Assignment Code: MPYE-001/TMA/2022-23

Course Code: MPYE-001

Assignment Name: Logic

Year: 2022-2023

Verification Status: Verified by Professor

Note:

I) Give answer of all five questions.

ii) All five questions carry equal marks.

iii) The answer of questions no. 1 and 2 should be in about 500 words.

iv) If any question has more than one part, attempt all parts.

1. What is dilemma? How can we avoid dilemma? 20

Ans) A dilemma is a situation that presents two options, neither of which is clearly preferable nor acceptable. The options are referred to as the horns of the dilemma, a clichéd expression but one that distinguishes the situation from other types of predicaments. Gabriel Nuchelmans claims that Lorenzo Valla first used the word "dilemma" in the 15th century, in a later edition of the logic book that is now known as Dialectica. According to Valla, it is the proper Latin translation of the Greek dilemmaton. According to Nuchelmans, George of Trebizond's logic text from around 1433 was his most likely source. Additionally, he came to the conclusion that Valla had revived an argument style that had been outmoded in the Latin West.

Valla's neologism did not immediately catch on because Cicero's use of the well-established Latin term complexion, with conversion being used to refer to the destabilisation of dilemmatic reasoning, was preferred. But by the end of the 16th century, dilemma had gained popularity thanks to Juan Luis Vives. If the dilemma is applied incorrectly and takes the form "you must accept either A, or B" (where A and B are propositions that each lead to a different conclusion), it is a false dichotomy, or fallacy. The sophism that gave rise to the Latin name cornutus was distinguished from the conundrum by traditional usage as a "horned syllogism." Nicholas Udall is credited with coining the term "horns" in his book Paraphrases in 1548, translating it from the Latin cornuta interrogation.

There are times when the dilemma is employed rhetorically. Hermogenes of Tarsus is credited with its isolation as textbook content in his book On Invention. Any argument that relies on an excluded middle is a dilemmatic argument, according to C. S. Peirce. A group of inference rules that are inherently valid rather than erroneous are subject to the dilemma in propositional logic. Each of them has the constructive and destructive dilemmas as well as three premises. These arguments can be disproved by demonstrating that the disjunctive premise, or the "horns of the dilemma," is false because it creates an artificial dichotomy. They ask you to choose between "A or B," but you refuse by demonstrating that's not all. "Escape through the horns of the dilemma" is the phrase used to describe effectively undermining that premise.

Dilemmatic reasoning is credited to Melissus of Samos, a Presocratic philosopher whose writings are only preserved in fragments, making it difficult to determine where the method first appeared in philosophy. It was founded with the help of Diodorus Cronus (died c. 284 BCE). Aristotle reported Zeno of Elea's paradoxes as dilemmas, but this may have been done to match what Plato said about Zeno's writing style. An actor is faced with a quandary regarding which moral principle to uphold when two seem to be in conflict. Cicero is credited with writing book III of his De Officials, which contains this kind of moral case study. An approach to the abstract ranking of principles that Bartolomé de Medina introduced in the 16th century was tarnished with the charge of laxism in the Christian tradition of casuistry, as was casuistry itself. A different strategy with legal roots is to highlight specific characteristics of a case, or the precise framing of the dilemma.

2. What is conditional proof method? Write an essay on the significance and the advantage of conditional proof method. 20

Ans) A conditional proof is one that asserts a conditional and demonstrates that the conditional's antecedent inevitably results in the consequent. The conditional proof assumption refers to the presumptive antecedent of a conditional proof (CPA). Therefore, the purpose of a conditional proof is to show that the desired conclusion would logically follow if the CPA were true. The only requirement for a conditional proof to be valid is that if it were true, the consequent would follow. In mathematics, conditional proofs are extremely important. There are conditional proofs connecting a number of otherwise unproven conjectures, making it possible for the validity of one conjecture to be inferred from the proof of several others. Showing a proposition's truth to follow from another proposition is sometimes much simpler than having to independently demonstrate it.

The NP-complete class of complexity theory is a well-known network of conditional proofs. There are many intriguing problems, and while it is unknown whether any of them have polynomial-time solutions, it is known that if some of them do, then all of them must have such solutions. The Riemann hypothesis has numerous consequences that have also been demonstrated. You may occasionally need to derive a conditional. When this is the case, using a method known as a "conditional proof" is practical (CP). A conditional that you need in a proof, either as the conclusion or as an intermediate step, can be derived using CP (hence the name). This method enables one to make an assumption and then extrapolate from it (and any other available propositions). A conditional statement is then used to connect the resultant proposition and the assumption. A CP is frequently more convenient, though it is typically possible to derive the resulting conditional in another way.

The idea is straightforward. The conditional "if P then Q" is proven if premise (P) is assumed, and another proposition (Q) can be shown to be derived from that Assumed Premise (P) using premise (P), given premises, and the application of inference and equivalence rules. The last line outside the bracket must always be a material implication, and the conditional proof must be bracketed from the assumed premise to the conclusion. Only the last line after the conditional proof is proven in a conditional proof. The horseshoe must be the dominant operator on the last line.

