Logic Definition and 1000 Threads

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. Y

    Engineering NAND Logic 4 input circuit with nonstandard output

    Homework Statement I have a weird boolean algebra expression for which I need to build a circuit using solely the NAND gates. There are 4 inputs A,B,C,D. From the truth table (I do not know how else to approach this question other than constructing the truth table and then trying to fit the...
  2. Jameson

    MHB Mano's question from Facebook on logic

  3. C

    Why Are Axiom Schemas Important in Propositional Logic?

    Actually, I have several questions: 1) Why are axiom schemas the way they are? What do they represent? I know that infinitely many axioms can be written using the axiom schema form. However, what's the formal definition of axioms in predicate calculus? I've heard that the formal definition of...
  4. N

    Logic Problem Solutions: Universal and Existential Formulas

    Homework Statement A universal (or ∀_1) formula is one of the form ∀x_1∀x_2...∀x_nθ, where θ is quantifi er-free. An existential (or ∃_1) formula is one of the form ∃_1∃_2...∃x_n θ. Let A be a substructure of B, and s valuation on A. a) Show that if ⊨_B ψ [s] and ψ is universal, then...
  5. C

    MHB Logic puzzle: students, cities, subjects and grades.

    This exercise is from my Discrete Mathematics course and it requires a step by step solution, backed up by logical equivalences or deductions made based on the rules of logic. 5 students, Oliver, Simon, Peter, Bob and Christie all seniors at Harvard, come from one of the following cities...
  6. D

    CprE logic gate problem, possibly unsolvable

    Homework Statement I was having a casual chat with one of my professors and he presented me with a problem that after 4 hours of attempting I believe is unsolvable. the set up is pretty simple. you have 3 inputs x,y,z that go into a system, the output is ~x,~y,~z. However you are only allowed...
  7. alane1994

    MHB Optimizing Prisoner Selection for Poisoned Wine Detection in Medieval Empires

    You are the ruler of a medieval empire and you are about to have a celebration tomorrow. The celebration is the most important party you have ever hosted. You've got 1000 bottles of wine you were planning to open for the celebration, but you find out that one of them is poisoned. The poison...
  8. C

    MHB Logic Exercise: Solving a Canoe Team's Seating Arrangement Dilemma

    Here I am with another problem... :) It's from a Discrete Mathematics course and it requires a step by step proof with logical equivalences and deductions made based on the laws of logic. A,B,C and D are all on the same canoe team. They don't know what sitting order they should make and all of...
  9. Logic Cloud

    Boolean algebras, sets and logic

    I know that propositional logic and Boolean algebra's are related in the sense that disjunction, conjunction and negation behave the same as join, meet and negation. Similarly, we also have union, intersection and complement when talking about sets. It's obvious that all these notions are...
  10. C

    MHB Discrete Math Exercise: Solving Logic Puzzle in Computer Science

    The Subject is Discrete Mathematics as part of a Computer Science major. The exercise is as follows: Erica, Stanley and Robert were all witnesses in a car crash. Their statements are contradictory to one another, and all of them claimed that someone else lied. Erica claimed that Stanley lied...
  11. A

    Why is this false? - Short Fitch proof using Herbrand logic

    Hello, folks. I'm taking my first formal logic class and some of the things seem contradictory; I know it's because I'm not fully understanding something, but I don't know what I'm not fully understanding—I hope someone can help me. The problem begins: The statement I'm having trouble...
  12. C

    Mathematical logic and statistics

    Let's say that a treatment A has been proven to have an impact on the levels of B with a given confidence interval. Let's also say that we know that the treatment C causes the treatment A to be imposed on our sample. Before the testing on the effects of C has been done, which statistical models...
  13. I

    3 input XOR gate using NAND gate only ( Logic )

    We can use NAND gate only to get ( XOR gate ) of 2 input ( A and B ) : By using (4) NAND gate : The output of 1'st NAND : (AB)' The output of 2'nd NAND : ((AB)'.A)' = (A'B) The output of 3'rd NAND : ((AB)'.B)' = (AB') The output of the hole circuit will be: ((A'B).(AB'))' = AB'...
  14. A

