What is Contradiction: Definition and 259 Discussions

In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's law of noncontradiction states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."In modern formal logic, the term is mainly used instead for a single proposition, often denoted by the falsum symbol



⊥


{\displaystyle \bot }
; a proposition is a contradiction if false can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i.e., a self-contradictory proposition). This can be generalized to a collection of propositions, which is then said to "contain" a contradiction.

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  1. S

    Proofs by contraposition and contradiction

    Homework Statement (a) Prove that if n is an integer and n2 is a multiple of 3, then n is a multiple of 3. (b) Consider a class of n students. In an exam, the class average is k points. Prove, using contradiction, that at least one student must have received at least k marks in the exam...
  2. T

    Angular Momentum Contradiction Due To Choice Of Origin

    Homework Statement Imagine a particle tracing a counter-clockwise circular path on a flat table with a certain speed. The particle is tied with a massless string of length R to a point P at the center of the circular path. Will the particle rotate about P forever at constant speed in the...
  3. M

    Magnetic Field Theory Contradiction? - Repost

    I've been thinking about Magnetic Fields, and I think that I've found a contradiction in conventional physics theory. While comparing the Left-Hand-Rule for Motors (LHR) and the Right-Hand-Rule for Generators (RHR), I found this contradiction: The Left-Hand-Rule states that if the Magnetic...
  4. J

    Contradiction in the definition and properties of the abs

    If the abs(a+b) = abs(a) + abs(b), so the abs(z) = abs(x+iy) = abs(x) + abs(iy) = abs(x) + i abs(y). However, the correct wouldn't be abs(z) = √[x²+y²] ? √[x²+y²] ≠ abs(x) + i abs(y) => abs(z) ≠ abs(z) It's no make sense. What there is of wrong with those definions?
  5. tomwilliam2

    Proof by contradiction for simple inequality

    Homework Statement I'm trying to show that if ##a \approx 1##, then $$-1 \leq \frac{1-a}{a} \leq 1$$ I've started off trying a contradiction, i.e. suppose $$ \frac{|1-a|}{a} > 1$$ either i) $$\frac{1-a}{a} < -1$$ then multiply by a and add a to show $$1 < 0$$ which is clearly...
  6. S

    Artin-Wedderburn theorem contradiction? Mind = Blown

    Homework Statement My question basically wants me to write the direct product of rings R = \mathbb{Z}_3 \oplus \mathbb{Z}_5 \oplus \mathbb{Z}_5 as a direct product of matrix rings over division rings. Homework Equations Relevant theorems http://img713.imageshack.us/img713/8471/g7pn.png...
  7. A

    Contradiction in laws of thermodynamics

    following is what little I know about energy:Energy: its the ability to do the work. First law says you cannot create nor destroy energy but can transform it to from one form to another. so its always 100 % conversation. but in a process you have 'work' done also. so does that mean 'work'...
  8. Z

    Proving Negative Real Numbers Using Contradiction

    Prove by contradiction that a real number that is less than every positive real number cannot be positive having troubles with this could someone give me a small hint to get started?
  9. M

    Proving n^2 is Divisible by 4: Contradiction Method

    Homework Statement Hello Guys, can you check my proof. Problem statement: Let n be an integer such that n2 is even. Prove that n2 is divisible by 4. Proof by contradiction: Suppose n2 is not divisible by 4, thus n is odd. Such that n=2p+1, and n2=4p2+4p+1. Factoring out 2 we have...
  10. T

    Increasing intensity causes a contradiction in classical physics?

    According to my prof, increasing intensity of the light source in a photocell for the photoelectric effect does not increase the kinetic energy of photoelectrons emitted. Instead, the number of electrons emitted (and current) increases. Changing the colour of the light causes an increase in...
  11. F

    Electrical Power Equation Contradiction

    We know that the rate at which electrical work is done (electrical power) is defined as: P = I2*R , or: P = V2/R The formula P = V2/R implies that if the resistance of an electrical component (R) (for example, a light bulb) is decreased, the power consumption (P) will increase hence the...
  12. S

    Contradiction in Wave Amplitude, intensity and Conservation of Energy?

    Hi guys, let's say we have a wave where the power P is proportionate to the square of its amplitude, which is A^2. If now we have 2 identical waves in superposition in phase, then we have an amplitude of 2A am i right? Next, we realize that because of the amplitude of the superposed waves is...
  13. T

    2nd Order ODE Contradiction ?

