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

    Logic axiom of simplification.

    Hello all I cannot find a simple explanation of the meaning of this axiom, probably because it is considered so obvioius that it needs no explanation. Can anyone explain in words. {a}\rightarrow{({b}\rightarrow{a})} Thanks. Matheinste.
  2. S

    Ask for Mathematical Logic Solutions

    Homework Statement 1.Assume the language has equality and a two-place predicate symbol. Given two structures (N;<) and (R;<), find a sentence true in one structure and false in the other. Can these two structures be elementarily equivalent? Can they be isomorphic? Why or why not? 2.The...
  3. C

    Logic gates (and and nand)

    Homework Statement Using only AND and NAND gates, design a circuit in which 4 switches must all be turned on before a switch in another circuit turns off. i think half the reason i don't understand may be the wording of his question, i know the tables but i am not sure how the logic gates...
  4. H

    Help with modal logic exercise

    I'm teaching myself modal logic, and I'm curious about the following exercise, taken from this book... Homework Statement Prove T:\; \Box A \rightarrow A using the following rules: Homework Equations Given the structure M=<W,R,P> in which W and P are as they are in a model, and R...
  5. T

    Logic: Deductive Science of All Possible Worlds

    How is this definition ( I made up): Logic is the deductive (as opposed to empirical) science of all possible worlds.
  6. T

    Uncovering the Truth: A Logical Approach to Determining Football Game Time

    Homework Statement You ask your friend if the football is on 9pm or 10pm. He sometimes tells the truth and sometimes doesn't. What should you ask him so that you will be certain which time the football will be on? The Attempt at a Solution ASK: Is it true that 'You are telling the truth...
  7. Shackleford

    Where Can I Find a Quality Logic Textbook for My Physics Degree?

    I'm not squeezing this into my degree (Physics), but I'd like to buy a good Logic textbook online or whatever. Thanks.
  8. K

    Full Adder Sum Formula (Logic)

    Hi, I've been trying to work out the formula for the sum for the full adder logic, however have come across a gap which I don't know how to fill. S = (¬A.¬B.C) + (¬A.B.¬C) + (A.¬B.¬C) + (A.B.C) S = ¬A.(¬B.C + B.¬C) + A.(¬B.¬C) S = ¬A.(B \oplus C) + A.( do not know what to do at this point to...
  9. T

    Modern 1st Order Logic Book: Clear Exercises & Solutions

    What would be such a book which is modern and clear with good excercises and solutions?
  10. H

    Does Proving a Contradiction Imply a Logical Relationship?

    Homework Statement Prove or disprove: "If you can prove ( y \wedge \neg c ) \rightarrow Contradiction , then y \rightarrow c must be right."Homework Equations My teacher used the sign \wedge , instead of \vee , like: "If ( a \wedge b \wedge \neg c ) \rightarrow Contradiction , then...
  11. Z

    Prove NOR is Complete: Logic Problem Solved

    how could i express using only the operator NOR (in logic) the rest of operation NOT(x) AND(x,y) OR(x,y) that is how i could prove that the Logic operator NOR is functionally complete
  12. H

    Prove that the law of excluded middle does not hold in some many-valued logic

    Hi, all. Wikipedia says: In logic, the law of the excluded middle states that the propositional calculus formula "P ∨ ¬P" ("P or not-P") can be deduced from the calculus under investigation. It is one of the defining properties of classical systems of logic. However, some systems of logic...
  13. J

    Proving Math Logic Problem: Puzzle 8 Configuration

    Homework Statement I am supposed to prove that if you have a puzzle 8 in this configuration A B C D E F H G I have to prove that no matter how many moves you make (as you would a normal puzzle moving one of the adjacent...
  14. M

    Design Logic Question: 3 Inputs, 3 Outputs | 2 Inverters

    Helloo every one, here is my qusetion: Design acircuit that has three inputs a,b,c and has three outputs a' ,b', c' . your circuit can only have two inverters and any number of AND and OR gates if some 1 could help me i'd appreciate it.
  15. T

    Solving Sum of n+1^n/n^(n+1) - Diverges?

    Can I use this logic? Homework Statement I'm wondering if I can use this kind of logic to solve: \sum\frac{(n+1)^n}{n^{(n+1)}} Converges or diverges The Attempt at a Solution \frac{(n+1)^n}{n^{(n+1)}} \geq \frac{(n)^n}{n^{(n+1)}} And \frac{(n)^n}{n^{(n+1)}} = n ...
  16. J

    How Do I Build a Logic Gate for a Voting System?

