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. icesalmon

    Engineering What is the output for the following logic circuit?

    I obtained the following result: ([(A xor B) xor 0]* AC)' ([(A'B + AB') xor (0)]*AC)' [([A'B + AB']*(0)' + 0)*AC)]' [(A'B + AB')*(1)*(AC)]' [A'ABC + AB'AC]' [AB'C]' A'+B+C' the solution to this problem is getting a different answer, I don't know why this solution isn't inverting the output AB'C
  2. W

    A Range of values for ##2^{\aleph_0}##

    Ok, so assume we have a model for ZFC where CH does not hold. What values may ##2^{\aleph_0}## assume over said models?
  3. C

    MHB Upper Bound of Sets and Sequences: Analyzing Logic

    Upper bound definition for sets: $ M \in \mathbb{R} $ is an upper bound of set $ A $ if $ \forall \alpha\in A. \alpha \leq M$ Upper bound definition for sequences: $ M \in \mathbb{R} $ is an upper bound of sequence $ (a_n)$ if $ \forall n \in \mathbb{N}. a_n \leq M$ Suppose we look at the...
  4. V

    First Order Logic: ∀x,y a(x) ∧ a(y)

    An example we were given is as follows: {ua|u∈∑*} (where ∑* is set of all words over ∑) so we have ∀x. last(x) → a(x). I am given {awa|w∈∑*} to do, and I know that I have to express that a is the first letter and last letter in a word. Could I write it as: ∀x,y ( a(x) ∧ a(y) ∧ x<y → ∃z(x<z<y))...
  5. entropy1

    I Applying Modus Tollens in Conditional Statements

    I can't figure out this: Say we have event B given α, denoted as {B|α}. If B happens, that implies that R happens: {B|α} → R. Now I want to apply modus Tollens. So if I do, do I get the result: ¬R → {¬B|α}? I mean, I hope I can keep the α unaffected. Is that the case? ¬X meaning X does not happen.
  6. S

    MHB Is Theorem 5.2 in SET THEORY AND LOGIC True or False?

    In the book: SET THEORY AND LOGIC By ROBERT S.STOLL in page 19 the following theorem ,No 5.2 in the book ,is given: If,for all A, AUB=A ,then B=0 IS that true or false If false give a counter example If true give a proof
  7. yucheng

    I Is ##\delta##-steady needed in this proof, given ##\epsilon##-steady

    In Tao's Analysis 1, Lemma 5.3.6, he claims that "We know that ##(a_n)_{n=1}^{\infty}## is eventually ##\delta##-steady for everyvalue of ##\delta>0##. This implies that it is not only ##\epsilon##-steady, ##\forall\epsilon>0##, but also ##\epsilon/ 2##-steady." My question is, why do we need...
  8. yucheng

    B ##A \subset B \iff \exists x \in B \land x \not\in A##?

    ##A \subset B## means that ##\exists x \in B ## such that ##x \not\in A##. Is this logically equivalent to ##\exists x \in B \land x \not\in A##? Formally, $$A \subset B \iff {(\exists x \in B \land x \not\in A)\land(\forall y \in A \land y\in B)}$$ I have tried consulting...
  9. L

    Calculus and Kinematic equations--- seeing the logic

    Details of Question: ds/dt= v which becomes ds=v dt, where s=displacement, t =time, and v=velocity Then we can integrate both sides of this equation, and do a little algebra, and turn the above equation into: s − s0 = v0t + ½at2 My main question is about the integration of...
  10. entropy1

    B Introduction to logic - website?

    Can anyone advise me on a good introductory website about (formal) logic? Preferably about logic in a broad sense.
  11. ThisTimeTomorrow

    I Logic question on relativity- no math

    Checking my understanding. Can it be said that it is the overlap of the reach and effect of each of the 4 forces, from each respective point of origin possible within the universe, that gives us universal general relativity? Like the most intricate gear set ever? Could it then also be said...
  12. S

    Understanding the Distribution of Negation in Propositional Logic

    Given that the negation is distributed across parenthesis, P become ~p and S gets double negation ~~S. Hence my solution was " I will not buy the pants but I will buy the shirt. (or and I will buy the shirt, since but can be used in the place of and). This is from How to prove things by...
  13. C