    2nd Order ODE "Contradiction"? To solve a 2nd order ODE, we can follow the steps as shown below. (Image 2 is a continuation from Image 1, apologies for the size difference.) :rolleyes: The method to obtain the solution is straightforward. Let's say \frac{d^2y}{dx^2}=ky If k = -1, a...
  14. X

    How to resolve the contradiction in twin clocks?

    I am pretty confused in the following situation: Two identical clocks moving at a constant speed v from each other in x-direction. If each clock is made up of a ball moving at a constant speed of 1 on a ruler in y-direction, then the position of the ball of a clock is the time of the clock...
  15. M

    MHB Troubling contradiction in Functional Analysis

    Hello I was doing an exercise that said: "If $P$ is a continuous operator in a Hilbert space $H$ and $P^2=P$ then the following five statements are equivalent". The first statement was that P is an orthogonal projection. Now this was suposed to be equivalent, under the condition of $P^2=P$, to...
  16. X

    Electricity contradiction? Which one is right

    Homework Statement The potential at a point is 20 V. Calculate the work done in bringing a charge of 0.5 C to this point. Homework Equations V = Ee / q W = -Ee The Attempt at a Solution Ee = 10 J W = -10 J But the answer is 10 J. Why isn't it negative? It's contradicting...
  17. F

    Faraday's law and gauss's law for magnetism apparent contradiction

    I was reading here https://www.physicsforums.com/library.php?do=view_item&itemid=294 gauss law's for magnetism says ∮B⋅dA = 0 but then faraday's law has d/dt ∮B⋅dA in it. Well if its 0 then d/dt of 0 is 0.
  18. Y

    Proof by Contradiction: Showing x ≤ 1 for x∈ℝ+ and t∈T

    Homework Statement Please check that the proof is correct or not. Let ℝ+ = {x\inℝ: x>0} and T = {x\inℝ: 0<x<1}. Let x∈ℝ+ and t∈T Prove: If x\leq xt then x\leq 1. * You may assume any common properties of log(x) as well as : if 0<a\leq b then log(a) ≤ log(b) Any help is appreciated...
  19. C

    Can Proof by Contradiction Be Validated through Logical Equivalences?

    Homework Statement Prove that if x^2 + y = 13 and y \neq 4 , then x \neq 3 .Homework Equations N.A.The Attempt at a Solution The proof itself is simple enough: suppose x^2 + y = 13 and y \neq 4 . Suppose for the sake of contradiction that x = 3 . Then \begin{align*} (3)^2 + y &= 13...
  20. Q

    Phase velocity contradiction

    So for the phase velocity of a massive particle we have Vph = Vg/2 for non-relativistic case Vph = c^2/Vg for the relativistic case Vg is the group velocity or particle velocity But there seems to be a contradiction in that for the non-relativistic case the phase velocity is...
  21. D

    Potential Energy of a Spring Contradiction?

    Hi everyone, I'm a bit confused about the concept of the potential energy. Let's say we have the following scenarios here: https://dl.dropbox.com/u/29312856/Springs.jpg In the first scenario, we have two identical springs with spring constant 5000 N/cm angled at 20 degrees below the...
  22. P

    MHB Understanding Proof by Contradiction in Logic and Mathematics

    Is this how proof by condraction works? Say we want to prove A-> B. We prove by showing the statement 'A and not B' implies some statement C which is false (since it contradicts a known fact). Therefore, anything which implies C must itself be false, so 'A and not B' is false. I.e. A implies B.
  23. K

    Proving: Proof by Contradiction

    I'm reading through a text's proof on proof by contradiction. But it makes inexplicable jumps and doesn't appear to use some of the things brought up. Here's the theorem and proof in the text (shortened with comment). \mbox{Theorem: If } \Sigma \cup \{\lnot P\} \vdash \{Q\} \mbox{ and...
  24. K

    Contradiction between equations of angular momentum

    Homework Statement Two equal, parallel and opposite forces at at both sides of a horizontal disk that lies on a smooth table, according to the picture. The mass is m and the moment of inertia is: kmR2 Angular momentum round the center point A: 2FR=kmR^2 \cdot \alpha. Angular momentum round...
  25. A

