Questions about logic and mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Consider using one of the following tags: (model-theory), (set-theory), (computability), (proof-theory) if they fit the question.

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9 views

Use Resolution to proove a sentence in First Order Logic

I was just wondering if anyone could tell me if I've solved this problem right. If wrong, I would like to know what I did wrong. "Use resolution to prove Green(Linn) given the information below. You ...
0
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0answers
21 views

Predicate logic ∀x.∃y.≥(x,y) [on hold]

∀x.∃y.≥(x,y) (1) ∃y.∀x.≥(x,y) (2) a) translate this in a sentence b) is (1) true, is (2) true c) is (1) ⊨ (2) true, is (2) ⊨ (1) true d) prove (1) ⊨ (2) thanks
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4answers
27 views

proof for a problem in propositional logic

I cant find a proof for given problem: $$p → ( q → p) ≡ ~p → ( p→ q ) $$ Please give proof to prove above statement.
3
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1answer
47 views

Are there non-equivalent cardinal arithmetics?

‎Generalizing a concept in mathematics is always a problematic situation. In most cases there are several ways to generalize a notion and it is not easy to decide if a particular generalization is ...
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2answers
44 views

Combinatorics homework problem [on hold]

In how many ways can $23$ different books be given to $5$ students so that $2$ of the students will have $4$ books each and the other $3$ will have $5$ books each?
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0answers
21 views

Did I do this big-Omega proof correctly?

Prove or disprove: 6n^3 – 4n^2 + 3n +2 is in Ω (5n^3 – n^2 + n +1). So I'm not sure if I did this right or not, any pointers or the correct steps would be helpful Ǝc ∈ ℝ+, ƎB ∈ ℕ, ∀n ∈ ℕ, n ≥ B ⇒ ...
2
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2answers
34 views

How would you prove in FOL that x is a member of {x} for all x?

How can I formulate and prove the following in first-order logic? $$\forall x (x\in \{x\})$$ I have the following two statements: member(x,$\alpha$) $\neg \exists y(\text{member}(y,\alpha )\land ...
4
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2answers
39 views

Formula for perfect squares spectrum.

I have been working on exercises from "A first Course in Logic" by S. Hedman. Exercise 2.3 (d) asks to find a first-order sentence $\varphi$ having the set of perfect squares as a finite spectrum. But ...
3
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2answers
34 views

origin of syntax for mathematical equations

Bear with me, I don't have any formal training in mathematics. I wonder if there is something that accounts for the syntax of mathematical equations, some deeper logic or reasons why I know that ...
1
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1answer
34 views

Proof of $p\rightarrow (\Box (\Box p \wedge p) \rightarrow (\Box p \wedge p))$

I need to prove: $$p\rightarrow (\Box (\Box p \wedge p) \rightarrow (\Box p \wedge p))$$ The system contains all propostional tautologies and the axiom scheme $\mathbf K$:$ \Box(p \rightarrow q) ...
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1answer
32 views

King Arthur and knights at the round table puzzle

Can you help me with this math problem: Each of the K knights from the round table needs to choose a card which is marked with a number from 1 to N, N >= K. The cards all have different number. ...
0
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1answer
30 views

Use mathematical induction to prove that any integer n>=2 is either a prime or a product of primes.

Use strong mathematical induction to prove that any integer n>=2 is either a prime or a product of primes. I know the steps of weak mathematical induction... basis step= p(n) for n=1 or any arbitary ...
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0answers
24 views

Given an open statement determine if their quantification is true

The Question My Work/Question My book says for part a, iv is true. I disagree. To show an existential statement is false we have to show that for all x that statement is untrue. There are no ...
0
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0answers
33 views

Calculation: Emotional Contagion [on hold]

Can someone help me with this calculation? If 1 person impacts the emotions of 100 others, and each of those other 100 also impacts the emotions of 100 others, and so on, how many people are impacted ...
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2answers
54 views

Consider $(\mathbb N, +)$ as a model for the language with one binary function $+$ . Are the following statements true?

