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 logical paradoxes.
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. D

    B Empty domains and the vacuous truth

    So, here's my question. I read somewhere that all universal truths on empty domains are vacuously true, whereas all existential are false. However, if all statements of the form (∀x∈A)(P(x)) , where A is an empty set, are vacuously true, then the statement (∃x∈A)(P(x)) should also be true...
  2. Florence

    Question about propositional logic

    Homework Statement I have to prove that ##(p \equiv q) \equiv ((p ∧ q) ∨ (¬p ∧ ¬q))## With no premisses In order to prove this, I first need to prove that: ##(p \equiv q) \to ((p ∧ q) ∨ (¬p ∧ ¬q))## And: ##((p ∧ q) ∨ (¬p ∧ ¬q)) \to (p \equiv q)## I was able to find the second implication...
  3. L

    Is my logic right for the answer to this inelastic car collision question?

    Homework Statement A 1000 kg car approaches an intersection traveling north at 20.0 m/s . A 1200 kg approaches this same intersection traveling east at 22.0 m/s. The two car collides and lock together. Ignoring any external forces , what is the velocity of the car immediately after the...
  4. T

    Wired Logic Gates: Exploring the Limitations and Velocity Factors

    Hi, why a wired-logic gate does not produce a physical second-level gate ? Is it due to the velocity of the components that create the "wired-connection" ? Thanks
  5. D

    B What is the converse statement of the given sentence?

    The sentence is : "For all real numbers there exists a natural number that is smaller". That is (∀x∈R)(∃n∈N)n>x. This is what I thought of: we can write this sentence as:"If x is a real number, then there exists a natural number n that satisfies n>x." So how would I make a converse statement...
  6. R

    Logic applied to making isothermal assumption

    Homework Statement A thin electrical heater is inserted between a long circular rod and a concentric tube with inner and outer radii of 20 and 40 mm. The rod (A) has a thermal conductivity of kA = 0.15 W/(m*K), while the tube (B) has a thermal conductivity of kB = 1.5 W/(m*K) and its outer...
  7. T

    MHB Logic - Natural Deduction HELP

    Hello guys, I really need help from you about natural deduction. I just can't get to solve theses problems. Can anyone give me the solutions ? I've been trying for hours... Thanks a lot ! Premise 1 : B, Premise 2 : C→(¬B∨A) Premise 3: ¬C→¬(A∨B) Conclusion : A≡C ____ Premise 1: (A∨B)∨C...
  8. Math Amateur

    Top Undergraduate Mathematical Logic Texts: Recommendations from MHB Members

    What mathematical logic texts do MHB members think are the best at undergraduate level ... that is the best introductory texts ... Peter
  9. W

    Green's functions: Logic behind this step

    Homework Statement Hi all, I came across these steps in my notes, relating to a step whereby, $$\hat{G} (k, t - t') = \int_{-\infty}^{\infty} e^{-ik(x - x')}G(x-x' , t-t')dx$$ and performing the following operation on ##\hat{G}## gives the following expression, $$[\frac{\partial}{\partial t}...
  10. K

    B Fitch's Paradox: Knowability of All Truths

    https://en.wikipedia.org/wiki/Fitch's_paradox_of_knowability It begins by assuming that 'All truths are knowable' and then logically proves that that assumption means 'All truths are already known.' The proof is like this: Suppose p is a sentence that is an unknown truth; that is, the sentence...
  11. S

    MHB Converting statements and logic

    Hello! I'm just starting out on this logic train and I'm not sure I'm grasping it correctly. I took a statement and attempted to convert it into the symbols below. ∀x∈ℤ,∃y∈ℤ,(E(x)∧E(y)) → x = 2y The original phrase is: Any even integer is equal to twice some other event integer. Which I...
  12. C

    MHB Writing a statement into symbolic logic

    Dear Everyone, I need to translate this following statement into a symbolic logical form of the statement: The square of every odd integer is one more than an integral multiple of 4. Thanks, Cbarker1
  13. V