    Simple Induction Direct Proofs regarding Induction

    Summary:: . When asked to prove by Induction, I'm asked to prove a statement of the form: Prove that for all natural numbers ##n##, ## P(n) ## Which means to prove: ## \forall n ( P(n) ) ## ( suppose the universe of discourse is all the natural numbers ) Then, I see people translating...
  14. C

    I Question regarding quantifier statement

    Suppose I have the following ( arbitrary ) statement: $$ \forall x\in{S} \ ( P(x) ) $$ Which means: For all x that belongs to S such that P(x). Can I write it as the following so that they are equivalent? ( although it is not conventional ): $$ \forall x\in{S} \land ( P(x) ) $$ Can I write...
  15. aspodkfpo

    Weird algebra and logic in a mechanics cart quesiton

    Logic and equations seem to have come out of nowhere in this question. I have been unable to understand where these equations come from and why they are used. Can someone describe the logic for the steps in the question?
  16. AndreasC

    Is Yuri Manin's A Course in Mathematical Logic generally unsound?

    A book that has really caught my attention recently is Yuri Manin's A Course in Mathematical Logic for Mathematicians. I am very interested in the foundations of mathematics and mathematical logic, plus I noticed that it had some chapters on quantum logic, so I started skimming through it...
  17. S

    I Computer languages as examples in formal logic

    A computer language is not a clear example of a formal language, a formal system, or a formal logic. Can we modify the rules for a computer language to create clear examples? My non-authoritative classification of topics in mathematical logc is: 1. Formal languages. A formal language is...
  18. Frigus

    What is the logic behind the mirror symmetry method of solving circuits?

    I recently encountered mirror symmetry method of solving circuits and by it solving circuits became very easy but problem I am facing with it is that I can't figure out logic behind it. For example if we try to simplify this circuit Then we say that if ##I## current flows from Point A to C then...
  19. Furthark

    MHB Attempting to add logic mathematics to a college paper - need your input

    Hello, I am a retired Marine that has decided to pursue an undergraduate degree in programming and am currently working on paper that discusses the future of Mobile Computing. In this paper I am introducing a hypothetical process that assists in forecasting the future of mobile computing. To do...
  20. S

    MHB Logic of Set Operations: Proof

    Given: x\in A\cap B\leftrightarrow x\in A\wedge x\in B x\in A\cup B\leftrightarrow x\in A\vee x\in B x\in A-B\leftrightarrow x\in A\wedge x\notin B A=B\leftrightarrow(\forall x(x\in A\leftrightarrow x\in B)) Then prove using only the above and the laws of logic that: ™ (A\cup B)-(A\cap...
  21. R

    I Who copied the assignment?

    If someone is lying, who copied the assignment? Alex: Cate copied the assignment. Cate: David copied the assignment. David: Cate is lying. Keil: I didn't copy.I think Cate is lying. If Alex is true and there is only one person who is lying, Cate and David can't be true at the same time. If...
  22. nomadreid

    I Quantum logic based on closed Hilbert space subspaces

    One proposal that I have read (but cannot re-find the source, sorry) was to identify a truth value for a proposition (event) with the collection of closed subspaces in which the event had a probability of 1. But as I understand it, a Hilbert space is a framework which, unless trivial, keeps...
  23. J

    Weight Scale Logic: Solving Problems without Calculation

    [mentor note: edited for clarity] The man weighs 100kg. All constructions, ropes, universal joints, rollers, fans, etc are massless. Friction between weight scale and feet or construction is enough to hold side forces... The red line is the rope. What will the weight scale show? <100kg...
  24. jisbon

    Engineering Circuit Analysis Question (Truth Table, Logic etc.)