Consider $(\mathbb N, +)$ as a model for the language with one symbol $+$ for a binary function. Are the following statements true? $(\mathbb N, +) \vDash \forall x \exists y \forall z\ x + y\neq z$ ...
2
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1answer
33 views

How to deal with long and tedious logic problem? [on hold]

I am always pretty bad at logic problems. Because most of the logics used aren't really logical (to me)So, as you might think, a long logic problem only adds to it already boring nature. The ...
0
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1answer
17 views

A predicate logic question about write down a sentence

Let $\mathcal{L}=\{f\}$ be a first-order language containing a unary function symbol f, and no other non-logical symbols. Write down sentences $φ$ and $ψ$ of $\mathcal{L}$ such that for any ...
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2answers
53 views

Let $\Gamma = \{\exists x \forall y (x\mathrel R y),\exists x \forall y(y\mathrel R x),\forall x\exists y(x\mathrel R y)\}$. Is $\Gamma$ consistent?

Consider the language consisting of one symbol $R$ for a binary relation. Let $\Gamma = \{\exists x \forall y (x\mathrel R y),\exists x \forall y(y\mathrel R x),\forall x\exists y(x\mathrel R y)\}$. ...
3
votes
1answer
29 views

How to negate $\forall A. \exists a,b. a \neq b \land a,b \in P(A)$?

$$ \forall A. \exists a,b. a \neq b \land a,b \in P(A) $$ My intuition tells me it is false, because given $A=\emptyset$, then $P(\emptyset) = \{\emptyset\}$, so $a=b=\emptyset$. I proceeded to ...
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1answer
115 views

SEVEN - NINE= EIGHT [on hold]

Things are not always what they seem. What is true from one point of view may be false from another and vice-versa, and here is a puzzle to prove it. Despite the fact that every arithmetic teacher in ...
0
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1answer
40 views

If $a+b \geq x$ is known to be true does that mean $a+b\geq x-1$ contradicts it?

So I was proving something and I'm wondering if this line of argument is correct. Suppose that it is true that given conditions $M,N,O$; $a+b\geq x$. That is given those conditions the minimum value ...
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0answers
20 views

How to recognize if a there is a logical entailment?

I have evaluated each of the formulas in gamma and they are not tautologies. Then, I have no idea... Can someone help?
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0answers
33 views

show that if A is creative then A is not computable

show that if $A$ is creative then $A$ is not computable? proof:If A is computable the $A$ andn the complement$A$ are computable enumerable. and $A$ is creative so there was a recsive function ...
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0answers
64 views

Let Γ = {p∧q,(¬p)∨q,p∨r}. Is it true that Γ ⊢ r?

I"m not sure how to solve this type of question. Here is the problem in more detail, and a similar problem: I know that given this set of formulas I'm supposed to show if its possible to deduce r ...
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0answers
63 views

P=NP and human thought [on hold]

I've been thinking about P=NP a lot, and wonder if there could be a new type of math created - like 23 would equal two plus three, two minus three, two times three, and two divided by three, etc. So ...
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0answers
21 views

Equivalent sentences using logical connectives

Using only logical connectives implication ($\to $) and negation ($\lnot $), write a sentence equivalent to the sentence: $$ (p \land q ) \lor r $$ Using logical connectives disjunction ($\lor$) ...
0
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1answer
27 views

what is the meaning of this predicate statement

This question appeared in the GATE exam 2011 Q.32 Which one of the following options is CORRECT given three positive integers x, y and z, and a predicate P(x) = ...
0
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2answers
58 views

Finding a formal deduction from an empty set of premises

I can't seem to make sense of any of this. I'm given a set of axioms schemes, modus ponens as the inference rule and I'm supposed to find a formal deduction. The question (question 1) is here. It ...
1
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0answers
32 views

Show that all recursively enumerable sets are definable in arithmetic [on hold]

This is taken to be a given in most proofs and textbooks - can somebody prove this. Thanks
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2answers
27 views

How to specify each digit of a real number in decimal representation in set theory?

So real numbers have decimal representations. If you want to say the $n$th digit of some real number, how do you say this formally in set theory?
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1answer
27 views

a formal logic proposition about real numbers

I have the following informal statement about real numbers: Every real number except zero has a multiplicative inverse. Can this be expressed as: $$ \forall x \exists y(x\neq 0 \implies xy=1) ...
0
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0answers
15 views

Find all n for which apply: If we shuffle pack 8 times we get the same order.

