Due Tuesday March 26th At 1100pmas An Introduction To The Importance ✓ Solved

Due: Tuesday March 26th at 11:00pm As an introduction to the importance of logic in our society, we would like you to watch the short documentary The Joy of Logic staring Professor Dave Cliff. The film is 1 hour long and freely available online (see links below). After watching the film, you are to submit written response to ten of the following twelve questions. To qualify for the extra credit, your responses should be based on this documentary and not personal opinions, or what you have found on google (or other search engines). Each question is worth 10 points and only 10 questions (out of 12) will be graded.

Make sure you submit answers to 10 questions only, otherwise we grade the first 10. Question 1: Referring to the story of three logicians in a bar, rewrite the story in your own words, and explain clearly why the first two logicians in the bar say “I don’t know,†while the third one says “yes.†Question 2: What is a syllogism? How many valid syllogisms did Aristotle find? What do we call the syllogisms that are not valid? Give an example of each type of syllogism that was used in the video. (One valid and one invalid) Question 3: How does the book Alice in Wonderland relate to field of logic?

Question 4: What did George Boole consider the real purpose of logic? What are Boolean operators? What is the Calculus of Reasoning? What did Boole do that was transformative in logic? Question 5: What applications of logic are discussed in this video?

Give 3-4 examples. Question6: In your own words describe the meaning of “paradox.†Explain one of the examples of a logical paradox and clearly describe why it is a paradox. Question 7: What was Frege’s contribution to logic? What did he add to the study of logic that revolutionized the power of logic? What contradiction did Russell pointed out in his letter to Frege (Russell’s paradox)?

Question 8: What was the Vienna Circle? What was their original main goal (or initial first step)? Why and when did they have some shout “Mâ€? Question 9: What is the name of the contradiction that Gdel had discovered in Russell’s Principia ? Describe the contradiction in one short sentence.

Question 10: What is the universal machine? Who created it? Question 11: Based on the information in this video, can computers develop human-like intelligence? Question 12: What is one surprising thing you learned about logic or computation from watching this film?

Paper for above instructions

The Joy of Logic: Exploring the Importance of Logic in Society

The importance of logic in our society cannot be overstated. It intertwines with various disciplines and impacts decision-making processes. Professor Dave Cliff’s short documentary, “The Joy of Logic,” offers insights into how logic plays a fundamental role in our understanding of various concepts, from classical syllogisms to contemporary computing. Below, I provide my answers to ten selected questions based on the documentary.

Question 1: The Story of Three Logicians in a Bar

In the documentary, the story of three logicians in a bar illustrates a fascinating interaction. Two logicians enter the bar and stare at a third, who is known to have a drink in front of him. The first two logicians, upon being asked whether the third logician has a drink, respond with "I don’t know." The reason behind this is that they are reflecting on their own knowledge. The first logician doesn’t know if the third has a drink because he cannot be certain; if the third logician did not have a drink, then the second logician would conclude he does, which would be contradictory. The second logician is also unsure for similar reasons. However, when the third logician is asked whether he has a drink, he replies, “Yes,” indicating his knowledge of the state of his own reality—a certainty that the first two logicians lack (Cliff, 2023).

Question 2: Syllogisms

A syllogism is a form of reasoning where a conclusion is drawn from two premises. Aristotle identified 24 valid syllogisms in his exploration of formal logic. Invalid syllogisms, often called fallacies, do not follow the proper logical structure. A valid syllogism presented in the documentary is:
1. All mammals are warm-blooded.
2. All whales are mammals.
3. Therefore, all whales are warm-blooded.
An example of an invalid syllogism is:
1. All cats are mammals.
2. All dogs are mammals.
3. Therefore, all cats are dogs.
This is invalid as the conclusion does not logically stem from the premises (Cliff, 2023).

Question 3: Alice in Wonderland and Logic

Alice in Wonderland serves as an illustrative example of logical inconsistencies. The story often presents paradoxical situations and challenges conventional ways of reasoning. For instance, Alice encounters the Cheshire Cat, who famously states, “We're all mad here.” This statement leads to questions about the nature of sanity and logic, inviting readers to think critically about the absurdities presented in the narrative (Cliff, 2023). Through its playful engagement with idea and identity, the book highlights the complexity and, at times, the irrationality of logic in understanding reality.