Any deductive claim can be expressed as a conditional proposition, regardless of whether it is true or false. What is more crucial to understand is that the original argument only holds water when the corresponding conditional statement satisfies the "tautology" condition. The conclusion in the Conditional Proof method is dependent upon the conclusion's antecedent. A different approach is known as the Strengthened Rule of Conditional Proof. In this approach, the antecedent of the conclusion is not always assumed when constructing the proof. This method's structure requires some clarification. At first, a presumption is made. It is not necessary to know whether an assumption is true or false because even if it is, the conclusion may still be correct. Furthermore, any part of any premise or conclusion may contain an assumption.

3. Answer any two questions in about 250 words each. (Word limit is only for theory related questions) 2*10= 20

b) Write an essay on the square of opposition. 10

Ans) This is an example of an immediate inference because it only draws its conclusion from one premise. Another word for a direct inference is education. When two propositions share the same subject and predicate but differ in quantity, quality, or both, they are said to "stand against" one another logically in terms of truth-value. Because these relations are represented by a square, traditional logic referred to this relation as the square of opposition. The Aristotelian system discusses four such relationships.

Contradiction

Contradiction is a relationship between two propositions that differ in both "quantity" and "quality," as in "All men are wise" (A) - "Some men are not wise" (O). Because they cannot both be true and false at the same time, it is the most complete form of logical opposition. If one is true, the opposite must also be true, and vice versa. This kind of internal inconsistency results from the conflicting nature of the statements. Similar contradictions exist between the statements "No men are wise" (E) and "Some men are wise" (I).

Contrariety

Contrary refers to the opposition between two universal claims that only differ in "quality," as in "All men are wise" (A) vs. "No men are wise" (E). According to definition, both contraries can be false at the same time, as in the case of the example given, but they cannot both be true. When one of them is false, the other could still be true or false, but if one of them is true, the other must necessarily be false.

Subcontrariety

Subcontrariety is the term used to describe an opposition between two specific propositions that only differ in "quality." "Some men are wise" (I) - "Some men are not wise," for instance (O). However, they cannot both be false at the same time. Subcontrary propositions can be true together, as in the example given. The other may be true or false if one of them is true, but if one of them is false, the other must necessarily be true. It is clear that "contrary" and "subcontrary" propositions should be read in reverse order.

c) Write a note on Boolean Algebra 10

Ans) In order to solve problems in mathematical logic or symbolic logic, George Boole urbanised Boolean algebra in 1847. The properties of bitable electrical circuits can be represented by Claude Shannon's "Switching Algebra," a two-valued Boolean algebra that he introduced in 1938. In order to design a switching network in 1939, Claude Shannon first used Boolean algebra. In Boolean algebra, values are used to represent the numbers "0" and "1." They are not quantifiable in numbers. In Boolean equations, the binary values "1" and "0" stand for high and low levels, respectively. Boolean algebra is distinct from elementary algebra in that the latter deals with logical operations while the former does not. Basic mathematical operations like addition, subtraction, multiplication, and division are used to express elementary algebra, whereas Boolean algebra deals with conjunction, disjunction, and negation.

In his books "The Mathematical Analysis of Logic" and "An Investigation of the Laws of Thought," George Boole first introduced and developed the idea of Boolean algebra. Boolean algebra has primarily been used in computer programming languages since its concept has been explained. Statistics and set theory both make use of it for mathematical purposes. By mathematically simulating market activity, Boolean algebra has applications in finance. A binary tree can be used, for instance, to represent the range of potential outcomes in the underlying security, which can help with research into the pricing of stock options. The Boolean variable in this binary options pricing model denotes an increase or a decrease in the security's price, where there are only two possible outcomes.

4. Answer any four questions in about 150 words each. (Word limit is only for theory related questions) 4*5= 20

a) Write a note on Complex destructive dilemma. 5

Ans) The three basic laws of logic—the law of contradiction, the law of the excluded middle (or third), and the principle of identity—have traditionally been referred to as the laws of thought. The laws of contradiction and excluded middle were two examples of axioms given by Aristotle. The complex proposition that either there will be a naval battle tomorrow or there won't is (now) true, he claimed, but not the simple proposition that there will be a naval battle tomorrow. He partially exempted future contingents, or statements about uncertain future events, from the law of excluded middle. This law appears as a theorem rather than an axiom in Alfred North Whitehead and Bertrand Russell's seminal Principia Mathematica (1910–13). It was a common belief among traditional logicians that the laws of thought serve as a sufficient basis for all of logic or that all other principles of logic are merely extensions of them.

b) State the differences between monadic and dyadic models. 5

Ans) The differences between monadic and dyadic models includes:

