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. 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...
  2. P

    Why does P -> Q hold true when P is false and Q is true in mathematical logic?

    When P -> Q, why is it true when P is false and Q is true, but why is it false when P is true and Q is false? If I suppose P mean "Jon is a guy" and Q mean "Mary is a girl". When both P and Q are true it does make sense that this proposition is true because Jon is a guy and Mary is a girl...
  3. kyphysics

    I How Often is First-Order and Modal Logic Used in Math?

    I'm a philosophy minor, who took first-order logic (philosophy department, not the math side) and I have a chance to take modal logic next semester (also under philosophy). It's going to be a much smaller class and will likely be harder. But it's also essential to doing work in and/or...
  4. Cocoleia

    How to go from micro operations to a logic diagram

    Homework Statement I am preparing for my final and I am stuck on the following question Homework EquationsThe Attempt at a Solution My best guess would be to make some kind of next state table, but I don't understand the relationship between xT1, yT2, yT3 and any kind of table.
  5. N

    Engineering Circuit Realization of XOR function with Three inputs

    Homework Statement This is from a past exam paper for logic design in my course. I have an exam coming up and would love to know how to solve this one. [/B] Develop a circuit realization of the XOR function with three inputs. You may use AND, OR and NOT-gates with not more than two inputs...
  6. OrlandoLewis

    How to toggle and latch using logic gates?

    We're implementing Dots and Boxes to logic circuits and the we would like to automate the process of changing turns. If you know the game, the game changes turn when one move doesn't score a point or form two or more boxes. I was able to create overlapping XOR gates that alternates from high to...
  7. Derek Hart

    Proving ƒ(x) is the Identity Function

    Homework Statement I have been going through a textbook trying to solve some of of these with somewhat formal proofs. This is a former Putnam exam question. (Seemingly the easiest one I have attempted, which worries me). Consider a polynomial function ƒ with real coefficients having the...
  8. C

    Implementing Logic Function with a ROM

    Homework Statement Homework EquationsThe Attempt at a Solution I'm a bit confused with this. I thought a ROM was just a non-volatile storage. I don't understand how this would work. The decoder can only select one of the 8 outputs depending on what it gets at its input. I still attempted...
  9. Mr Davis 97

    What would the other villager tell me to do in order to reach my destination?

    Homework Statement Each inhabitant of a remote village always tells the truth or always lies. A villager will only give a "yes or a "no" response to a question a tourist asks. Suppose that you are a tourist visiting this are and come to a fork in the road. One branch leads to where you want to...
  10. U

    Discrete Mathematics logic questions

    Homework Statement 1. Why is the statement: " Vicky is not clever" Not a mathematical proposition? Provide examples please 2. Why is the statement: "a^2+b^2=c^2 an indeterminate proposition?" 3. Why is the negation of " If a triangle has two equal angles it is isosceles" = "Not all triangles...
  11. Mr Davis 97

    Solve Yes/No Logic Puzzle & Save Explorer: How?

    Homework Statement An explorer is captured by a group of cannibals. There are two types of cannibals--those who always tell the truth and those who always lie. The cannibals will barbecue the explore unless he can determine whether a particular cannibal always lies or always tells the truth. He...
  12. Mr Davis 97

    Logic puzzle with 100 statements

    Homework Statement We have 100 statements, where the nth statement says that "exactly n of these statements are false." What conclusions can you draw about the truth values of the statements? Homework EquationsThe Attempt at a Solution Immediately we see that the 100th statements can't be...
  13. doktorwho

    Can you help me solve this Boolean algebra problem?

    Homework Statement Prove that $$(\bar{a} + b)(b+c) + a\bar{b}$$ where ##a,b## can be from the set ##B\in\{0, 1\}## equals $$a+b+c$$ Homework Equations Rules of Boolean Algebra 3. The Attempt at a Solution [/B] My attempt: ##\bar{a}b + \bar{a}c + bb + bc + a\bar{b}## ##b(\bar{a} + 1+c) +...
  14. Mr Davis 97

    I When are statements in propositional logic true or false?