    Hi all, Having some problems digesting on electric circuits. Below is an example of a question and I would like to ask how do I go ahead in solving this. Firstly, for these types of questions: I have understood how to write a function table, and it goes something like this: Now what I am...
  25. D

    MHB Request to Solve Symbolic Logic Questions Using Strengthened Conditional Proof

    Sir/madam, I request you to solve 2 questions ( q-3 and q-5 ) of symbolic logic ( Strenthened method of conditional proof ). These questions are taken from I.M.Copi's 'symbolic logic' ( edition -5, sec. 3.8, pg- 61 ) File is being attached. thank you yours truly Deep Kumar Trivedi
  26. Mikaelochi

    I Intro to Symbolic Logic: Replacement Rules

    Basically the problem starts with these given premises: 1. ~ (A ∨ (B⊃T)) 2. (A ⋅ C) ∨ (W ⊃ ~D) 3. ~(P ∨ T) ⊃ D 4. ~P ≡ ~(T ⋅ S) And from these premises, I must prove ∴ ~W. This is what I have done so far: 5. (~P ⊃ ~(T ⋅ S)) ⋅ (~(T ⋅ S) ⊃ ~P) B.E. 4 6. ~P ⊃ ~(T ⋅ S)...
  27. nomadreid

    I "Theory" in multi-valued logic?

    As in the summary: is the term "theory" only used for binary-valued logics? If not, how is it defined for multi-valued logics?
  28. R

    Other Intro Logic Books (differences between Peter Smith, J.J. Smith and P.Teller)

    I want to get started with FOL and decided to get through some very basic book first. Currently looking at: A Modern Formal Logic Primer - Teller An Introduction to Formal Logic - P.Smith Logic: The Laws of Truth - J.J. Smith These 3 books are frequently recommended I just don't know which...
  29. S

    What did Hans Bethe think of von Neumann's quantum logic?

    Nobel laureate Hans Bethe was a friend of mathematician-physicist John von Neumann, and he once said: "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man" and "[von Neumann's] brain indicated a new species, an evolution beyond man"...
  30. WMDhamnekar

    MHB Thank you.Best regards,Sriram

    $\left\{ \begin{array}{rcl} x+y+z &=& 1\\ x^2+y^2+z^2 &=& 2\\ x^3+y^3+z^3 &=& 3 \\ x^5+y^5+z^5 &=& ? \end{array}\right.$How to find out $x^5+y^5+z^5=?$
  31. J

    MHB Find Proofs for the following 5 propositional logic statements

    i came acroos the below while studying propositional Logic, can anyone find the proofs 1) P ⊢ P 2) P → Q, Q→R ⊢ P → R 3) P → Q, Q→R, ¬R ⊢ ¬P 4) Q→R ⊢ (PvQ) → (PvR) 5) P →Q ⊢ (P&R) → (Q&R)
  32. S

    A The concepts of "quasimanual" and "manual" in logic

    The paper https://www.whitman.edu/Documents/Academics/Mathematics/klipfel.pdf (beginning page 2``1) describes a model for experiments based (it says) on the book An Introduction To Hilbert Space and Quantum Logic https://www.amazon.com/gp/product/1461388430/?tag=pfamazon01-20. This approach...
  33. Nim

    I Modal Logic - Possibility

    One of the ways we use the term possibility is "We can go to Mars". Another way is "We may learn how to travel to Mars faster than light as our understanding of physics progresses". What exactly does it mean in modal logic? Would a statement like 1+1=2 be considered possible in modal logic or...
  34. P

    MHB Infinite Natural Numbers: First-Order Logic Formula Explained

    There are an infinite number of natural numbers. Why is that? Well this follows from the following facts: (i) There is at least one natural number. (ii) For each natural number there is a distinct number which is its successor, i.e., for each number $x$ there is a distinct number $y$ such that...
  35. P

    MHB Proving First Order Logic in Machover's Text

    Trouble working through Set theory, Logic, and their Limitations by Maurice Machover. Particularly these 1. $\sigma \vDash \alpha \rightarrow \forall x\alpha$ where $x$ does not occur in a free $\alpha$ 2. $\sigma \vDash s_1 = t_1 \rightarrow ... \rightarrow s_n = t_n \rightarrow...
  36. P

    MHB Please help with bool algebra and logic things

    Hello! I'm a beginner in discrete math and don't actually know how to solve the bool algebra and logc problems. Sorry for errors in formulas - it's my first post here. I have some tasks that I want someone could help for me to solve. Step-by-step solutions would be really good, to really know...
  37. S

    MHB "How Much Money Do I Need for Transport to Visit Chris and John?

    If I go to town Iwill visit chris or john. But i own to chris 3 dollars and 5 to john.Hence if I visit them i have to pay my depts. So if I go to town how much money do i have to spend Whatever is your answer prove it
  38. S

    I What did Omnès mean with this?

