Questions tagged [logic-translation]
For translating between natural language expressions and logic expressions.
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Do ∃x(Dog(x)) and ∃x(¬Dog(x)) contradict each other? [closed]
Formally, ∃x(Dog(x)) and ∃x(¬Dog(x)) look like they contradict each other. However, in the real world,
there exist objects which are dogs i.e. ∃x(Dog(x))
there exist objects which are not dogs i.e. ∃...
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On the tautology $(P \implies Q) \vee (Q \implies P)$
The logical statement
$$(P \implies Q) \vee (Q \implies P)$$
is an example of a tautology. However, if I choose logical statements for $P$ and $Q$, it is not always true that either $Q \implies P$ or $...
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When to use implication arrow versus equivalence arrow?
In my class we've been asked to complete an exercise and choose whether to use implication or equivalence arrows: "The equation $2x−4=2$ is fulfilled only when $x=3$."
I understand that we ...
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Finding the relationship (equivalence or implication) between two expressions
I am trying to find the relationship between $$\exists X \; (p(X) ∧ q(X))$$ and $$\exists X \; p(X) ∧ \forall X \; q(X).$$
I believe that quantifiers cannot be used in forming truth tables, after all ...
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Two arguments with the same form, one valid, one not
A convertible car is fun to drive. Isaac’s car is not a convertible. Therefore, Isaac’s car is not fun to drive.
Letting $C(x)$ be "$x$ is a convertible car" and $F(x)$ as "$x$ is fun ...
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Is "William only eats icecream when the sun is shining" a biimplication?
William only eats icecream when the sun is shining
Let $P(t)$ be the sun is shining at time $t.$
Let $Q(t)$ be William is eating an icecream at time $t.$
Which implication is there between $P(t)$ and $...
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Using quantifiers to rewrite some simple mathematical statements
Have I used quantifiers correctly to rewrite these sentences?
The equation $x^3=7$ has at least one root.
$∃x∈\mathbb R\;\;x^3-7=0$
The equation $x^2-2x-5=0$ has no rational roots.
non$(∀x∈\mathbb ...
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Translating "there is no number between two consecutive numbers"
There is no number number strictly between two consecutive number.
Is my translation $$∀x ∈ Z | ∀y ∈ Z | ¬∃z ∈ Z \;(z<y ∧ z>x ∧ y=x+1)$$ correct?
Is $$∃x | ¬∃n ∈ Z \;(n<x<n+1) $$ also ...
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Not knowing how to translate a theorem to logic [closed]
Proposition: for every positive integer $n$, there do not exist four positive integers $a,b,c,d$ with $ad=bc$ and $n^2 <a<b<c<d<(n+1)^2$
I understand how to prove this by looking for a ...
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Difference between “for some $k$” and “for some arbitrary $k$”
I am told that the “for some” and “for some arbitrary” are different.
For example, when proving the statement “if n is odd, then $n^2$ is odd”, one of the steps includes writing $$\text{$n = 2k+1,\:\:...
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Symbolising the set of pupils who do not like both subjects
If $A= \{\text{pupils who like Science}\}$, $B= \{\text{pupils who like History}\},$ then is the set of pupils who do not like both subjects $$(A\cap B)^\complement$$ or $$(A\cup B)^\complement\,?$$
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What does the word 'with' mean in this theorem?
If $a, b$ and $c$ are positive integers with $a, b ≥ 2$, then equation (1.1) has at most one solution in positive integers $x$ and $y$ with
$b^y ≥ 6000 c^{1/δ∗(a,b)}.$
I'm unclear about the above ...
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Translation of : "the sum of two prime numbers each greater than 3 is even" in First Order Logic
I have used these symboles :
$Z(x) : x$ is a prime number
$f(x,y) : x + y$
$L(x,y) : x\ge y$
$t: 3$
$E(x) : x$ is even
$$∀x∀y(Z(x)∧Z(y)∧L(x,t)∧L(y,t))\rightarrow E(f(x,y)))$$
Is this a correct ...
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Formalising the sentence "someone is plotting against me"
Someone is plotting against me.
Can the above sentence be translated into predicate logic without using the existential quantifier? If not, is it because the sentence is self-referential?
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How to translate SAME and DIFFERENT in predicate logic?
Betty told every secretary a lie.
I was told that this sentence is ambigous as it can be intrepreted in 2 different ways.
These are my 2 interpretations:
"Betty told every secretary the same ...