    Homework Statement I'm studying logic gates and trying to create my first one. Homework Equations 3. In a TV programme, the panel votes on new performers. The panel consists of a Chairman (C) and three others (X, Y and Z) who each press a button to register their vote. The SUCCESS light is...
  17. K

    Proving the Occurrence of Coinciding Clock Hands at Noon and Midnight

    Hi. So a classic logic puzzle goes like this: At noon the hour, minute, and second hands coincide. In about one hour and five minutes the minute and hour hands will coincide again. What is the exact time (to the millisecond) when this occurs. (Assume that the clock hands move continuously.) If...
  18. B

    Engineering Double negation in logic circuit

    Probably a very simple answer here... I'm creating a logic circuit using only NAND gates. I have 4 inputs, A B C and D, but I am just using a single 'path'/part of the equation for this question, that being A'B = F, F being the output. Full equation... AB'C + AB'D' + A'B + A'C = F F...
  19. T

    Logic Order of All Physics- Help

    Can someone tell me what order someone should learn physics until they can say that they totally know physics (please don't write something like you can never learn everything in physics) Thanks!
  20. W

    What is a good logic book for a newbie?

    I`m studying Engineering -not mathematics-, so what is a good book to start with?
  21. M

    Theory behind a logic puzzle ?

    Hi, I am helping my children with a computer game based on Logic and Reasoning. The player has to solve different puzzles as the game progresses. I am struggling with a matching puzzle where the player has to form a string of the game characters based on their traits. Please see the attached...
  22. J

    DeMorgans laws and rules of logic

    Homework Statement (p -> q) has an unambiguous meaning both in logic and in natural language. The DeMorgans laws tell us what is meant by the negation of a conjunction or the negation of a disjunction, but what is the negation of a conditional such as p -> q? Use the rules of logic to produce...
  23. T

    I cant see the logic of thes definitions

    http://img21.imageshack.us/img21/3070/11650282.th.gif on the limsup case each time we take out the biggest member of the sequence . so it goes to the smallest memeber how its supposed to be the supremum of the limits
  24. K

    Predicate Logic and Relations on sets

    Homework Statement Here are a few questions from an exercise sheet that I need help on. I really don't have a clue on how to start them. Could anyone help me attempt at each a) for each question? 1. Use (nested) quantifiers (∀ and ∃) (and propositional junctors) and only equality ``=''...
  25. J

    First-order logic without sets?

    I have been trying to study first-order logic to have a sound basis on mathematical language. The main target is to have a clear path: I start with first-order logic (the language), then I go and study set theory, which is in fact a series of axioms (ie, a series of statements of the language)...
  26. T

    I cant follow the logic of this proof

    W_1 and W_ 2 are subspaces of V of inner product V. prove that if W_1\subseteq W_2 then W_1^\perp \supseteq W_2^\perp the proof is: we take v\epsilon W_1 so <v,w>=0 for every w\epsilon W_1^\perp and because W_2^\perp \subseteq W_1^\perp (i can't see why the "viven expression is...
  27. H

    Is S^{c} the key difference in this logical equivalence?

    Is it equivalent? ( \forall x \in S \forall y P(x) ) <=> \neg ( \exists x \in S^{c} \exists y \neg P(x) ) Attempt at solution I think it should be ( \forall x \in S \forall y P(x) ) <=> \neg ( \exists x \in S \exists y \neg P(x) ) The diiference to the above statement is S^{c}.
  28. S

    Proof of a simple logic statement

    Homework Statement \forall x \in S <-> \exists x \not \in S The Attempt at a Solution The statement is clearly false. I will try to show that by the proof of contradiction. Let P: \forall x \in S and Q: \exists x \not\in S The negation of Q is negQ: \forall x \not\in S and the negation...
  29. C

    Showing linear independence, correct logic?

    Homework Statement Let u and v be two nonzero vectors in R^2. If there is no c E R such that u = cv, show that {u, Bv} is a basis of R^2 and that R^2 is a direct sum of the subspaces generated by U = <u> and V = <v> respectively. Homework Equations Clearly, u and v are linearly...
  30. G

    What is the difference between digital and integrated circuits?

    Hi, I'm trying to build a digital circuit that will perform one of three logic functions on a set of three inputs A B C based on the input value of a counter that counts mod-3 (00, 01, 10). I can use basic chips such as mux, demux, 555 timer, logical chips Thanks
  31. T

    What are the negations of complex statements in statement logic?

    1. Negate the following statements (i): \forallx\inR \existsy\inR such that x+y=0 (ii): introduction: Each of us got, let's say, 20 bags of green apples. Actual Statement: At least one of us (each) found at least one red apple in at least one bag (each). (Each person and each one of it's...
  32. Shackleford

    Courses Should I take a Logic I course?

    I'm trying to pick my last elective. My others are PDEs. What's a Logic I course like? I'm considering this one course/professor, and he has good reviews on ratemyprofessor for this course.
  33. A

    Logic Question: Mathematician's Life Depends on Next Statement to Cannibals

    Honestly, I have no clue how to answer this question. A professor sent this out in a email, but I don't have a class with her so I don't think its homework. Care to throw some ideas? Is it really simple...? Apparently its a logic question, but it seems illogical to me. "A mathematician is...
  34. B

    Top PhD Programs in Mathematical Logic in the US

    I cannot find any sort of comprehensive list of top schools in mathematical logic. It seems that U Wisconsin should be good, though I don't know if that is the case now that Barwise has passed. UC Berkeley is clearly a good school for logic. All of the places that show up are in Europe. I would...
  35. R