    Understanding Big Oh: Exploring the Best Function for a Given Equation

    Let me start by saying that this is from a 30 question assessment on Big Oh, Big Theta, and Big Omega. I understood every other question, however, even after being given the correct answer, I do not understand why my answer was wrong for this one. If you could point me in the direction of any...
  14. Y

    Understanding the Logic Behind KCL: An Exploration

    Homework Statement Ok so I remember learning KCL. Current is either entering or leaving the node. There's only two possible option. You can call entering the node positive or negative, you can call current exiting the node either positive or negative. As long as what you call entering the node...
  15. V

    Ht12e/d input output logic question

    Good day. I'm currently in the soldering phase of an RF circuit designed to operate an L293D motor chip. On the breadboard I managed to get it to work, but after soldering (and checking for shorts) I am not getting proper high/low signals sent through the encoder/decoder from 2 2p2t switches to...
  16. Nikhil N

    How to make a ladder logic program for Profinet

    I have a SIMATIC S7-1200 CPU and Anybus card. I need to make communication between these. Can anybody tell how to make the ladder logic program for this in tia portal?
  17. M

    I The truth value of ##P(x)## → ##Q(x)##

    I'm reading Velleman's book titled "How to Prove it" and I'm very confused when I'm reading about conditional statements. I understand that there exists some issue with the conditional connective and I accept that because that's the cost of espousing a truth-functional view. I came here to ask...
  18. M

    Propositional function problems

    1. Suppose P(x) and Q(x) are propositional functions and D is their domain. Let A = {x ∈ D: P(x) is true}, B = {x ∈ D: Q(x) is true} (a) Give an example for a domain D and functions P(x) and Q(x) such that A∩B = {} (b) Give an example for a domain D and functions P(x) and Q(x) such that A ⊆ B...
  19. M

    Need help with a mod-6 counter using JK flip flops w/ control bit

    Homework Statement [/B] I am currently working on an assignment in which I create a six-state up/down counter. The purpose is to understand the functionality of flip-flops within a circuit among other things. There are 3 input bits (Q2,Q1, Q0), and a 4th control bit (C) which determines the...
  20. T

    This is a logic problem about tennis.

    Homework Statement Pat beat Stacy in a set of tennis, winning six games to Stacy’s three. Five games were won by the player who did not serve. Who served first? source: https://ocw.mit.edu/courses/mathematics/18-s34-problem-solving-seminar-fall-2007/assignments/hw8.pdf Homework Equations N/A...
  21. victorhugo

    Biocentrism - To me this lacks some logic

    I'm very interested as far as I've gone into the book (just over 100 pages) and there is a lot of stuff that I've pondered about myself. However, quite a few things don't make sense to me. The main one I'd like to ask your opinions on is at 9:15 until about 12:00 (he talks about it here: )...
  22. Tukhara

    Books on mathematical logic, foundations, and philosophy

    Hello, all. I am looking for some good books to start becoming invested in mathematical logic, the foundations of the field of mathematics, and also basically in general the philosophical heart of this wide subject which has interested me greatly. Now I have already read Shoenfield and Halmos...
  23. S

    Where is the faulty logic in my solution to this problem?

    I was working on https://www.hackerrank.com/contests/101hack50/challenges/even-and-odd-boxes for 2.5 hours and couldn't get it right. The programming competition is over, so now it's ok to discuss. I can't figure out where I was going wrong. static void OnesToFront(List<int> list) {...
  24. P

    Logisim question -- Simulation of a small logic circuit is not working

    I am trying to implement the following circuit in Logisim, yet all I get is y=0. Any idea what might be the problem? First image is what I am trying to implement; second it my Logisim attempt which yields 0. I'd appreciate some feedback.
  25. W

    A Transcription from SQL to FOL (First Order Logic)

    Hi All, When a query is made in SQL , it is transcribed into FOL in the back end , and, if the transcription is a wff , the models, if any, are returned as the answer to the query. I have an idea of how the transcription works for basic statements, but, does anyone know the actual "...
  26. radouani