    Summary: What did Omnès mean with this? I found an old article by Roland Omnès which analyzes the EPR paradox and offers a solution to it (https://www.sciencedirect.com/science/article/abs/pii/0375960189900182). At some point, the article says: "Some macroscopic systems do not satisfy the...
  39. T

    Other Advice on books for logic puzzle questions for software engineers

    Summary: Advice on books for software engineer logic puzzles for interviews and improve overall skills I am looking to getting into software engineering, and am having trouble with find the logic puzzle books or websites to prep for interviews. Now of course there lots of logic puzzle type...
  40. F

    I Is there a violation in logic when we define notions of mass and force?

    Which part of Newton's second law is definition and which part is law content?It seem that there is a violation in logic, because we define the notion of force through the notion of mass, then we define the notion of mass through the notion of force when we consider the second law.
  41. S

    I Is the holographic principle based on fuzzy logic?

    Physicist Craig Hogan has proposed that the universe is based on holographic principle. To prove that the universe is a "hologram" he (and other physicists) have designed an experiment named "The Holometer" to measure quantum fluctuations that would become fuzzy at Planck scale...
  42. J

    A First order logic and set theory: who comes first?

    Goldrei's Propositional and Predicate Calculus states, in page 13: "The countable union of countable sets is countable (...) This result is needed to prove our major result, the completeness theorem in Chapter 5. It depends on a principle called the axiom of choice." In other words: the most...
  43. J

    A Second order logic and completeness

    Goldrei's Propositional and Predicate Calculus states (in my words; any mistake is mine) that first-order logic is complete, i.e. any logic deduction from a set of axioms (written in first-order logic) is equivalent to proving the theorem for all models satisfying the axioms. Completeness is...
  44. K

    MHB Intro to Logic (prove sequents)

    I am stuck on these questions and don't really know how to start/solve them. prove the following sequent: 1. $(\exists x) Fx \to (\forall x) Gx \vdash (\exists x)(Fx \to (\forall x)Gx)$ 2. $(\forall x)(Fx \to (\forall y)\neg Fy) \vdash \neg(\exists x)Fx$ 3. $(\exists x)Fx, (\forall x)(Fx \; à...
  45. N

    Numpy multidimensional array assigment unknown logic error

    I'm confused as to why this bubble sort I'm implementing is setting _all_ items to one of the items (I have no idea which one as array is too big) the data type is of ... [[1128 1026 1192 1023]] [[ 771 195 858 196]] [[ 953 1799 955 1738]]] when I have an array of int, this same algorithm...
  46. E

    MHB Isomorphism of logic, arithmetic, and set theory

    Has anybody ever heard of this? I learned about it in a discrete math class in grad school, and I've never heard of it anywhere else !? For example, logical disjunction (OR) and set-theoretic UNION are isomorphic in this sense: 0 OR 0 = 0. {0} UNION {0} = {0}. Similarly, logical AND & set...
  47. snoopies622

    What is the logic behind Lagrangian mechanics?

    I like using the Euler–Lagrange equations to solve simple mechanical systems, but I'm not perfectly clear on the theory behind it. Is it derived by assuming that action is minimized/stationary? Or does one define a system's Lagrangian according to what makes the Euler–Lagrange equations...
  48. R

    Where is my logic wrong (lottery problem)

    I don't understand why I can't answer this question as a bernuli trial. There are 6 possible correct integers out of 40, and 34 incorrect integers out of 40. I'd assume it would look like this: (6c1)(6/40)(34/40)^5 I guess, it's because when you choose and incorrect or correct integer, the...
  49. A

    Did I find an error in the logic of Evolution with just 2 questions?

    1. If we descended from an unknown common ancestor that was a step up from chimpanzees and bonobo's, then by biological definition, shouldn't chimpanzees and bonobos be much more wise and more intelligent than humans as they are the ones who came first through evolution before our unknown common...
  50. Avatrin

    A Logical foundations of smooth manifolds

    Hi I am currently trying to learn about smooth manifolds (Whitneys embedding theorem and Stokes theorem are core in the course I am taking). However, progress for me is slow. I remember that integration theory and probability became a lot easier for me after I learned some measure theory. This...
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