    I am studying propositional logic, and have studied how propositions can be combined with logical connectives and such, and truth tables can be used to analyze the resulted truth values, depending on the truth values of involved variables. However, when not talking in the theoretical, how do we...
  15. Nipuna Weerasekara

    How Does the Condition x > -1 Influence the Inequality x² + 1/(x²+1) ≥ 1?

    Homework Statement Let ##x\in \mathbb{R} ## Prove the conditional statement that, if ## x>-1## then ## x^2 + \frac {1}{x^2+1} \geq 1## 2. The attempt at a solution Suppose ## x>-1## is true. Then ## x^2>1## Then ## \frac{1}{2}>\frac {1}{x^2+1}## Then ##x^2+ \frac{1}{2}>x^2+\frac...
  16. moenste

    Logic gates, truth tables, circuits

    Homework Statement (a) The NOR gate combination in Figure 1 acts as an AND gate. Copy and complete the truth table. (b) An aircraft door is locked by two bolts A and B each of which operates a sensor giving a logic 1 when the bolt is fully inserted. A further sensor C gives a logic 0...
  17. moenste

    Construct a truth table for the combination of logic gates

    Homework Statement Construct a truth table for the combination of logic gates shown above. 2. The attempt at a solution First of all the combination of logic gates with added letters to the combination: The truth table: I think it should be correct. But just want to be sure.
  18. moenste

    Simple Logic Gate Homework: Complete Truth Table

    Homework Statement Complete the truth table for the simple combination of logic gates shown above. 2. The attempt at a solution I think this should be correct, but just want to check, since I don't have a book answer to compare:
  19. S

    Where is my logic wrong in this dynamic programming problem?

    I'm trying to solve https://www.hackerrank.com/challenges/summing-pieces and I think I have come up with an O(n) solution although the numbers aren't coming out exactly right. The problem is essentially to find a sum like ABCD ---> All sets of sets of contiguous subarrays spanning ABCD is {...
  20. Nipuna Weerasekara

    Can a triangle be formed with these length constraints?

    Homework Statement There is a triangle with sides $$ 3,3r,3r^2 $$ such that 'r' is a real number strictly greater than the Golden Ratio. Is this statement true or false...? Homework Equations $$Golden \space Ratio = \phi = 1.618... $$ The Attempt at a Solution Actually I have no clue at all...
  21. A

    What is your answer to this Logic question? (Mine included)

    Homework Statement The rules for a certain frequent-flyer club include the following statements: "Any member who fails to earn any mileage during the first twelve months after enrollment in the program may be removed from the program. Except as otherwise provided, any member who fails at any...
  22. R

    Digital Logic 4-bit Multiplier/Adder

    Homework Statement Build a circuit that either adds or multiplies two 4-bit numbers based on a control input C(C is 1 add, C is zero multiply). Numbers are positive and negative so use two's complement. You can use muxes, Full Adder Circuits, and logic gates Homework Equations Multiplier...
  23. Ling Min Hao

    I Is for every x in domain D, Q(x) false if Q(x) is false for all x in D?

    "For every x in domain D , Q(x) "is false if Q(x) is false for some x in D . "For every x in domain D , Q(x) "is false if Q(x) is false for all x in D . Which of the following is correct ? Or both are correct ?
  24. Euler2718

    Is This Logical Argument Valid?

    Homework Statement Determine whether the following is valid: p \rightarrow \neg q , r \rightarrow q , r, \vdash \neg p Homework Equations Modus Ponens, disjunctive syllogism, double negation. The Attempt at a Solution I've boiled it down to p \rightarrow \neg q , q, \vdash \neg p...
  25. K

    Is the Function f(x) = x^2 Injective?

    Homework Statement Prove that ##f: \mathbb{R}\to\mathbb{R}, f(x) = x^2## is not injective. Homework Equations Definition of an injection: function ##f:A\to B## is an injection if and only if ##\forall a,b \in A, f(a) = f(b) \Rightarrow a = b##. The Attempt at a Solution ##f...
  26. Olive1923

    Dropping Beads and their velocity?

