# What is Logic: Definition and 1000 Discussions

Logic (from Greek: λογική, logikḗ, 'possessed of reason, intellectual, dialectical, argumentative') is the systematic study of valid rules of inference, i.e. the relations that lead to the acceptance of one proposition (the conclusion) on the basis of a set of other propositions (premises). More broadly, logic is the analysis and appraisal of arguments.There is no universal agreement as to the exact definition or boundaries of logic (see § Rival conceptions). However, the scope of logic (broadly construed) includes:

The classification of arguments.
The systematic analysis of logical forms.
The systematic study of the validity of deductive inferences.
The strength of inductive inferences.
The study of faulty arguments, such as fallacies.
The study of syntax and semantics of formal languages.
The study of the concepts of meaning, denotation and truth.Historically, logic has been studied mainly in philosophy (since Antiquity), mathematics (since mid-19th century), and computer science (since mid-20th century). More recently, logic has also been studied in linguistics and in cognitive science. Overall, logic remains a strongly interdisciplinary area of study.

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1. ### B Implication vs Equivalence

I've seen a lot of people use implication and equivalence logic incorrectly. For example, when solving equations (i.e. ##x - 2 = 3 \implies x = 5##). Implication is not reversible, thus it only works in one way. By saying, ##x - 2 = 3 \implies x = 5##, you are essentially saying that it is...
2. ### I How to find this equivalent of the material conditional?

Hi there, It's well known that for two assertions A and B : A → B is equivalent to (nonA or B). The only proof I know of this equivalence relies on the truth table, one just brute forces all the possible combinations of truth values and then notice they're the same every time with A → B and...
3. ### How to motivate ‘unless’ = ‘if not’, with etymology?

My students still reckon Lande’s 2 motivations for ‘unless’ = ‘if not’ (below) too abstract, formalistic! Thus I need a 3rd simpler, motivation with merely etymology. But I never studied linguistics! How do I 1. use these etymology quotations, to motivate ‘unless’ = ‘if not’? 2. teach why...
4. ### I Contrapositive of theorem and issue with proof

Consider the following theorem: First, I don't know how to take the contrapositive of this statement. I'm not sure if the opening hypothesis, that is, ##f## is continuous, remains outside. Because the way this is proved in the text I'm reading is by assuming ##f(X)## is disconnected and then...
5. ### Simplifying Boolean expressions

This is the first time I am doing logic so the question is probably stupid, but could I just factorise (p∨t)∧[(r v¬r] or perhaps you cannot do that in Boolean algebra?
6. ### I Logical representation of prime numbers

One of the discussion questions for my class this week was to express the condition "p is a prime number" using formal logic. The answer my uni's got is (∀d∈N)[(d>=1∧p≠1)∧(d|p⇒((d=1)∨(d=p))]. My interpretation of this is all the natural number d that (satisfy d>=1 and prime number is not equal...
7. ### Figuring out Natural Deduction problem but can't find mistake

I'm in undergrad and I unexpectedly began taking a discrete math class, everything was sunshine and rainbows until this chapter.... All in all, Below is the Natural deduction problem with its premise and intended conclusion and these are my steps. I can't see where I'm going wrong, any ideas...
8. ### What is the best way to learn logic of computer language?

Hi I want to give an introduction of programming to my grand daughter who's going to college next year. She never learn any language. I want to play with Audino Kit with her. It's mostly C++. But it's too difficult for her to learn as first language and she doesn't have time as she's still very...
9. ### B How Can I Calculate Negative Multiplication Without Following Traditional Rules?

Say I have 6 pencils. I want to times this by negative two. Now ignoring the rules that your teacher taught you work this out. 6 pencils negative 2 times. Negative one time would be 0 and another negative times would be -6 right? So 6 x - 2 = -6 according to simple logic. The calculator will say...
10. ### I Understanding Absorption Laws (Boolean Algebras)

I can't understand how absorption law is obtained. I get following steps.##a∨(a∧𝑏) = (a∧⊤)∨(a∧𝑏)## ##=(a∨a)∧(a∨b)∧(⊤∨a)∧(⊤∨b)## then, I come up with ##=a∧(a∨b)∧⊤∧⊤## so ##=a∧(a∨b)## But, I cannot get ##a∧(⊤∨𝑏)##, as shown on here, therefore ##a##. Can you help me? I cannot obtain ##a∧(⊤∨𝑏)##...
11. ### A What are your insights on The Hardest Logic Puzzle Ever?