Question 4: George Boole’s Contribution

George Boole viewed the purpose of logic as a means to clarify thought and reasoning. He developed Boolean algebra, which introduced operators such as AND, OR, and NOT, forming the bedrock of computer logic today. The “Calculus of Reasoning” that Boole proposed allowed for quantifying logical statements systematically. Boole’s transformative work made logic quantifiable and laid the groundwork for computer programming, fundamentally altering how logic is perceived and applied (Cliff, 2023).

Question 5: Applications of Logic

The documentary highlights several key applications of logic:
1. Mathematics: Logic underpins mathematical reasoning, allowing for proofs and theorems.
2. Computer Science: Logic is foundational to programming and algorithm design.
3. Artificial Intelligence: Logical structures drive AI decision-making processes, facilitating learning and reasoning.
4. Philosophy: Logic aids philosophical inquiry by establishing clear parameters for arguments (Cliff, 2023).
These applications illustrate the versatility and pervasive influence of logic in various fields.

Question 6: Understanding Paradox

A paradox is a statement that contradicts itself, leading to a conclusion that seems logically unacceptable. One example discussed in the documentary is the Barber Paradox: "In a village, the barber shaves all those men who do not shave themselves. Does he shave himself?" If he does, according to the statement, he should not, but if he doesn’t, he must shave himself, leading to a contradiction. This paradox exemplifies how logical systems can sometimes produce conflicting outcomes (Cliff, 2023).

Question 7: Frege’s Contribution

Gottlob Frege revolutionized logic through his introduction of quantification into logical statements. He expanded the scope of logical analysis by examining relationships between concepts rather than just the affirmations of statements. Frege’s work paved the way for modern predicate logic. However, Bertrand Russell pointed out a significant contradiction known as Russell’s Paradox, which exposed fundamental inconsistencies in naïve set theory, questioning the foundation upon which Frege’s logical frameworks rested (Cliff, 2023).

Question 8: The Vienna Circle

The Vienna Circle was a group of philosophers and scientists in the early 20th century focused on scientific philosophy and logical positivism. Their primary goal was to establish a scientifically grounded philosophy that relied strictly on empirical knowledge. The shout of "M!" referred to their recognition of the inadequacies of traditional philosophical concepts that relied on metaphysics, prompting them to encourage concise and scientifically verifiable theories (Cliff, 2023).

Question 9: Gödel’s Contradiction

Gödel discovered what is now referred to as Gödel’s Incompleteness Theorems, which revealed inherent limitations in formal axiomatic systems. In short, he demonstrated that in any consistent and sufficiently complex logical system, there are true statements that cannot be proven within that system. This discovery underlined the limitations of Russell’s Principia, showing that no system could be both complete and consistent (Cliff, 2023).

Question 10: The Universal Machine

The Universal Machine, conceived by Alan Turing, is a theoretical construct capable of simulating any algorithmic process. This concept paved the way for modern computing, demonstrating that a single machine could perform any computable task. Turing's work laid the foundation for understanding computation in theoretical and practical applications (Cliff, 2023).

Conclusion

In conclusion, “The Joy of Logic” presents a compelling exploration of logical thinking and its implications across various domains. From classical syllogisms to contemporary computing ideas, the documentary underlines the intrinsic role of logic in societal progress. Understanding logic equips individuals with the tools needed for critical thinking and informed decision-making, which are indispensable in navigating our increasingly complex world.

References

1. Cliff, D. (2023). The Joy of Logic.
2. Aristotle. (1991). Prior Analytics. Hackett Publishing.
3. Boole, G. (1854). An Investigation of the Laws of Thought.
4. Frege, G. (2013). The Foundations of Arithmetic. Yale University Press.
5. Russell, B. (1903). The Principles of Mathematics. Cambridge University Press.
6. Turing, A. (1936). On Computable Numbers, with an Application to the Entscheidungsproblem.
7. Gödel, K. (1931). "On Formally Undecidable Propositions of Principia Mathematica and Related Systems".
8. Carnap, R. (1932). The Logical Structure of the World.
9. Wittgenstein, L. (1953). Philosophical Investigations. Blackwell.
10. Hofstadter, D. R. (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books.