    Discover the Best Books on Symbolic & Pure Mathematical Logic

    I'm trying to find a good book on Symbolic, Pure, Mathematical Logic. Anyone know such a book? I'd prefer it have no mention to number theory, set theory, etc. since I have books on those already and I find that they detract value from the books since it's just less time spent on the main topics.
  36. B

    Good books in Set theory and Mathematical Logic

    I am more precisely looking for a book on mathematical logic which presupposes only minimal exposure to set theory. Preferably something which includes an introductory chapter delineating relevant set theoretic principals. I am familiar with only basic set theory. More precisely this means...
  37. Y

    Math High impact on society, reality tested, and heavy use of logic and symbolic math

    I am looking to make a career shift. I am looking for a career track with the following characteristics: 1) Heavy use (and need to learn) advanced mathematical concepts (especially symbolic in nature). I want to be working with equations and logic in a symbolic manner on a daily basis. 2) A...
  38. J

    Having trouble with Logic Derivation -

    Hey everyone, Thanks for the help in advance. I am having trouble deriving a tautology from no premises. I am trying to derive: (X <-> Y) v (X <-> -Y) "-" meaning not. However, I keep getting stuck. It seems very similar to the Law of the Excluded Middle but I am having trouble...
  39. E

    Convergent series. Is my logic correct?

    Show \sum_1^\infty\frac{x^n}{1+x^n} converges when x is in [0,1) \sum_1^\infty\frac{x^n}{1+x^n} = \sum_1^\infty\frac{1}{1+x^n} * x^n <= \sum_1^\infty\frac{1}{1} * x^n = \sum_1^\infty x^n The last sum is g-series, converges since r = x < 1
  40. T

    Solve Digital Logic Homework: Simplify w/ K-map, Sum of Prods & NAND Gates

    Homework Statement Using k-map simplify the boolean function: f(w, x, y, z) = \prod(1, 3, 4, 5, 7, 11, 12, 13, 15) into sum of products form product of sums form implement using nand gates Homework Equations The Attempt at a Solution I have the k-map, I understand that...
  41. M

    Solving a Logic Circuit Puzzle: Minimizing to SOP Form

    Homework Statement Here is an image of the problem (Write the circuit in min SOP form.) http://img211.imageshack.us/my.php?image=ddtl0.jpg Homework Equations The Attempt at a Solution I've looked at it and gone over notes but I don't get it. If you would like I can scan my...
  42. T

    Validity SD Logic Homework Question

    Show that the following argument is valid in SD. (I will use "⊃" to show conditional.) (E v (L v M)) & (E ≡ F) L ⊃ D D ⊃ ~ L ---------------------- E v M I'm only allowed to use the basic derivation rules of SD: Reiteration (R) Conjunction Intro. (&I) and Conjunction Elim. (&E)...
  43. T

    SD Logic Homework (Derivability)

    The question is: Show that the following derivability claim holds in SD. (I'll use ">" to stand in for conditional.) {(A > F) & (F > D), ((M v H) v C) > A, ~(M v H) & C} entails D I'm only allowed to use the basic derivation rules of SD: Reiteration (R) Conjunction Intro. (&I) and...
  44. N

    Quantum logic gate measurement?

    Do quantum logic gates perform measurements? (collapse the wavefunction) For example: If I apply a cNOT gate to a pair of cubits with wavefunction 1/\sqrt{2} |10> + 1/\sqrt{2} |00> what would I expect as the result? 1/\sqrt{2} |11> + 1/\sqrt{2} |00> ? or 1|11> or 1|00> or...
  45. J

    Help with logic proof (this should be really basic)

    help with logic proof (this "should" be really basic) Hey I am new to logic (I've just started a masters) I can't for the life of me prove this using only the natural deduction inference rules: A --> B |- ¬A v B I can do it easily in the other direction but I cannot do it in the above...
  46. S

    Multiple Choice question about Proposition Logic

    The question is here: http://dl.getdropbox.com/u/175564/logic.png The correct answer is D. My attempt to the problem: Let the beginnings of the sentences be A. In a, A is B -/-> C is D. In b, A is B -/-> C is D. These first two do not make sense. in c and d, the word "only" suggests...
  47. R

    Set Theory Book Reviews: Halmos Edition

    I'm looking for a book on Set Theory, currently. I've found one by Halmos which looks good, but I'd like some input on it.
  48. K

    Who Started Logic: Plato or Aristotle?

    Was it Plato or Aristotle? I don't need to be told that logic was an evolution that was merely advanced by these two Greeks. But who really gave us the rules of the game? I've read some of Plato. Not much, but from his dialogues I gather more ethics and "what does what mean" sort of...
  49. D

    What is Hennessy-Milner Logic and Where Can I Find References?

    Homework Statement What is Hennessy-Milner logic? I need references asap. Wikipedia is empty. Homework Equations The Attempt at a Solution
  50. M

    Does a Valid Rule of Inference Always Lead to a True Conclusion?

    Must a valid rule of inference always lead to a true conclusion?
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