Hello! I’m an assistant of a mathematical scientific researcher, and my research programme evolves around finding and developing all the (possible) solutions regarding all unsolved mathematical, logic, exact, and IQ puzzles ever created. If you search on the internet for: “The hardest unsolved...
12. ### Studying Programming is coming to me but I forget how I solved the problem...

I'm surprised that automatically programming is coming to me. It blows my mind. I could not think of any logic in the past months and now I am able to solve puzzles in programming. But the toughest part is I forget how I solved the problem too quickly. For example, the blackjack program. I...
13. ### B I have an issue with Cantor's diagonal argument

I'm pretty bad at maths, got an A at gcse (uk 16 years old)then never went any further, I've been looking into cantors diagonal argument and I thing I found an issue, given how long its been around I'd imagine I'm not the first but couldn't any real number made using the construct by adding 1 to...
14. ### Logical Proof: Theorem (Truths of Logic) A iff ~~A

My thought was to break up the sentence into its equivalent form: (A ->~~A) & (~~A -> A) From there I assumed the premise of both sides to use indirect proofs, so: 1. ~(A -> ~~A) AP 2. ~(~A or ~~A) 1 Implication 3. ~~A & ~~~A 2...
15. ### I [(Linear) Temporal Logic] Are the given sentences consistent or not?

Consider the following set of sentences that represent the requirements of a multi-threaded system for two threads t1 and t2: □¬[(t1 active) ∧ (t2 active)]. □[(t1 active) ⊕ (t2 active)]. □[(t1 active) → O(t2 active)]. □[(t2 active) → O(t1 active)]. (Pretend the O letters are the circles...

17. ### [Mathematical logic] prenex normal form and skolem normal form

The attached picture below is the note I solved halfway through. Please tell me the entire process of getting to the correct answer.
18. ### Comp Sci Digitial Logic and Binary Counter

1 000 0 1 000 0 0 001 1 0 001 1 1 0 1 0 0 1 On the left is my input, in the middle is my output and on the right is the decimal. 0 000 0 0 000 0 1 001 1 0 001 1 1 010 2 Here is another example I was studying where the input is 00101 and I...
19. ### Comp Sci Digital Logic 8 Bit Full Adder Circuit

I am unsure of how to approach this problem, we begin with two 8-bit full adders, is the goal to just combine them in some way so that it becomes one entire circuit while also adding NAND and NOR gates where necessary? By the end will we then have 16 1-bit full adders?

21. ### Prove the 2nd axiom of mathematical logic using the Deduction Theorem

prove: The 2nd axiom of mathematical logic 2) $((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))$ By using only the deduction theorem
22. ### Use mathematical logic to prove this proposition

Given the following axioms: 1) ##P\implies(Q\implies P)## 2) ##((P\implies(Q\implies R))\implies((P\implies Q)\implies(P\implies R))## Where ##P,Q,R## are any formulas 3)##(\neg P\implies\neg Q)\implies (Q\implies P)## then prove: ##\{A\implies B,B\implies C\}|- A\implies C## Without using the...
23. ### B Vacuously true statements and why false implies truth

We say that an implication p --> q is vaccuously true if p is false. Since now it's impossible to have p true and q false. That is we can't check anymore whether the contrary, p being true and q being false,can be.Since p being true is non-existent. So we take the implication as true. For eg...
24. ### Interpreting logic levels for level translator SN54SLC8T245-SEP

Hello all, I plan on using SN54SLC8T245-SEP for level shifting of single-ended signals from 1.5 V to 1.8 V, 1.5V to 3.3 V, 1.8 V to 1.5V and 3.3V to 1.5V. First time using level translators for mostly SPI interfaces, so I would like some guidance. I am having difficulties interpreting the logic...
25. ### I Logical implication vs physical causality