    Proving √n Irrational: A Proof by Contradiction

    The problem reads as follows: Let n be a positive integer that is not a perfect square. Prove that √n is irrational. I understand the basic outline that a proof would have. Assume √n is rational and use a proof by contradiction. We can set √n=p/q where p and q are integers with gcd(p,q)=1...
  26. H

    Proof by Contradiction || Prove that an equation can never be a square number

    Homework Statement Prove that for any integer n n^2+n+1, can never be a square number. Homework Equations None. The Attempt at a Solution We could put the equation to a^2, (where a^2 is a square number) and solve for n and show that n can not be an integer. I tried quadratic...
  27. A

    Prove by Contradiction: For all integers x greater than 11

    Homework Statement Prove by Contradiction: For all integers x greater than 11, x equals the sum of two composite numbers. Homework Equations A composite number is any number that isn't prime To prove by contradiction implies that if you use a statement's as a negation, a contradiction...
  28. J

    Contradiction between Henry's Law and Le Chatlier's Principle

    Le Chatlier's Principle is used to determine the direction of a reaciton based upon a stress put on the system. In addition, Henry's law states that the solubility of a gas is related to the partial pressure of that gas. Therefore I present a seemingly contradictory setting: Example for...
  29. A

    Prove by Contradiction: For all Prime Numbers a, b, and c

    Homework Statement Prove by Contradiction: For all Prime Numbers a, b, and c, a^2 + b^2 =/= c^2 Homework Equations Prime number is a number whose only factors are one and itself. Proof by contradiction means that you take a statement's negation as a starting point, and find a...
  30. C

    Proof by contradiction for statement of the form P->(Q and R)

    Say I have a statement like this: P implies (Q1 and Q2). If I wanted to prove this by contradiction, I would assume P and not(Q1 and Q2)=[(not Q1) or (not Q2)] both hold, and try to find a contradiction. My question is... Am I done if I find a contradiction while assuming P and [(not Q1)...
  31. G

    Simple Apparent contradiction?

    I came across this when doing another problem: Suppose we have 2 numbers, (a+b) and (c+d), which both equal 0. a+b=0 c+d=0 Then a+b=0=c+d, Thus, a+b=c+d However, a+b+c+d=0 Thus, a+b=-c-d Therefore, a+b=c+d AND a+b=-(c+d) How is this possible?
  32. M

    Confusing Contradiction: dE-TdS+PdV=0?

    My textbook says that dE-TdS+PdV<0 for all irreversible processes. However, for reversible processes, the author says that dE-TdS+PdV = 0, and in fact this equation can be applied to irreversible processes because E is a state function. I'm confused -- this seems to be a contradiction.
  33. T

    Divisibility rules and proof by contradiction

    I posted this in the number theory forum to no success... so I figured maybe the homework help people would have some input Let x,y,z be integers with no common divisor satisfying a specific condition, which boils down to 5|(x+y-z) and 2*5^{4}k=(x+y)(z-y)(z-x)((x+y)^2+(z-y)^2+(z-x)^2) or...
  34. T

    Trouble with deducing the contradiction

    Let x,y,z be integers satisfying a specific condition, which boils down to 5|(x+y-z) and 2*5^{4}k=(x+y)(z-y)(z-x)((x+y)^2+(z-y)^2+(z-x)^2) or equivalently 5^{4}k=(x+y)(z-y)(z-x)((x+y-z)^2-xy+xz+yz) I want to show that GCD(x,y,z)≠1, starting with the assumption 5 dividing (x+y), (z-y), or...
  35. A

    Justifying Proof by Contradiction

    It seems to me (though I would be *extremely* glad to be proven wrong here) that in mathematics we often blindly assume that the theorems we attempt to prove/disprove are either true or false. Such an assumption is implicit in every proof by contradiction. We eliminate the possibility of the...
  36. T

    Bertrand's and Earnshaw's theorems contradiction

    I think the title is self-explanatory. The first theorem states that gravitational forces (1/r potentials in general) are able to produce stable orbits, whereas the second excludes stability! Can somebody help me to clear this out?
  37. C

    How Many Cases Do I Need to Consider for Proof by Contradiction?