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Translating "There are two different students in your class who between them have sent an e-mail message to or telephoned everyone else in the class"
Let $M(x,y)$ be “$x$ has sent $y$ an e-mail message” and $T(x,y)$ be “$x$ has telephoned $y$”, where the domain consists of all students in your class. Assume that all e-mail messages that were sent ...
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To prove that for every $x$, $(x\in Z\implies x\in R),$ is it ok to write "For any $x,$ suppose $x\in Z$. Then... Then $x\in R$"?
To prove that for every $x$, $(x\in Z\implies x\in R),$ is it ok to write "For any $x,$ suppose $x\in Z$. Then... Then $x\in R$" ?
For example, in the above proof of $$\forall x{,}y\left(\...
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"There are two different people who have visited exactly the same websites"
Let $W(x,y)$ mean that student x has visited website y, where the domain for $x$ consists of all students in your school and the domain for $y$ consists of all websites. Express the statement $∃x∃y∀z((...
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Which of these translates the universal and existential quantifiers better?
Consider this proposition (which I know is false):
$$(\exists y{\in}\mathbb Z)\,(\forall x{\in}\mathbb Z)\,(y > x).$$
I am wondering whether the analogy of picking a variable value according to the ...
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Translating "Boris hasn't tried anything better than chocolate"
Boris hasn't tried anything better than chocolate.
The above sentence needs to be converted to first-order logic. The given domain is candies, and Boris is a sentient piece of candy.
$B(x,y)$: $x$ is ...
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Formalising "every number other than $0$ has a unique multiplicative inverse"
The domain is the set of all real numbers.
Express "Every number other than 0 has a unique multiplicative inverse" as a logical expression.
My solution is: ...
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Hows my symbolization for the following Argument?
Every philosophical empiricist admires Humes. Some philosophical idealists like no one who admires Humes. Therefore, some philosophical idealists like no philosophical empiricists.
$\forall x(Ex \...
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I am Confused, mixing predicates and propositional letters?
In the exercises at the end of each section the author gives you the Arguments and ask you to determine the validity of it, he then goes on and gives you the proper predicates to use which are located ...
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Is my symbolization of the following argument correct?
Formalise the following argument, then prove or disprove its validity:
Some psychologists admire Freud. Some psychologists like no one who admires Freud. Therefore some psychologists are not liked by ...
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Translating "None of the paintings is valuable except the battle pieces."
Example 1:
No intelligent person who drinks to excess also eats to excess.
I am stuck on deciding whether this means
a) $\forall x(Ix \implies -(Dx \lor Ex)$
or
b) $\forall x(Ix \land Dx \implies -...
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Translating "If there were no computers with antivirus every computer would work fine."
"If there were no computers with antivirus every computer would work fine".
Use: O(x) = x is a computer, A(x) = Computer x has an antivirus, F(x) = x works fine.
My take is $$∀x((O(x)∧¬A(x))...
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Negation of converse of a theorem related to linear independence
Theorem $T:\quad$ Let $S = \{v_1, v_2,..., v_p\}$ be a set of vectors in $\mathbb R^m$. If $p>m$, then this set is linearly dependent.
Theorem $T$ can be proved to be true.
Converse $T_c$ of ...
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What does $\exists x\left(L\left(x,x\right)\wedge \forall z\left(L\left(x,z\right)\rightarrow \left(z=x\right)\right)\right)$ mean?
I translated "There is someone who loves no one besides himself or herself" as $$\exists x\left(L\left(x,x\right)\wedge \forall z\left(L\left(x,z\right)\rightarrow \left(z=x\right)\right)\...
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"No integers $x$ and $y$ exist for $28x+7y=8$"
What is the logical structure of this statement?
No integers $x$ and $y$ exist for $28x+7y=8.$
I'm not sure, but I think the answer is
$$¬∃x\;∃y\;(x ∈ \mathbb Z ∧ y ∈ \mathbb Z ∧ 28x + 7y = 8).$$
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How do I translate "is not sufficient" into symbolic logic?
Take the proposition "it is not sufficient for the monkey to dance in order for me to get an A on the test"
m = the monkey dances
a = I get an A on the test
It makes sense why I can ...
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"Neither Ana nor Bob can do every exercise but each can do some."
I’m a bit confused as to how I should translate the following sentence:
Neither Ana nor Bob can do every exercise but each can do some.
I've identified the atomic sentences $A$ = Ana can do every ...
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Expressing the instance of a ZF Set Theory axiom for a given property
I am currently in the process of studying ZF Set Theory (without the Axiom of Choice) and I have come across a type of question that is unclear to me. The basic format of the question is to "...