There is something I don't understand that I want to ask quantum physics experts here: Suppose the happening of event X results logically speaking in the happening of event A. So we could for instance have the following logical implication ##X.happens \rightarrow A.happens##. If this is...
26. ### I Questions about SMT logic solvers

Are smt-solvers (like z3) theoretically able to (always correctly) check consistency of any 1.-order logic formula? Does it follow from The undecidability of first order logic, that an algorithm, that could check consistency of any 1.-order logic formula, does not exist? How does the algorithm...
27. ### Choice, Chance and Sensible Logic

Three non-concurrent and non-parallel lines divide an infinite plane in 7 parts. and four non-concurrent and non-parallel lines divide an infinite plane into 10 parts. Is this statement correct? If correct, we can proceed further to answer this question. If incorrect, suggest me the...
28. ### MHB Debugging Algorithm Logic: 2 Iterations & Finding Right/Wrong Answers

https://lh3.googleusercontent.com/J-M_R31KFl76aXCRE4sxzAWI6iAldXG9fMI2k4YRMZo6vZY0w1DPUfV7tLc1w5IsVQX8uUkjgThzROZrbb3bc6kluAEmyiH63Z4BsXKT5Xr7X7mLxOwbR1L0y3ttoLitS6bOnwz- Here is a solution, part of which doesn’t make any sense to me-: ->Why did we select c3,c1 and c3,c2 for 2nd generation...
29. ### Comp Sci How to learn resolution in predicate logic?

This is not really a homework question so don't bother answering them. It is more of a guidance problem. This is what I find the hardest out of all topics.. Unfortunately, this topic is a fixed 10 marks question in our 80 marks exam. Comes every time. The types of questions that I need to deal...
30. ### MHB How to learn resolution in predicate logic?

This is not really a homework question so don't bother answering them. It is more of a guidance problem. This is what I find the hardest out of all topics.. Unfortunately, this topic is a fixed 10 marks question in our 80 marks exam. Comes every time. The types of questions that I need to deal...
31. ### Comp Sci Mistake while removing implications in first order predicate logic-:

As you can see I am not getting correct result. What have I messed up? I want to learn it. https://slideplayer.com/slide/4942120/ Here is full slide in case anyone wants to refer to it.
32. ### Comp Sci What I am supposed to write in this question about fuzzy logic?

There are 2 questions commonly asked(rarely though) in my exams. 1) Explain about mamdani fuzzy inference method with example. 2) explain fuzzy inference with suitable example. Now I am self studying. I have 4 books with me, but none of them have this content. IDK why tho..There are other...
33. ### MHB What I am supposed to write in this question about fuzzy logic?

There are 2 questions commonly asked(rarely though) in my exams. 1) Explain about mamdani fuzzy inference method with example. 2) explain fuzzy inference with suitable example. Now I am self studying. I have 4 books with me, but none of them have this content. IDK why tho..There are other...
34. ### Engineering Simulating this cascoded NMOS logic circuit

I need help for following exercise from Rabaey - Digital Integrated Circuits: A Design Perspective Size and simulate the circuit (IMAGE BELOW) so that it achieves a 100 ps delay (50-50) using 0.25 μm devices, while driving a 100 fF load on both differential outputs. (VDD = 2.5V) Assume A, B and...
35. ### B Traditional logic and its usefulness in the past

Hi, I was reading this article, https://writings.stephenwolfram.com/2015/11/george-boole-a-200-year-view/, and the following excerpt is taken from it. George Boole came up with his Boolean logic around 1847. I think that it was Claude Shannon who showed that any numerical operation can be...
36. ### Python What does this code signify in relation to boolean logic?

my_age = 10 if my_age >= 100: print("One hundred years old! Very impressive.") elif my_age <= 3: print("Awwww. Just a baby.") else: print("Ah - a very fine age indeed") https://www.fullstackpython.com/blog/python-basic-data-types-booleans.html Article says-:
37. ### Python-What does this code signifies in relation to boolean logic?

my_age = 10 if my_age >= 100: print("One hundred years old! Very impressive.") elif my_age <= 3: print("Awwww. Just a baby.") else: print("Ah - a very fine age indeed") https://www.fullstackpython.com/blog/python-basic-data-types-booleans.html Article says-:
38. ### MHB What logic is used to build context free grammar for L= a^m b^n; m>n

I have the solution but I am unable to decode the solution, how did we come up here? can you tell me that?
39. ### Comp Sci How to build logic to make context free grammar?