    Homework Statement Suppose I want to prove the following statement by contradiction: P \longrightarrow (Q \land Z) Homework Equations If (Q \land Z) is false, then either: (i) Q is false and Z is true; (ii) Q is true and Z is false; (iii) Q and Z are false. The Attempt at a...
  38. J

    Apparent contradiction: Two charges moving parallel to each other

    Hello I need help with a problem: If I see two identical charges moving in the same direction parallel to each other with the same constant velocity, my intuition tells me that the magnetic field generated by their movemente will cause them to attract much like what happensa with two wires with...
  39. F

    Conservation of momentum and energy contradiction?

    Suppose we have two objects and we're only talking about rectilinear motion. Initially, one object has mass m and is moving at velocity V. The other has mass M and is standing still. Then they hit each other and suppose that all kinetic energy is conserved and they stick together and move at...
  40. B

    Gauss's law and surface electrons on a charged conductor, contradiction?

    My textbook (Halliday & Walker) explains that a charged conductor (a solid, of an arbitrary shape) in electrostatic equilibrium will have the electric field inside be 0 and all electrons will be on its surface. It proves this by saying that if the electric field inside was not 0, the free...
  41. M

    Ostensible Contradiction b/w Continuity & Cartan's Magic Formula

    Continuity equation is dj+\partial_t\rho_t=0 where j and \rho are a time-dependent 2-form and a time-dependent 3-form on the 3-dimensional space M respectively. (see e.g. A gentle introduction to the foundations of classical electrodynamics (2.5)) If we use differential forms on the...
  42. X

    Can I prove the statement if n^2 is odd, then n must be odd by contradiction?

    I never understand the proof by contradiction, because somewhere in the middle I always lost myself. In this https://www.physicsforums.com/showthread.php?t=523874 there's an example of proof by contradiction. We assume that if n^2 is odd than n is odd. This means that if n^2 is even, n...
  43. P

    Proving Statements by Contradiction: Understanding the Logic Behind It

    Hi, I have a question about proofs by contradiction in general. Without getting into the mathematical details, suppose we had the statement: For every (condition A), B is true. If we want to prove this by contradiction, we want to assume the negation of this statement, and then prove it to...
  44. I

    Proof by contradiction,

    Homework Statement If a,b and c are integers and a2+b2=c2, then at least one of a and b is even. [b]2. Contradiction statement There exist an integer a,b,c such that a2+b2=c2 and a or b is odd The Attempt at a Solution I am not sure if my contradiction statement is correct...
  45. O

    Contradiction of statement regarding monotonicity

    Hi all! We were given to proove or falsify the following statement: Given f(x)>0 \,\ ,\,x>0 \,\,\,\,,\lim_{x\to\infty}f(x)=0 Then f(x) is strictly decreasing at certain aεℝ for every x>a Now in their solution they contradicted the statement with: \newcommand{\twopartdef}[4] {...
  46. D

    Is there a contradiction here.

    list of contradictory situations. 1. Age of universe is determined by rate of proton decays at some place taking into consideration half life of proton. But since ALL protons were created after big bang, how can protons even decay. Considering the fact that their half life is greater than...
  47. S

    Contradiction in Common BJT Configuration: KVL vs Early Effect

    Now as you can see in the picture a common base configuration of BJT. The green markings in the picture are resistances. :p My question is: By KVL : VBB = VBE + IE*RB which means that IE is esentially a constant because VBB , VBE, RB are constants. Now If Voltage Vcb is increased Early...
  48. M

    This is the contradiction that proves that l = m.

    Prove: If f approaches l near a and f approaches m near a, then l = m. ...Im skipping to the end of the proof... " to comlete the proof a particular ε>0 has to be choses for which the two conditions |f(x) - l|< ε and |f(x) - m|< ε cannot both hold if l=/=m." if l=/=m so that |l - m|> 0...
  49. S

    How to do a delta-epsilon proof by contradiction?

    Homework Statement Prove that lim x→1 of x2 does not equal 1+10-10. You could use a proof by contradiction. (It is question 2.b here) Homework Equations δ-ε proofs! The Attempt at a Solution Given ε > 0, there is some number δ > 0 such that if: |x - a | < δ |x - 0 | < δ |x| < δ Then: |...
  50. khurram usman

    Lim x->0 (sin x / x) =1 contradiction?

    lim x->0 (sin x / x) =1...contradiction? sin(x)/x =1 (limit x to 0) this is an identity proved by using geometry and squeeze theorem ...right? now today i came across another question and doing it my way ...gives me two answers;) the question is limit x-->0 of [ x*sin(1/x)] my first...
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