    How to simplify this boolean algebra (logic)

    This is related to logic circuits. How do I simplify this boolean algebra(logic) into its simplest expression? h = (abc + abd + acd + bcd)’ My first attempt would be to change it into product-of-sum. h = (abc)'(abd)'(acd)'(bcd)' and then what next?
  15. D

    Lebesgue outer measure of a set of countably many points is 0 - logic check

    So I know that the Lebesgue outer measure of a set of only countably many points is 0. An example of this is the rationals as a subset of the reals.I want to make sure my intuition behind this is correct. The process: Now, if we are going to take the Lebesgue outer measure of the rationals, we...
  16. M

    Set problems(Of which one includes propositional logic)

    I have a few questions regarding 2 set problems. Exercise 1: Homework Statement 1. the set A = P(empty) (the powerset of the empty set); 2. the set B = P(A); 3. the set C = P(B). 2. The attempt at a solution 1. A= {empty} 2. B = {empty, {empty}} 3. C = {empty, {empty}, {{empty}}, {empty...
  17. K

    Using induction technique in mathematical logic

    Homework Statement Prove the following LEMMA: For every proposition A[P_{1}, \dots, P_{n}] and any two interpretations v and v', if v(P_{i})=v'(P_{i}) for all i=1, \dots,n, then v^{*}(A)=v'^{*}(A). Homework Equations The Attempt at a Solution Sure this is obviously an...
  18. F

    Basic physics logic help - kinematic equations and forces

    Homework Statement A small object, of mass 0.02Kg is at the top of a tall structure 160m high, it is dropped from rest. Ignoring air resistance.. 1. Find the speed of the object as it hits the ground 2. find the time taken Homework Equations So far we have learned the equations...
  19. andrewkirk

    Logic puzzle - trying to justify a step in proof of Godel's theorem

    I am trying to fill in some of the gaps in wikipedia's proof of Godel's diagonal lemma, which is a crucial step in proving his first incompleteness theorem. There is a step that the wiki proof just takes for granted, perhaps because it looks obvious. But I am unable to find a formal proof of...
  20. andrewkirk

    Problem in Logic - Hilbert Systems

    Homework Statement In a Hilbert System, prove: \phi[x|m] \rightarrow\forall y((y=m)\rightarrow\phi[x|y]) where \phi is a formula, y, x are variables and m is a constant. \phi[a|b] denotes the formula obtained by substituting b for a in \phi This problem crops up in my attempting to prove...
  21. C

    Mathematical logic & its branches

    What branches of mathematical logic are there? I've taken formal logic (that is, the logic where one has various operators, like conjunction, disjunction, etc). What other fields of logic should I take to become better at mathematical logic? Which fields of logic are the most useful and essential?
  22. A

    Which is more advantageous: sequential or combinational logic?

    There are some circuits which can be made only using sequential logic like registers and counters. But some like adders can be made both by combination logic and sequential logic. So which one is better in such cases? I guess sequential logic is advantageous but what advantage does it offer?
  23. Mathelogician

    MHB A question on consistency in propositional logic.

    Hi everybody! We have a theorem in natural deduction as follows: Let H be a set of hypotheses: ==================================== H U {~phi) is inconsistent => H implies (phi). ==================================== Now the question arises: Let H={p0} for an atom p0. So H U{~p0}={p0 , ~p0}. We...
  24. R

    Makeing logic gate using only NAND Gates

    Homework Statement How to make a logic gates only using NAND gates of the following diagram. How many NAND gates I would have to use for that diagram Homework Equations The Attempt at a Solution I know that A bar bar + B bar bar =AB A bar bar B bar bar = A+B De-morgans Theorem
  25. M

    Stepper Motor Anatomy and Logic

    Hello all, I've been learning and trying to reverse-engineer the logic, nature and anatomy of a stepper motor. In my quest I am trying to figure out what any given stepper motor will draw at any given rpm with given specs about the motor. These motors come with certain specs as far as...
  26. J

    Y=ln(3-x) How to transform it? What is wrong with my logic?