    Homework Statement A straight rope is dropped at a height where is it not touching the ground. The rope has 10 beads spaced out on it and tied so they do not move. From the bottom to the top, the space between the beads increase, where the distance is the smallest at the bottom and largest at...
  27. L

    Creating a simple boolean logic calculator in C

    #define _CRT_SECURE_NO_DEPRECATE #include <stdio.h> int main() { int A; int B; printf("This is a simple boolean truth value calculator.\n\n"); printf("Please input the boolean value for statement A\n\n\n"); scanf("%d", &A); printf("Press enter to continue\n\n\n"); getchar()...
  28. J

    I Logic Game: Abelard & Heloise - Naturals xu-yt > v^z

    The question is about a logic game between two competing players "Abelard and Heloise", Abelard is the name given to the "For all" quantifier and "Heloise" is the name given for the "There exists" quantifier. The statement is: Is ##(\forall x)(\forall y)(\forall z)(\exists u)(\exists v)(\forall...
  29. wololo

    Is Person A a Knight or a Knave on the Island?

    Homework Statement A person can either be a knight (always tells the truth) or a knave (always tells a lie). On an island with three persons (A, B and C), A tells "If I am a knight, then at least one of us is a knave". Homework Equations Truth tables, logic rules. The Attempt at a Solution...
  30. PhotonSSBM

    Base 8 Logic Puzzle: Finding Solutions to Divisibility Rules

    Homework Statement We call a seven digit number in base eight X, whose digits are given by ##abcdefg## No digits are zero, and none of them are the same. The following divisibility rules are true: (I)The number ##ab## is divisible by 2 (II) The number ##abc## is divisible by 3 (III)The...
  31. M

    I Proving the Implication of p and (p -> q) to q without Truth Tables

    Hello everyone! I want to proof that: ##p \land (p \to q) \Rightarrow q## I know this is a quite trivial problem using truth tables, however, I want to do it without it. As I'm learning this myself, is this the correct approach? ##p \land (p \to q)## ##\iff p\land (\neg p \lor q)## ##\iff (p...
  32. Mr Davis 97

    I Weird statement of conditions in propositional logic

    So I am studying conditionals in proposition logic, and I have discovered that there are a variety of ways to phrase a conditional "if p, then q" in English. Some of the harder ones are... p is sufficient for q a necessary condition for p is q q unless ~p (where ~ is the not operator) p only...
  33. T

    MHB Logic Math Calculator App: Helping Students Learn Discrete Math

    Hello! Some time ago I asked the admins for permission to post a link to my Logic Calculator app for android. They allowed me to post in this section so, here I am! I am a software engineering student from Buenos Aires and in my freshman year I started to develop this app as a project for a...
  34. L

    A Why Can't the Continuum Hypothesis Be Decided Using Standard Real Number Models?

    I know that there are several models of the real numbers, some where the Continuum Hypothesis holds, others where it does not. I would like to understand why the usual definition of the reals, limits of Cauchy sequences of rational numbers under the usual absolute value norm, isn't one of these...
  35. I

    B Can you reason out a differential equation using logic?

    Let us suppose that we are modeling a particle traversing some distance in space & that f(x)=x². Let the x-axis be time (hrs) & the y-axis be position in space (miles). What is the instantaneous velocity of said particle at t=4 hours? Let's assign variables to each coordinate in our x, y...
  36. W

    I Temporal Database and its Relational Algebra

    Hi all, Just curious on whether temporal databases are designed in the same way as "standard" relational databases, so that queries on temporal databases are wffs in temporal logic , i.e., using Until, Release and all other unary operators , and the query outputs is the set of models for the...
  37. B

    MHB Percentage Change Logic Question

    Question regarding different percentage changes. If I have a monthly rental fee of \$100 in July with a discount of 60%. I will pay \$40. In the month of August the monthly rental fee is \$100 with a discount of 45%. I will pay \$55. My discount has decreased by %15. However my...
  38. L

    Game of battleship logic puzzle

    Homework Statement -Admiral, your orders are to deploy the fleet into battle formation. -your fleet contains 1 battleship, 2 cruisers, 3 destroyers and 4 corvettes. -battleship requires 4 squares deployment, cruiser requires 3, destroyer requires 2, and corvette requires 1 -each ship...
  39. P