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Predicate Logic "John drinks any type of drink"
For the following English statement,what would be the correct predicate logic translation:
John drinks any type of drink.
1.$\forall x {(Drinks(John,x))}$
2.$\forall x(Drink(x) \to Drinks(John,x)\\$
I ...
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Have I misunderstood this argument about truth values? [closed]
7.2.1 Every truth value is confirmed by intuition that affirms it.
$(∀x)(Tx\to Ix)$
7.2.2. Every intuition that affirms truth value is a feeling of correctness.
$(∀x)(Ix\to Cx)$
7.2.3. The feeling ...
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Mixing predicate logic with propositional logic
In this paper in legal philosophy (https://doi.org/10.3790/rth.40.1.1) on p. 25, the author writes:
P(x) ↔ (x → ¬h)
This is supposed to be a formalisation of the following statement: "Every ...
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How is "so" correctly translated into predicate logic?
I have come across an exercise that asks to have “There is only one ball, so you need to have it” translated into predicate logic. Using the predicates $\text{Ball}(x)$ for $x$ is a ball and $\text{...
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Why is implication used instead of conjunction in this instance of translating English into a logical statement?
Can you please help me understand why implication was used instead of conjunction in the answer to this practice question? I have been struggling with nested quantifiers and when to use implication ...
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Translating an English statement into a nested quantification with 3 variables.
I am trying to translate:
"There is a student in this class who has been in every room of at least one building on campus."
My solution was:
$$ \exists x \exists b \forall r(B(x, b) \implies ...
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Problems with nested Predicate in FOL
In my teacher's lecture, its have a problems like this: execute statements based on the following base predicates
$L(x)$: $x$ is a logician
$f(x)$: a function that return values is a friend of $x$
...
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Translating English to predicate logic with units
I am having difficulty translating this fragment from a larger sentence into predicate logic: three pets bathe together.
Let pets be P(x) and B(x,y) be bathe togehter.
How would I deal with the number ...
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"More than two" in First Order Logic
I'm trying to express the following statement which has "More than two" in first order logic.
More than two people work at Google
Using the following:
Name: Google
Property: People
Relation: ...
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Quantifiers in first order logic
I am trying to express the following sentence in first order logic with quantifiers:
All cats like to eat any mouse
Using the following properties and relation:
Properties: Cat, Mouse
Relation: ...
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Representing "unless" in propositional logic [closed]
I am trying to represent the following statement, which contains "unless", into propositional logic:
If it's snowing when I am outdoors I get cold unless I am wearing a coat.
Using the ...
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For every $x \in A$, $x<5$, there is a number in set $B$ greater than $x$ in mathematical notation
I would like to ask why were my answers wrong. I was supposed to decide which notation has the same meaning as the text (title).
$(x<5 \Rightarrow (\exists y \in B)(y>x))$ - This is not correct,...
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First Order Logic simplification for sentences with multiple quantifiers
Background
I’m having problems translating some more complicated questions using First Order Logic. I’m wondering if there are any ways to simplify sentences that are very long with lots of ...
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“Every human needs a shelter”
Given a signature $\Sigma$ consisting of an empty set of functions and a set of predicates $P=\{human,\ shelter,\ need\},$ where $need$ is of arity two and $human$ and $shelter$ have arity one, I want ...
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There is an odd number between any two even numbers
Write the complete mathematical logical expression of the following: there is an odd number between any two even numbers.
My attempt:
$E(x): x$ is even.
$O(x): x$ is odd.
$$∀m∀n∃k[(E(m) ∧ E(n) ∧ O(k))...
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Are opinions considered propositions?
I was wondering the thought. My textbook says:
A proposition is still a proposition whether its truth value is known to be true, known to be false, unknown, or a matter of opinion.
"It is a ...
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How to read $Q_j = \{k:\beta_k = j\}$?
In my work, I came across the following expression:
$$Q_j = \{k:\beta_k = j\},\tag1$$
where $j = \mathcal{J} = \{1,...,J\}$ set of cell-phone tower,
$k = \mathcal{K} = \{1,...,K\}$ is a set of users, ...
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Why is $\exists x \forall y \exists z \left(\left( y = x + z \right) \lor (z \leq x) \right)$ false?
I am trying to wrap my head around the proposition
$$\exists x \forall y \exists z \left(\left( y = x + z \right) \lor (z \leq x) \right),$$
where $x, y, z \in \Bbb N^+$.
The proposition is false, but ...