    A First order logic : Predicates

    I have a small problem with the first order logic, in particular, predicate logic Let us take this sentence as an example: Each teacher has given a form to each student. From this sentence, can we have different reading? This is my try to solve such problem, I did not know if this is the...
  27. thebosonbreaker

    B Sweets in a bag probability problem

    Andrei has a bag of x sweets. He removes two sweets from the bag simultaneously (without replacement). He now removes a third sweet. The probability that the third sweet is red is (x/2) - 1. How many red sweets were in Andrei's bag to begin with? Could somebody please tell me if (and how) it is...
  28. PetSounds

    GEB: Exploring Douglas R. Hofstadter's Strange Loops

    Lately, I've been hooked on Douglas R. Hofstadter's book Gödel, Escher, Bach. In it, he discusses the idea of "strange loops"—often apparent logical paradoxes—and argues that they are the key to understanding consciousness. He includes witty dialogues, as well as examples of "strange loops" in...
  29. J

    Engineering Designing 3-Stage Async Counter & Logic Circuit in PSpice

    Homework Statement The block diagram of FIGURE 3 shows a three-stage asynchrononous counter that is used to count a series of randomly occurring input pulses. The ‘Q’ outputs of the counter are used to drive a logic circuit that gives the output shown in TABLE 1. (a) Design the counter using...
  30. R

    Specialization-generalization(mathematical logic)

    Specialization as defined in Wikipedia: "Concept B is a specialisation of concept A if and only if: every instance of concept B is also an instance of concept A; and there are instances of concept A which are not instances of concept B" We then call B as special case of A, it seems evident...
  31. moriheru

    A A "Proof Formula" for all maths or formal logic?

    I was wondering whether or not there could be a "master formula" . What I mean by a master formula is, maybe not even a formula, some mathematical expression that would allow mathematicians to prove statements simply by plugging in some numbers into a formula. So I guess in a way a I am...
  32. J

    Courses Intro to Philosophy: Knowledge and Reality or Philosophy: Practical Logic?

    Which course do you guys think would be more interesting/helpful for a physics major? They are both 100 level, Intro consists of two lectures and a seminar a week, and Practical Logic consists of a three hour lecture once a week.
  33. N

    Help me Prove that such a set does NOT exist

    Let Q denote the theory of Robinson Arithmetic. A theory T is nice iff T is consistent, is p.r. adequate and extends Q. The fixed-point lemma states that for all nice theories T, for any formula φ, there is a sentence σ such that T ⊢σ↔φ("σ")...
  34. T

    Relation closures proof

    Homework Statement Suppose R1 and R2 are relations on A and R1 ⊆ R2. Let S1 and S2 be the transitive closures of R1 and R2 respectively. Prove that S1 ⊆ S2. Please check my proof and please explain my mistakes. thank you for taking the time to help. Homework Equations N/A The Attempt at a...
  35. Auto-Didact

    A Undecidability and multivalent logic

    Are undecidable statements, such as the provability of the continuum hypothesis, natural examples of statements that require a multivalent logic in order for them to be adequately described and/or even properly understood? (NB: by properly I am taking this to mean that undecidable matters such...
  36. T

    MHB How to Translate 'There Exists Exactly One Happy Person' into Predicate Logic?

    How to translate "there exists exactly one happy person" into predicate logic? I came up with $$ \exists x : happy(x) \implies \forall y: happy(y) \land y = x$$. But this is incorrect. I also tried $$\exists x: happy(x) \land \forall y: happy(y) \land x = y$$. This is also incorrect. The...
  37. P

    MHB Solve Logic Puzzle: Predicates & True/False Explained

    I'm unsure about these three, here are my attempts. Please also explain the difference between a predicate and true/false. I assumed it is a predicate when it can be either true or false. a) Predicate. Negation is ¬(∃n ∈ N n²>n) b) True. Negation is, "When x<0 there is y such that y^2=x c) No...
  38. B