    Hi, my name is Tim, I am new to this forum. Here's the description: I don't understand how they've done it. I realize I can plot this, but what is the logic. As a graph, I drew ln(-x) first, then added 3, so move left by 3 but that was not correct I can see they've added 3 first, move...
  27. H

    Best place to study mathematical logic uk

    I was wondering where the best place to study logic at postgraduate level in the UK. As far as I know it's Manchester but I could be wrong? Thanks for any help
  28. T

    Engineering Need Help Identify Digital Logic Circuit

    Homework Statement On the data sheet, name the device in Figure 9, and appropriately name the four pins A, B, C, and D according to what the device is. Check your Digital Logic book for some guidance...
  29. B

    MHB Steps and logic on how to properly show discount

    Hi, I have the following math problem which I'm a hardtime finding the best way to reflect/calculate. If anyone can help I would really appreciated. Problem:I have an agreement with a supplier that he will provide a bottle per case (12 bottles in a case) and I will provide a bottle per case...
  30. L

    Logic behind the number of combinations of numbers

    Hey, so i think this is a fairly simple question but i'd like to get it firmly understood in my head. How do you figure out the amount of combinations of digits in say a 4 digit code. with numbers 0-9... I can't think of a good way to say it, but for example you could have 0,1,2,3 or...
  31. B

    Propositional Logic: Edit Protected Wikipedia Entry Only as Admin

    Homework Statement You cannot edit a protected Wikipedia entry unless you are an administrator. Express your answer in terms of e: “You can edit a protected Wikipedia entry” and a:“You are an administrator." I thought the answer would be a\rightarrow\neg e; but the actual answer is...
  32. JJBladester

    Engineering Very simple logic circuit question

    Homework Statement Draw a logic circuit for the case where the output, HOLD, is LOW only if the input, LOAD, is HIGH and the input, READY, is LOW. Homework Equations Truth tables for AND, OR, NAND, NOR, XOR, XNOR are known (I'm not re-typing them here). The Attempt at a Solution...
  33. D

    The logic of the 11 dimensions in M theory

    According to M-theory as I understand, the 7 additional spatial dimensions to our familiar 3 are curled up all around us but they are too small for us to see. However I have difficulty in understanding how this constitutes a "dimension" because dimensions allow additional degrees of freedom...
  34. G

    Food for thought Sets logic and applied science

    I was just reading an article the other day about the debate in public schools about teaching evolution as an absolute truth as it has been taght for the past umteen years. Not saying that I'm a proponent of creationism or even that I'm not, but there are some serious flaws in teaching it as the...
  35. T

    Problem with Propositional Logic

    Hi, I've been set an assignment, part of which is to come up with a formal proof for (p \wedge q) \Rightarrow p. I have to show that the formula is either a tautology or contradiction, or contingent. If it is contingent, I have to show the smallest possible equivalent expression that uses only...
  36. PerpStudent

    Does Relativity Allow for Faster-than-Light Travel?

    Suppose we have two points A and B, separated by ten light years. Now I shoot off a proton at .999c from A to B. From the perspective of the proton the distance between A and B is now about .3 light years. Would it get from A to B in less than four months in the time frame of the proton? If not...
  37. A

    Introduction to digital logic design Video Lectures

    Hi, Can anybody recommend me a good video lecture series on Introduction to digital logic design? I want to watch these videos before going to class.
  38. B

    Solve 3 Bit Grey Code Decoding: Output Logic

    Homework Statement Given that the variables A[2:0] form a three bit grey code to represent numerical values: i) Write down all binary combinations for A[2:0], in ascending order of numerical value ii) A three bit grey code to decimal decoder is to be designed, where the outputs X[0:7]...
  39. J

    What's wrong with my logic on this combinatorial/probability question?

