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. J

    MHB Predicate Logic: Semantics and Validity

    Hello, Given the domain as: D = {a,b}; ~Ba & Bb & Laa & ~Lab & Lba & ~Lbb Why is the interpretation false? (∀x)[Bx ⊃ (Lxx ⊃ Lxa)] I am having trouble understanding why that is the case because (Lxx ⊃ Lxa) evaluates to true in any case as long as Lxa is true in all cases, so the overall...
  2. nomadreid

    I Lattice of truth values for a paraconsistent logic?

    Usually the truth values of propositions of a logic are structured into a lattice, with 0 (False) on (say) the bottom and 1(True) on (say) the top, and the connecting lines being implication. In paraconsistent logics, there is at least one node which is not implied by 0. Can one safely say that...
  3. nomadreid

    I Cardinality of a set of constant symbols (model theory)

    First, I want to be pedantic here and underline the distinction between a set (in the model, or interpretation) and a sentence (in the theory) which is fulfilled by that set, and also constant symbols (in the theory) versus constants (in the universe of the model) Given that, I would like to...
  4. R

    Can random, unguided processes produce a rational brain?

    I am fascinated by Einstein’s quote that the most unbelievable aspect of the universe was that it was intelligible. So my question is does anyone know whether it is so unlikely as to be absurd to suppose that random unguided processes could produce a rational brain in man in as little as 3...
  5. GlassBones

    Prime factors of odd composites

    Homework Statement Let ##n## be odd and a composite number, prove that all of its prime is at most ##\frac{n}{3} ## Homework Equations Some theorems might help? Any ##n>1## must have a prime factor if n is composite then there is a prime ##p<√n## such that ##p|n## The Attempt at a Solution...
  6. R

    Is Validity the First Step in Evaluating Arguments?

    My school uses Rosens book which is an awful book, would not recommend. Can anyone please help me solidify if what I am saying is true: If you are presented an argument, to even take this argument into consideration, you must prove whether the argument is valid or not. After this is done, you...
  7. A

    MHB Solutions book to Copi, "Symbollic Logic"

    Does anyone know how where one may find a book of exercise solutions to Copi, "Symbolic Logic", 5th edition. Since these are probably the same across all editions, another one may be acceptable. Thanks for all help.
  8. YoungPhysicist

    B How Do You Solve This Mysterious Number Puzzle?

    A logic puzzle I just found online: $$ 5+3+2 = 151022\\ 9+2+4 = 183652\\ 8+6+3 = 482466\\ 5+4+5 = 202541\\ 7+2+5 = ? $$ [/SPOILER] Edit: The answer to the problem is not 14. Please post how long did the problem take you to solve:wink:
  9. L

    MHB Understanding Ke Logic Rules & Finding Contradictions

    Hi, the question and Ke logic rules are attached. This is my attempt at the question. $1. P \land (R\implies Q) $ Premise $2. ( P \land Q ) \implies \lnot S) $ Premise $3. ( P \land S) \implies R) $ Premise $4. \lnot S $...
  10. F

    Engineering Design a logic circuit to add two 2-BCD decade numbers

    Homework Statement Homework Equations - The Attempt at a Solution Here's my work : When I added 0000 0001 to 0000 0000 , I didn't get the correct answer . Could someone check where is my mistake please ?
  11. Auto-Didact

    Logic as a dynamical system?

    This thread is a shoot-off from this thread. Assuming some relation between human language and logical reasoning, how would this relate, let's say, to the arrival and evolution of logic in human and animal psychology? I would presume that some logic, for example classical logic, can be more or...
  12. Q

    I Predicate logic and modal logic

    I have a question about predicate logic and modal logic. Namely, do any of them overlap with one another? To give an example, does existential quantification apply to counterfactual statements? A counterfactual statement can be something like "A possible world where I won the lottery." I...
  13. F

    Engineering Combinational Logic Circuits using Logic Gates

    Homework Statement Design a circuit with four inputs and two outputs which realizes the functions : f1 = Σm(0,2,4,5,6,14) f2 = Σm(0,1,4,6,12,14) Homework Equations - The Attempt at a Solution Here's my work : 2 levels (NAND / OR ) 10 gates 17 gate inputs 2 levels ( NAND / OR ) 7 gates 11...
  14. nomadreid