    [Discrete] Prove that |nZ| = |Z| for any postive integer n

    I have been studying discrete mathematics for fun and I am kind of stuck on this bijection problem. 1. Homework Statement I wanted to apologize in advance if i put this homework question in the wrong part of the forums. Discrete Math and much logic math is a computer science type math of...
  40. E

    I First order logic - help with translation algorithm between

    given a dictionary \Sigma = \left \{f(),g(),R(,),c_0,c_1,c_2 \right \} and a sentence \phi over \Sigma, I need to find an algorithm to translate \phi to \psi over \Sigma' where \Sigma' = \left \{Q(,,,), = \right \} (Q is a 4-place relation symbol), so that \psi is valid iff \phi is valid. I...
  41. W

    MHB How Can Logic Laws Remove Implications in Expressions?

    Hello Everyone, This is my first post, so please me kind :D I have an assignment and have been stuck on this thing for 2 days. I have no problems working this out with a truth table but we are not allowed to use this. So this it the expression. I understand that I have to remove the...
  42. C

    MHB Prop Logic Proof Help: (pv~q)vr; ~pv(q.~p)/q>r

    i need help with a proof: (pv~q)vr ~pv(q.~p) / q>r this is some propositional logic thanks all
  43. A

    MHB How to Approach a Logic Proof Involving Conditionals?

    hi! I've been studying formal proofs for my logic final and i had a question on this one that i found in my textbook: 1.)~P\supset(horseshoe)U 2.)P\supsetF 3.)F\supsetU\thereforeU i just have had a little trouble in the past getting started with proofs. Can anyone give me a little push and...
  44. M

    MHB Logic: Reductio Ad Absurdem Assumption

    When you are proving a valid argument using the assumption reductio ad absurdem, can you assume anything in the conclusion, and then put a not in front of it, or do you have to assume the whole conclusion with a not in front of it? For example: 1. p\supset q 2. p \lor q \therefore p and q To...
  45. C

    B Does Quantum Physics really defy logic?

    I heard a lot of this "Quantum theory defies the human logic, it's totally illogical but it works!" and I'm trying to understand whether this is true or not. I don't mean by logic some mind blowing stuff, like teleportation is possible. I mean the kind of logic that would be difficult or...
  46. chwala

    Logic question on finding age of someone

    Homework Statement Homework EquationsThe Attempt at a Solution the solution according to blog is Neither May nor June is possible because they have unique dates - that means Albert couldn't know for sure that Bernard doesn't know. If Albert had May for example, Bernard might have the number...
  47. H

    Check logic for Watts calculation

    Thanks to anyone kind enough to help check my logic. Using 13 gauge copper wire and a magnet I can generate .05 watt every time the magnet passes the coil (according to the voltmeter). Am I correct in assuming that if the magnet passes the coil 60 times a minute, then (.05 watts X 60rpm X 60...
  48. J

    Digital Logic Help: Understanding Q1+ and Q0+ from a Truth Table

    I cannot figure out where the values from Q1+ and Q0+ are coming from. What truth table are these coming from?? I would greatly appreciate any help. Thank you.
  49. T

    I Prove Logical Equivalence of P->(Q or R)

    From the text it says (P -> Q) or (P -> R) is equivalent to P -> (Q or R) I tried to see if this is true so I tried (P \to Q) \lor (P \to R) \\ (P \lor \neg Q) \lor (P \lor \neg R) \\ P \lor \neg Q \lor \neg R \\ P \lor \neg(Q \land R) \\ P \to (Q \land R) and P \to (Q \lor R) \\ P \lor...
  50. T

    I How Does the Distributive Law Simplify Logical Expressions?

    Im just reading this one example and i am stumped at this one step. (R\to C) \land (S \to C) \\ (\neg R\lor C) \land (\neg S \lor C) \ \ \ \ \ \textrm{by conditional law}\\ (\neg R\land \neg S) \lor C \ \ \ \ \textrm{by distributive law} I don't understand how it went from the second step to...
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