    Logic: exclusive-OR statements to form AND/OR statements

    Homework Statement Using only NOT and XOR, construct a compound statement having the same truth table as: (a) p OR q (b) p AND q Homework Equations XOR is "exclusive OR." p XOR q = (p OR q) AND NOT (p AND q). I have been working under the assumption that I can use parentheses. The...
  39. P

    MHB Help with Logic Questions | 65 Characters

    Hi there, I desperately need your help :P I was very ill for the past week and missed out on class time when we were taught the content. I attempted the homework questions but because I've had to breeze through the content I'm not sure if I have answered all the questions correctly or as well...
  40. T

    I Problems with Existential Instantiation

    Why is it required to use a "fresh name/variable"? And because of that requirement, Existential instantiation always precedes universal instantiation. What I am thinking is, If we are picking elements at random from our universe of discourse then why can't universal instantiation pick that...
  41. T

    Using existential generalization

    Homework Statement is my method valid? ∃x¬R(x,a) --> ¬∃R(a,x) ¬R(a,a) thus, ¬R(a,b) Homework Equations N/A The Attempt at a Solution ∃x¬R(x,a) by existential gen. of ¬R(a,a) ¬∃R(a,x) by modus ponens ∀x¬R(a,x) by identity of ¬∃R(a,x) ¬R(a,b) by universal instantiation of ∀x¬R(a,x)
  42. A

    MHB What's the Logic of this Application using Math

    Here i attached some mobile application screenshots I want to know the logic of this application 1:- It have four screens and every screen have numbers. 2:-Application asked on every screen "Do you Know your age is here in these numbers". 3:-If i see the number is on the screen then i pressed...
  43. C

    Designing a NAND Gate Logic Circuit

    Homework Statement Draw the logic circuit to represent the following Boolean expression using only NAND gates: Homework EquationsThe Attempt at a Solution Is this correct? Thanks
  44. T

    Quantifier logic translation

    Homework Statement All married couples have fights Homework Equations let x be a person and y be another person M(x,y): x is married to y F(x,y): x have fought y The Attempt at a Solution ∀x∃y(M(x, y) → F(x, y)): i read my translation as "For all x, there is some y such that if x is married...
  45. T

    Using symbolic logic in mathematical proof?

    is this a practical way of proving math theorems? i asked because when i tried, it seemed difficult for me to decide as to how exactly i should translate theorems and given statements into logical forms and since there are so many different ways, i do not know which one is correct. For example...
  46. M

    MHB Formulas of predicate logic

    Hey! :o I want to formulate the following statements into formulas of predicate logic. If a bird cannot fly, then not all birds can fly. What Donald cannot do, can no one do. John likes everyone, that is older than $22$ years old and that doesn't like those who are younger than $22$...
  47. T

    Is Trestone's Layer Logic the Next Breakthrough in Philosophy?

    Hello, I am a hobby philosopher digging at the fundaments of thinking. Unfortunatly I dedected something like a new logic ("layer logic Trestone"), so I had to do a lot of formal definitions and studies. As I am rather lazy and my university time is some thirty years ago, ten years later it is...
  48. F

    BCD to 7 segment display logic minimisation

    Homework Statement A 7-segment display is used to show a decimal digit and it is driven from 4-bit input. Each bar is assumed to light up when a logical 1 is applied to it. Draw the truth table to drive segment (d) of the display.Using the truth table you have obtained, draw a Karnaugh map to...
  49. nomadreid

    I Existential Import paper by Corcoran and Massoud

    In the paper John Corcoran & Hassan Masoud (2014): Existential Import Today: New Metatheorems; Historical, Philosophical, and Pedagogical Misconceptions, History and Philosophy of Logic, http://dx.doi.org/10.1080/01445340.2014.952947, already in the introduction it says, as self-evident, that...
  50. M

    Convert statements into first order logic

    Given are the following predicate symbols: Member(x) : x is a member of the bicycle club Chairman(x) : x is the chairman of the bicycle club Bicycle(x) : x is a bicycle Brand(x, y) : the brand of x is y Owns(x, y) : x is the owner of y a. Statement: every member of the bicycle club has the same...
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