    I "Laws of Form" by G. Spencer-Brown (1969)

    I have received (unasked) a digital edition of "Laws of Form" (1969) by G. Spencer-Brown; I have glanced at it, and also at the Wikipedia article https://en.wikipedia.org/wiki/Laws_of_Form. OK, another logical system; logical journals (e.g. by ASL) are full of them, and I am not sure whether...
  15. F

    Engineering Write the Boolean expression of a given logic circuit

    Homework Statement In the logic circuit shown below , write the Boolean expression for the output function X . Which gate(s) could be removed without affecting the function X ? Homework Equations - The Attempt at a Solution [/B] So , we can get rid of ##A \overline{B} C ## . gate # 4. Right ?
  16. T

    I Truth, lie and random confusion

    So I am very, very new to logic based questions, and in the past have solved some with relative ease but whilest scrolling through the next to find some example stuff I came across a website that gives a question and hints to the question if stuck, so I thought this would be good practice. But...
  17. entropy1

    B Does retrocausality follow from logic?

    I can't remember where this subject came forward in my topics, so I created a new topic. Suppose that: If X happens, we observe A, and: If Y happens, we observe B. Could we then say: If we observe A, Y did not happen, and: If we observe B, X did not happen, if we apply this to...
  18. C

    B Does anybody think that there is an underlying reality?

    It seems like we keep chasing "reality", and by "reality" I think Physicists would mean the apparent rules of quantum physics which we hope would (if applied) lead to all the apparent known rules of macro-physics. However ... It seems like we have had to create a few things to do that: 1. Ideas...
  19. Avatrin

    I Understanding Implication Problems: Examples & Explanations

    I just saw this proof... And, I don't understand why this is true. How am I supposed to think about problems like this one? Edit: Here's another one: The only steps here I understand are 1 to 5. I don't know why 6 and 7 are true.
  20. W

    How to Interpret the Basic Logic Problem with Predicate P(x,y,z)

    Homework Statement I refer to part G of this little problem: I don't see how to arrive at any conclusion, especially when I can't even see how ##z## comes into play. Assistance in interpreting the problem is appreciated!Homework EquationsThe Attempt at a Solution I know that the answer for...
  21. A

    MHB Help with Logic Proof: Establishing 7.

    Consider the following proof from Copi's "Symbolic Logic", p. 109: 1. (∃x) ¬Fx Assumption 2. ¬Fy Assumption 3. (∀x) Fx Assumption 4. Fy...
  22. E

    Proof by Induction of shortest suffix of concatenated string

    Homework Statement Wherein α, β are strings, λ = ∅ = empty string, βr is the shortest suffix of the string β, βl is the longest prefix of the string β, and T* is the set of all strings in the Alphabet T, |α| denotes the length of a string α, and the operator ⋅ (dot) denotes concatenation of...
  23. S

    A Multiverse theory with impossible universes?

    I found an article written by physicist George Ellis that confused me a little. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.498.4569&rep=rep1&type=pdf At some part, he says: 3.2 Non-uniqueness: Possibilities There is non-uniqueness at both steps. Stating “all that is possible...
  24. nomadreid

    I Why the emphasis on first-order logic?

    It seems to me that, despite several systems developed for higher-order logics, almost all the attention in Logic is devoted to first-order. I understand that higher-order logics have some drawbacks, such as the compactness theorem and the Löwenheim-Skolem theorems and other such not holding in...
  25. Posty McPostface

    I Loophole in Godel's Incompleteness Theorem?