    Homework Statement The Attempt at a Solution If at least 3 balls must be blue, then the possible ways of selecting at least 3 blues are: r, b, b, b b, r, b, b b, b, r, b b, b, b, r b, b, b, b where r = red ball, and b = blue ball So there are 5 ways to draw 3+ blue.
  40. H

    Formal logic question from beginner

    Hello, I am a complete novice at formal logic and so far removed from mathematical study that I have no idea how the following should be interpreted, or read, and what its implications are: (a∨~a)∨~a⇒a∨(~a∨~a)⇒(a∨~a)⇒((a∧⊤)∨~a)⇒((a∧(b∨~b)∨~a)⇒(((a∧b)∨(a∧~b))∨~a)⇒...
  41. D

    Logic Gates/ Truth Tables Trouble

    Hey guys I am having todo a resit paper for a maths module i failed and I've come up against a question on my paper where I am stomped and don't know if I am down the right road or not. Its truth tables from Boolean equations. Question was: K = ( X + Y) . (X . Y) I Came up with: K X...
  42. M

    Find a counter example/predicate logic

    Homework Statement Find a counter example to show that the following is invalid: ∃xP(x) ∧ ∃x(P(x) → Q(x)) → ∃xQ(x) Homework Equations - The Attempt at a Solution Well, I have tried to find something related to properties of numbers but found nothing... Then I tried to mess...
  43. N

    NOT Gate: Inverting Inputs & Current Flow - Help Clear Confusion!

    From what I learned at school today.. Afaik NOT gate acts as an inverter, right? Which means if input is 0, output will be 1 and vice versa. My question is, if input is 0 then output will be 1. From where does the gate gets its current flow? Since input is 0. Or, it cannot be used like...
  44. V

    Currently working through Apostol need a check on my logic

    Homework Statement Prove the following theorem. If a≠0, then (a-1)-1=a Theorems proven before/axioms I am allowed to use: Existence of reciprocals axiom: there exist real numbers x and y where x≠0 such that xy=1 Possibility of division Thm: basically a-1=1/a Homework Equations I just need a...
  45. A

    Logic: difference between very similar statements

    What is the difference between the following two questions: (a) For every positive real number x, there is a positive real number y less than x with the property that for all positive real numbers z, yz ≥ z. (b) For every positive real number x, there is a positive real number y with the...
  46. O

    Exploring the Possibility of Laser-Driven Logic Gates

    I'm sure someone has tried this, and I'm sure there's a reason it's not something that is widely used, but here's a question: Can a logical NOR gate be created by shining two lasers (an A and B input) through tiny slits (like the double-slit experiment), and determining an output at a known...
  47. B

    Extension of Binary Connectives to n-valued Logic?

    Hi, All: Just curious as to whether there is some sort of canonical extension of the standard binary connectives: and, or, if, iff, etc. , to n-valued logic. I imagine this may have to see with Lattices, maybe Heyting Algebras, and Order theory in general. Just wondering if someone...
  48. W

    Proving Logic Statements with Truth Tables and Laws

    Homework Statement Sometimes i got a question with homework if i can prove something. I got a book where the tolled me to use truthtables to see the outcome : For example : (notP V Q) <=> (P => Q) P Q | notP | notP v Q | P=>Q | (notP V Q) <=> (P => Q) 0 0 | 1 | 1...
  49. B

    Predicate logic to prove equality of two formulas

    Hi All, I'm trying to write a program that will prove that a certain mathematical formula that is entered by a student is the same as the goal formula that I want. For example, if the goal formula is: 2 * X + Y then any of this student's answer can be considered correct: -> X + X + Y...
  50. J

    Logic in Computation: Exploring Deviations from Classical Logic

    Since I know that this will get lost in the Liar's paradox thread I wanted to separate it out from the thread: In post 18 of the liar's paradox thread Hurkyl writes:" and in post #47 he writes: I would like to know where he is going with this line of thought. A related concept I've seen...
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