I already have the solution. I also have another solution for this-: S->AB A->aA/a B->ab/aBb But my issue is I am memorizing them, I am not understanding how this came up. Please explain the logic building behind this.
40. ### I [Logic] Order of quantifiers and brackets

I asked a question in yesterday's lecture about whether we can change the order of quantifiers and move it around in statements involving and, or, & iif. My instructor said no and after the lecture, my fellow student made this post below, intending to demonstrate that the instructor was right...
41. ### I Quantum logic spectroscopy in a quadrupole trap

Hello! I see that most experiments applying quantum computing techniques for high precision spectroscopy use a quadrupole trap. I don't know much about the experimental implementation of these kind of traps, so any help (or suggested paper) would be appreciated, but I was wondering how well can...
42. ### A Can Quantum Logic Techniques Be Applied to Complex Molecular Ions?

Hello! I read several papers about using quantum information techniques in order to do precision spectroscopy on ions. Usually they trap 2 ions in a quadrupole trap and the 2 ions are coupled through their vibrational motion along the trap axis. However most of these results involve atomic ions...
43. ### Interesting video on: Why are 2 logic gates faster than 1?

Just saw an interesting video on youtube explaining Logical Effort very nicely. https://en.wikipedia.org/wiki/Logical_effort
44. ### MHB What is the Tautology in the Given Logical Equivalence?

Hi guys I can't figure this one out. I tried to use truth tables, but never found an equivalence , no matter which of the 5 options I tried. It is given that $\alpha$ is logically equivalent to $\alpha \rightarrow \sim \beta$ . Which of the following is a tautology ? 1) $\alpha$ 2) $\beta$...
45. ### DP: proving existence of optimal substructure for "Sherlock and Cost"

I was attempting to solve the "Sherlock and Cost" problem from HackerRank using DP: But before I went to come up with a recursive relation, I wanted to find if the problem possesses an optimal substructure, and I was following these steps as written at CLRS book: Mentor note: Inline images of...
46. ### I have a few questions about Formalization & Pseudo-code

Hello, I have a few questions and I'd appreciate if you can please help me. 1. If I want to say "for every ## i \in \Bbb N ## and ## 0 \leq j \leq i ## define ## A_{i,j} := i ## and ## B_{i,j} := i \cdot j ## ", then is the logical formula used for this is as such?: ## \forall i \in \Bbb N...
47. ### B Translating Abraham Lincoln quote into symbolic logic

Can anybody check my work regarding these three statements, the third, in particular, please? I think I got the first two statements down, but I think that I'd feel safer if I got a second opinion. I think I also have a correct translation of the third statement down, but only because I...
48. ### A Search for CPT violation with protons/antiprotons using quantum logic

Just wanted to share a cool proposal paper from the BASE collaboration. I found this article dense and the theoretical aspects are way above my pay grade, so please chime in if you think I get anything wrong. For the record, I have no ties to this group and hadn't heard of them before this...
49. ### Showing continuous function has min or max using Cauchy limit def.

Problem: Let ## f: \Bbb R \to \Bbb R ## be continuous. It is known that ## \lim_{x \to \infty } f(x) = \lim_{x \to -\infty } f(x) = l \in R \cup \{ \pm \infty \} ##. Prove that ## f ## gets maximum or minimum on ## \Bbb R ##. Proof: First we'll regard the case ## l = \infty ## ( the case...
50. ### I Translate compound proposition p → q (implication) to p↓q question

I hope someone can help me or point me in the right direction. I am reading Discrete Mathematics with its Applications by Rosen. I am trying to self learn discrete math. I am actually able to do most questions but I have a question about a solution (not the question itself.) The question is...