    Gödel's incompleteness theorem only applies to logical languages with countable alphabets. So it does not rule out the possibility that one might be able to prove 'everything' in a language with an uncountable infinite alphabet. Is that a loophole in Godel's Incompleteness Theorem? Doesn't...
  26. Delta2

    Ohmic Resistance and Logic gates

    Is ohmic resistance a "necessary evil" in order for transistors to be able to function as logic gates? I mean, I have seen some circuits for the NAND gate in RTL and TTL and they both seem to involve ohmic resistances. Can we make a super conducting NAND gate that will have total ohmic...
  27. T

    MHB Tricky Logic Puzzle with 26 Variables

    Can you figure out what the answer of (x-a)(x-b)(x-c)...(x-z) is? This problem seems very tricky and you might think you need to expand one by one, but if you think carefully, you will find out that the answer is very simple! Solution:
  28. C

    MHB Trouble determining truth value of logic statements

    For the folowing two problems determine the truth value of each statement: assume a and b are true and c and d are false. not(a V b) -> s not( T V T) -> s F -> s T r -> [(d -> w) <-> (a ^ c )] r -> [(F -> w) <-> ( T ^ F)] r -> [T <-> F] r -> F I am fairly certain I did the first one correct...
  29. Wi_N

    Who wants to be a Millionaire logic question....

    So i was thinking if i was a contestant and i got a question i had no clue what the answer was. and i wanted to use Ask the Audience and 50-50 lifelines, which one should I pick first? Is there a math logic to this? Lets pretend its a question very doubtful the majority of the audience would...
  30. akaliuseheal

    Logic gate question -- Tying both inputs of a gate together

    Hello, I don't know much about logic gates and how they work. I was wondering if this (Image) can work. (Is something like this possible?) The idea is that if we have logic 1, AND gate gives 1 while NOR gives 0. And vice versa.
  31. J

    A Does the Frauchiger-Renner Theorem prove only MWI is correct

    Hello all, I have only seen this paper brought up here once before based on the search function 2 years ago, and the thread devolved into something off topic within the first page. I am asking in reference to this paper: https://arxiv.org/pdf/1604.07422.pdf Which claims to show that single...
  32. M

    MHB What is the simplified form of (p ∧ q) ↓ q using basic propositional logic?

    Please help me with this thing. I'm so frustrated I can't understand propositional logic Demonstrate this: (p ∧ q) ↓ q ≡ ¬q PLEASE.
  33. S

    A About the “Axiom of Dependent Choice”

    I learned something new today: the “Axiom of Dependent Choice”: The axiom can be stated as follows: For every nonempty set ##X## and every entire binary relation ##R## on ##X##, there exists a sequence ##(x_n)_{ n \in \mathbb{N} }## in ##X## such that ##x_nRx_{n+1}## for all ##n \in...
  34. Speakpigeon

    B Logic in Science: Exploring Real-Life Applications of Formal Logic

    I'm working on a research project on logic and would be interested in real examples of the use of the theory of formal logic for any scientific or technological research. I'm well aware that logic, to begin with our intuitive sense of logic, is necessary to just about everything we do in life...
  35. Speakpigeon

    B What is the usefulness of formal logic theory?

    What exactly is the usefulness of formal logic theory? And how much useful is it? I believe that most of us have an intuitive sense of logic, which has to be very useful throughout our lives and just about for every situation we can think of. Given that, I came to wonder what more does formal...
  36. Phylosopher

    Foundations Logic in philosophy and mathematics Textbook

    Hello,Lately, I have been interested in philosophy and in particular the branches of philosophy where there is a tie with mathematics and physics. From this point, I think the best branch to start studying would be the most robust one for scientists. i.e Logic. But I am having a problem finding...
  37. M

    Logic Behind a Proof: Injective Function G

    Homework Statement suppose I have a function defined as: G: ℚ--->ℚ f(x)= { 2/ 3x if x does not equal to 0, 0 if x=0} Homework Equations Injective:if for all x,y in ℚ, f(x)=f(y) then x=y. or if x does not equal to y then f(x) does not equal to f(y)The Attempt at a Solution I am confused as to...
  38. S

    Circuit Diagrams versus Logic Diagrams

    Why do we sometimes neglect resistors and power sources when we draw circuits and other times not? Is there different terminology for diagrams that are only concerned about the logic involved and diagrams where the voltage and resistance is accounted for?
  39. O

    Is it possible to prove (P→Q)↔[(P ∨ Q)↔Q] without using truth tables?

