq. A disjunction is false if and only if both statements are false; otherwise it is true.
A disjunction may be a mere lack of connection between two things, or a large gulf. There’s often a huge disjunction between what people expect from computers and what they know about them, and the disjunction between a star’s public image and her actual character may be just as big. We may speak of the disjunction between science and morality, between doing and telling, or between knowing and explaining. In recent years, disjunction seem to have been losing out to a newer synonym, the noun disconnect.
Ans. Implication, in logic, a relationship between two propositions in which the second is a logical consequence of the first. In most systems of formal logic, a broader relationship called material implication is employed, which is read “If A, then B,” and is denoted by A ⊃ B or A → B.
The truth or falsity of the compound proposition A ⊃ B depends not on any relationship between the meanings of the propositions but only on the truth-values of A and B; A ⊃ B is false when A is true and B is false, and it is true in all other cases. Equivalently, A ⊃ B is often defined as ∼(A·∼B) or as ∼A∨B. This way of interpreting ⊃ leads to the so-called paradoxes of material implication: “grass is red ⊃ ice is cold” is a true proposition according to this definition of ⊃.
Ans. Contrary is the relationship between two propositions when they cannot both be true (although both may be false). Thus, we can make an immediate inference that if one is true, the other must be false.
The law holds for the A and E propositions of the Aristotelian square of opposition. For example, the A proposition ‘every man is honest’ and the E proposition ‘no man is honest’ cannot both be true at the same time, since no one can be honest and not honest at the same time. But both can be false, if some men are honest, and some men are not. For if some men are honest, the proposition ‘no man is honest’ is false. And if some men are not honest, the proposition ‘every man is honest’ is false also.
Note that the A and E propositions are contraries only if the A proposition, ‘every man is honest’, is understood to mean ‘there is at least one man and every man is honest’, unlike the standard interpretation of the formula of modern logic with the universal quantifier, Modern presentations of the square of opposition and variants usually make this explicit.
e) Modus Tollens
Ans. Modus tollens takes the form of “If P, then Q. Not Q. Therefore, not P.” It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.
The history of the inference rule modus tollens goes back to antiquity. The first to explicitly describe the argument form modus tollens was Theophrastus.
Modus tollens is closely related to modus ponens. There are two similar, but invalid, forms of argument: affirming the consequent and denying the antecedent.
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