# Logic Definition and 105 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 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. ### B Vacuously true statements and why false implies truth

We say that an implication p --> q is vaccuously true if p is false. Since now it's impossible to have p true and q false. That is we can't check anymore whether the contrary, p being true and q being false,can be.Since p being true is non-existent. So we take the implication as true. For eg...
2. ### I Logical implication vs physical causality

There is something I don't understand that I want to ask quantum physics experts here: Suppose the happening of event X results logically speaking in the happening of event A. So we could for instance have the following logical implication ##X.happens \rightarrow A.happens##. If this is...
3. ### DP: proving existence of optimal substructure for "Sherlock and Cost"

I was attempting to solve the "Sherlock and Cost" problem from HackerRank using DP: But before I went to come up with a recursive relation, I wanted to find if the problem possesses an optimal substructure, and I was following these steps as written at CLRS book: Mentor note: Inline images of...
4. ### I have a few questions about Formalization & Pseudo-code

Hello, I have a few questions and I'd appreciate if you can please help me. 1. If I want to say "for every ## i \in \Bbb N ## and ## 0 \leq j \leq i ## define ## A_{i,j} := i ## and ## B_{i,j} := i \cdot j ## ", then is the logical formula used for this is as such?: ## \forall i \in \Bbb N...
5. ### Showing continuous function has min or max using Cauchy limit def.

Problem: Let ## f: \Bbb R \to \Bbb R ## be continuous. It is known that ## \lim_{x \to \infty } f(x) = \lim_{x \to -\infty } f(x) = l \in R \cup \{ \pm \infty \} ##. Prove that ## f ## gets maximum or minimum on ## \Bbb R ##. Proof: First we'll regard the case ## l = \infty ## ( the case...
6. ### I Translate compound proposition p → q (implication) to p↓q question

I hope someone can help me or point me in the right direction. I am reading Discrete Mathematics with its Applications by Rosen. I am trying to self learn discrete math. I am actually able to do most questions but I have a question about a solution (not the question itself.) The question is...
7. ### 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?
8. ### I Modus Tollens on {B|α} → R?

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.

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...
10. ### 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...

12. ### 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...
13. ### 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...
14. ### 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...
15. ### 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)...
16. ### 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"...
17. ### 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...
18. ### 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...
19. ### 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...
20. ### 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...
21. ### 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...
22. ### 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...
23. ### 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...
24. ### 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 web site that gives a question and hints to the question if stuck, so I thought this would be good practice. But...
25. ### 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...
26. ### 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...
27. ### I Implication problem

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.
28. ### Basic Logic Problem P(x,y)

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 Equations The Attempt at a Solution I know that the...
29. ### 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...
30. ### 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...
31. ### 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...
32. ### 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...
33. ### 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...
34. ### 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...
35. ### Logic Proposition Proof

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 truth The Attempt at a Solution Im unable to demonstrate the Tautology and obtain (¬Q) as solution. I start by facing the right...
36. ### 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.
37. ### I Induction axiom

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)...
38. ### Setting up a truth table

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)...
39. ### B Empty domains and the vacuous truth

So, here's my question. I read somewhere that all universal truths on empty domains are vacuously true, whereas all existential are false. However, if all statements of the form (∀x∈A)(P(x)) , where A is an empty set, are vacuously true, then the statement (∃x∈A)(P(x)) should also be true...
40. ### B What is the converse statement of the given sentence?

The sentence is : "For all real numbers there exists a natural number that is smaller". That is (∀x∈R)(∃n∈N)n>x. This is what I thought of: we can write this sentence as:"If x is a real number, then there exists a natural number n that satisfies n>x." So how would I make a converse statement...

https://en.wikipedia.org/wiki/Fitch's_paradox_of_knowability It begins by assuming that 'All truths are knowable' and then logically proves that that assumption means 'All truths are already known.' The proof is like this: Suppose p is a sentence that is an unknown truth; that is, the sentence...
42. ### Big Oh for a Fraction of 'n'

Let me start by saying that this is from a 30 question assessment on Big Oh, Big Theta, and Big Omega. I understood every other question, however, even after being given the correct answer, I do not understand why my answer was wrong for this one. If you could point me in the direction of any...
43. V

### Ht12e/d input output logic question

Good day. I'm currently in the soldering phase of an RF circuit designed to operate an L293D motor chip. On the breadboard I managed to get it to work, but after soldering (and checking for shorts) I am not getting proper high/low signals sent through the encoder/decoder from 2 2p2t switches to...
44. ### I The truth value of ##P(x)## → ##Q(x)##

I'm reading Velleman's book titled "How to Prove it" and I'm very confused when I'm reading about conditional statements. I understand that there exists some issue with the conditional connective and I accept that because that's the cost of espousing a truth-functional view. I came here to ask...
45. M

### Propositional function problems

1. Suppose P(x) and Q(x) are propositional functions and D is their domain. Let A = {x ∈ D: P(x) is true}, B = {x ∈ D: Q(x) is true} (a) Give an example for a domain D and functions P(x) and Q(x) such that A∩B = {} (b) Give an example for a domain D and functions P(x) and Q(x) such that A ⊆ B...
46. M

### Need help with a mod-6 counter using JK flip flops w/ control bit

Homework Statement [/B] I am currently working on an assignment in which I create a six-state up/down counter. The purpose is to understand the functionality of flip-flops within a circuit among other things. There are 3 input bits (Q2,Q1, Q0), and a 4th control bit (C) which determines the...
47. ### Logic problem about tennis

Homework Statement Pat beat Stacy in a set of tennis, winning six games to Stacy’s three. Five games were won by the player who did not serve. Who served first? source: https://ocw.mit.edu/courses/mathematics/18-s34-problem-solving-seminar-fall-2007/assignments/hw8.pdf Homework Equations N/A...
48. ### Foundations Books on mathematical logic, foundations, and philosophy

Hello, all. I am looking for some good books to start becoming invested in mathematical logic, the foundations of the field of mathematics, and also basically in general the philosophical heart of this wide subject which has interested me greatly. Now I have already read Shoenfield and Halmos...
49. ### A First order logic : Predicates

I have a small problem with the first order logic, in particular, predicate logic Let us take this sentence as an example: Each teacher has given a form to each student. From this sentence, can we have different reading? This is my try to solve such problem, I did not know if this is the...
50. ### B Sweets in a bag probability problem

Andrei has a bag of x sweets. He removes two sweets from the bag simultaneously (without replacement). He now removes a third sweet. The probability that the third sweet is red is (x/2) - 1. How many red sweets were in Andrei's bag to begin with? Could somebody please tell me if (and how) it is...