1. Monadic, a relation or function having an arity of one in logic, mathematics, and computer science

2. 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

3. Monadic, in computer programming, a feature, type, or function related to a monad (functional programming)

4. Monadic or univalent, a chemical valence

The common-fate model and the mutual feedback model are two additional dyadic models. The interdependency in the outcome errors is caused by an external factor that affects both dyad members in the common-fate model because dyad members do not influence one another. For instance, in the illustrative example, a traumatic event that affected both dyad members may be the cause of the correlation in depressive symptomology. According to the mutual feedback model, each person's performance has an impact on that of their partner, and vice versa. One dyad member's depressive symptoms, for instance, may influence their partner's depressive symptoms, which may be the root of the interdependency in the outcome errors. The mutual feedback or common-fate models have not been estimated by multilevel methods.

c) Write a note on the problems of induction. 5

Ans) In Popper's view, the problem of induction as it is typically understood asks how to justify theories given that they cannot be justified by induction. According to Popper, seeking justification "begs for an authoritarian answer" and is not even necessary. Popper argued that the proper course of action is to search for and correct errors. In sharp contrast to the inductivist theories of knowledge, Popper thought that theories that had withstood criticism were emphatically less likely to be true, even though they had received more and harsher criticism. According to Popper, seeking out theories with a high likelihood of being correct is a misguided objective at odds with the pursuit of knowledge. Science should look for hypotheses that are highly likely to be false on the one hand (i.e., highly falsifiable, meaning there are numerous ways they could be wrong), but which have so far resisted all attempts at actual falsification (that they are highly corroborated).

d) Write an essay on the fallacy of presumption. 5

Ans) These fallacies rely on the truth of a few unproven hypotheses. Often, such assumption goes unnoticed. Therefore, to disprove such a fallacy, it usually suffices to draw attention to the smuggled assumption and to its uncertainty or falsity. Of this type, there are three prevalent fallacies. P1. An accident occurs because the definitions of the terms used are unclear. There are two types: I The direct or simple fallacy of accident involves asserting that something's true in normal circumstances means that it will also be true in unusual circumstances. Take the statement, "Freedom is the birth right of man; no one should be imprisoned," as an example. While normally true, a man who has committed a serious crime is an exception. More instructive is another illustration. When someone jumps into the water to save someone from drowning, they should be fined for disobeying a "No Swimming" sign.

5. Write short notes on any five in about 100 words. 5*4= 20

a) Denotation 4

Ans) The number of people to whom a term is applied or extended is referred to as the term's denotation. For instance, the term "society" can refer to a political society (or a State), the Society of Jesus, a philanthropic society, the human society, etc. Extension is a synonym for "denotation." The full meaning of a term, as expressed by the sum of all of its essential characteristics as opposed to accidental ones, is what we refer to as the connotation or intention of a term. Consider the term "society," which denotes (a) a group of people who are (b) linked by a shared interest. Crowd lacks these qualities, so it does not have the same meaning as society.

b) Disjunction 4

Ans) When two statements are connected by the connector OR, a disjunction is created. P or q stands for the disjunction "p or q." If and only if both statements are untrue, a disjunction is false; otherwise, it is true. A disjunction can be as small as a gap or as wide as a complete lack of connection between two things. People frequently have very different expectations for computers compared to what they actually know about them. A similar gap may exist between a celebrity's outward appearance and her true nature. We can discuss the conflict between knowledge and explanation, between doing and telling, or between science and morality.

c) Implication 4

Ans) A compound statement of the form "if the train is late, then we will miss the connection" is a conditional statement. Disjunctions also include conditional statements in the dictionary's strict sense. We will, however, limit ourselves to the aforementioned form for the time being. Inference and hypothetical proposition are other terms for conditional statements. Here, they will both be used equally. The part that comes before "then" is known as the antecedent (implicant, or infrequently protasis), and the part that comes after "then" is known as the consequent (implicate, or infrequently apodosis). In an implication, it is said that the antecedent implies the consequent or that the antecedent implies the consequent.

d) Contrary 4

Ans) When two statements can't both be true, their relationship is contrary (although both may be false). As a result, we can conclude right away that if one is true, the other must be false. The Aristotelian square of opposition's A and E propositions are subject to the law. Since no one can be both honest and dishonest at the same time, the A proposition that "every man is honest" and the E proposition that "no man is honest" cannot both be true at the same time. Both, however, may be untrue if some men are sincere while others are not. Because if some men are sincere, the statement "No man is honest" is untrue. Additionally, if some men are dishonest, the statement "Every man is honest" is false.

e) Modus Tollens 4

Ans) If P, then Q is the form of the modus tollens. Not Q. Consequently, not P. It is a practical application of the axiom that if a statement is true, then its contrapositive is also true. The form demonstrates the validity of the inference from P implies Q to the negation of Q implies the negation of P. The inference rule modus tollens has a long history that dates back to antiquity. Theophrastus was the first to specifically define the argument form modus tollens. Modus ponens and modus tollens are related concepts. Arguments that affirm the consequent and deny the antecedent are both similar but fallacious.