    Homework Statement Need to demonstrate this proposition: (P→Q)↔[(P ∨ Q)↔Q] . My textbook use truth tables, but I'd like to do without it. It asks me if it's always truthThe Attempt at a Solution Im unable to demonstrate the Tautology and obtain (¬Q) as solution. I start by facing the right side...
  40. S

    Looking for tricky and interesting physics and maths questions

    I want to do tricky physics and maths questions that require only A-level knowledge of maths and physics to solve. Questions that don't have a straightforward answer and will help me to develop my logic and problem-solving skills. I'm looking books, online resources and etc.
  41. T

    Rectifying my logic of Gauss's Law

    Homework Statement Homework Equations The Attempt at a Solution E4Πr2 = Q/∈0 49000⋅4Π4.12 =Q/∈0 Q = 91.6 μC Qshell = Q = 91.6 μC Qshell = Qinner + Qouter 91.6 = -3.3 + Qouter Qouter = 94.9 μC Can someone point out the error? I have skipped too many lectures and I am catching up right...
  42. T

    Confirming my logic on potential difference

    Homework Statement You’ve decided to protect your house by placing a 5.0 m tall iron lightning rod next to the house. The top is sharpened to a point and the bottom is in good contact with the ground. From your research, you’ve learned that lightning bolts can carry up to 50 kA of current and...
  43. B

    MHB Predicate logic with multiple quantifiers

    Hello everyone. This is my first post on this forum. Thank you for taking the time to help me with my question. I have no idea where to start. :( Question 1: Find an example of a predicate P(x,y) where the domain of x and y are D such that $\forall x \in D, \exists y \in D, P(x,y)$ is true but...
  44. ertagon2

    MHB More Logic statements and arguments

    So once again this is my homework: If you could please check these answers and help me with Q.9 part 1 Q.8 is just checking if both expressions give the same values right ? Q.9 part 1 I'm in the blind here Q.9 part 2 seems logical Q.10 So if i get this right... for the statement/argument to...
  45. D

    I Understanding the Induction Axiom: Notation & Equivalence

    So , what I was wondering about was a slight difference in notation, for which I am not certain if correct (mine, in particular.). The induction axiom says: If M is a subset of ℕ, and if holds that: a)1∈M b)(∨n∈ℕ)(n∈M→s(n)∈M) then M=ℕ. Now my question is: why do we write (∨n∈ℕ)(n∈M→s(n)∈M)...
  46. M

    (Boolean Algebra) Did I write this logic expression correctly?

    Homework Statement My solution, is this correct? This is what I came up with. Y=A+((A*B)+B+C'+(B+C'*D)+D) Is it safe to say that it is correct or did I make a mistake?
  47. Mr Davis 97

    Interpreting a statement in first order logic

    Homework Statement Rewrite the following statements in symbolic form: a) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a solution. b) If ##a## and ##b## are real numbers with ##a \ne 0##, then ##ax+b=0## has a unique solution. Homework EquationsThe Attempt at a...
  48. Myimposter

    Using Wolfram Alpha to Create Truth Tables for Logical Statements

    Homework Statement [/B] "Use the truth table method in Boole to determine whether the conclusion is a tautological consequence of the premises." (Tet(a) ^ Small(a)) v Small(b) ------ Small(a) v Small(b) Homework Equations [/B] Taller(claire,max) v Taller(max,claire) Taller(claire,max)...
  49. Myimposter

    Liberal Studies Degree Student

    Hello, Student looking for help and absorbing as much information as I can to get me through a Symbolic Logic course that is currently kicking me butt. Using tools such as Tarski's World, Boole and Fitch. Anyone with experience feel free to contact me. Thanks
  50. N

    A Entropy and derivations - is my logic faulty?

    It is assumed that entropy increases in the universe. However, the fluid and acceleration equations are derived assuming that. TdS=dE+pdV where dQ = TdS. But dQ is usually set equal to zero to derive these equations. Hence since T is non zero, dS should be zero and so there would be no...
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