We have pack of cards 2n. Each mixing changes the order of cards of $a_1$, $a_2$, ..., $a_n$, $b_1$, $b_2$, ..., $b_n$ to $a_1$, $b_1$ ,$a_2$, $b_2$, ..., $a_n$, $b_n$. Find all n ...
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0answers
19 views
0
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2answers
45 views

what happens in a universal implication when the premise is false

I have just started learning Mathematical logic and couldn't figure out the answer to the above question . my question is what happens to the truth value if the premise in a universal implication is ...
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0answers
61 views

Logic puzzle of two numbers

The puzzle goes like this.. ...
0
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1answer
34 views

Proof for $∃xA⇔¬∀x¬A$

I want to prove, that $∃xA⇔¬∀x¬A$, using classic axioms. I think, I have to start with the following step: $∃xA⇔∃x¬¬A$ But I do not know, how to make this step, using axioms: $∃x¬¬A⇔¬∀x¬A$
1
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2answers
61 views

Hilbert's style proof (FO logic)

I am stuck with this question to check whether the following formulas are valid and if they are valid, then derive them using Hilbert's axiom schema and Modes Ponens for First Order Logic. ...
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0answers
23 views

Help with logical equivalence proof regarding a lemma for equivalence relations.

So the question is this: Suppose A is a set. Let ~ be an equivalence relation on A and let a,b be elements of A. Then Ta = Tb if and only if a ~ b. I need to prove this statement to be true. I know ...
1
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3answers
108 views

When does $\,2x=14\iff x\neq7\,$ hold?

I am a non-mathematiciam who is taking some classes in computer engineering. My question is: For which real numbers $x$ does the following hold? $$\,2x=14\iff x\neq7\,$$ I am not interested in ...
1
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2answers
20 views

Question regarding the arithmetic hierarchy notation used in the corollary of Post's theorem

A set $B$ is $\Delta_{n+1}$ if and only if $B \leq_T \emptyset^{(n)}$. More generally, $B$ is $\Delta^C_{n+1}$ if and only if $B \leq_T C^{(n)}$. This is from ...
1
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1answer
44 views

Proof of Tarski's self-reference lemma

In http://www.math.hawaii.edu/~dale/godel/godel.html, Tarksi's self reference lemma is mentioned but the proof is omitted. Tarski's Self-Reference Lemma. For any formula $p(x)$ in an adequate ...
0
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2answers
39 views

Is this theory complete?

I have a language $L=\{P\}$ with equation, where $P$ is binary predicate symbol. Language's formulas are: $\varphi \equiv \forall x \forall y (\neg P(x,x) \land (P(x,y) \to P(y,x)))$, $\psi \equiv ...
4
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2answers
82 views

Is it possible to prove that some point belongs to Mandelbrot set? Is this an example of Gödel's theorem?

Everybody knows about Mandelbrot set drawing computer programs. Program takes some point, builds sequence from it, and if found that sequence goes out of circle with 2 radius, then knows that this ...
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1answer
63 views

Which if the following three propositions are logically equivalent? [on hold]

Which if the following three propositions are logically equivalent? $(p \wedge q) \Rightarrow (p \wedge r)$ $p \wedge (q \Rightarrow (p \wedge r)) $ $(\lnot p) \vee (\neg q) \vee (r \wedge p)$ ...
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0answers
64 views

Determine the truth value of the following proposition: “If it is sunny, then it is raining if and only if it is snowing.” [on hold]

Determine the truth value of the following proposition: "If it is sunny, then it is raining if and only if it is snowing."
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0answers
24 views

What is principle of duality?

What is principle of duality? What is difference between principle of duality and De Morgan's law?
2
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2answers
26 views

When there is a proposition $(P\rightarrow Q)$, which row in the truth table of $\rightarrow $ should I use?

I solved one question in a book of analysis, and although I used an informal method to check it, I'd like to know more about what should be done. The question was the following: $A\subset X$ ...
2
votes
4answers
80 views

Question about logical implication $P\to Q$ [duplicate]

Having come across mathematical logic, a question suddenly came into my mind. We commonly know that the truth value of $P\to Q$ given as: $\begin{matrix} P&Q&P \Rightarrow Q \\ ...
1
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1answer
27 views

$\sigma \in\sum^\prime \leftrightarrow \sum^\prime \vdash \sigma$ [on hold]

$\sum^\prime$ maximal consistent $\sigma$ is a sentence $\sigma \in\sum^\prime \leftrightarrow \sum^\prime \vdash \sigma$ answer:$\rightarrow$ if ...
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0answers
25 views

$\exists G \in L'. G \iff \mathtt{True}(gn(\neg G))$ in the language $L'$ with Godel numbering $gn$ and $\mathtt{True}$ predicate?

I am reading a paper Definability of Truth in Probabilistic Logic . Given a language $L$ with the Godel numbering $gn:L \to \mathbb{N}$ the authors extend it with a predicate $\mathtt{True}$ to a ...