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Questions tagged [predicate-logic]

Questions concerning predicate calculus, i.e. the logic of quantifiers.

Some well-known formal systems covered by this term are

  • first-order logic, containing the quantifiers $\forall$ and $\exists$
  • second-order logic
  • many-sorted logic
  • infinitary logic
4144 questions
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How do you layout the proof for $\forall \in \Bbb R, \exists \in \Bbb R ∶ − ^3 = 0$ being true?

So I know it's true and I understand why it's true but I don't understand how I'm supposed to give the answer, as the answer to this practice question is the following: How exactly am I supposed to layout the answer for this?
tinkertailor
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Instantiation of the statement : $\forall x(P(x)) \vee \forall x(Q(x))$

$\forall x(P(x)) \vee \forall x(Q(x))$ I am currently reading a logic book by Patrick J. Hurley, and in the book the author says that we can't universally instantiate a statement like statement 1. Specifically, he says that universal instantiation…
desoana
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Instantiation of a universally quantified statement.

~∀x(A(x)) Let A represent the category of things that are apples. Then, statement 1 is saying: it is not the case that everything is an apple. This means that we can have things that are apples and things that aren't apples. For example, 80%…
أحمد الدسوقي
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Kleene $\models^{x_1 x_2 ... x_n}$

Mathematical Logic by Kleene page 107 As I understand for one variable $\forall x.A(x)$ it means that instead of considering all possible logical functions for $A(x)$ we need to choose only the function $I(x)=true$ for all values of…
user142248
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Finding out whether a predicate logic formula is true or false.

I'm given the formula: ∃x∃y∃z(P(x, y) ∧ P(z, y) ∧ P(x, z) ∧ ¬P(z, x)). The universe is all natural numbers. R is the relation corresponding to P. R = { : x >= 0} When putting values into the formula I've thought that if x = x then y = x + 1.…
bigboss37
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What is the difference between interpretations of the following predicates?

I need to translate 2 sentences into predicate logic using the following definitions:- A(x, y): x admires y B(x, y): x attended y P(x): x is a professor S(x): x is a student L(x): x is a lecture m: mary Now I want to translate the following 2…
Likhit
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Writing "Someone has visited every country in the world except Libya" using quantifiers.

Someone has visited every country in the world except Libya. Let $B(x,y)-x\,has\,visited\,country\,y$ Domain for x is all people and for y is all countries is $\exists x \forall y [B(x,y) \land (y \neq Libya)]$ the correct representation?
user3767495
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What is the precise definition of a free and a bound variable?

I've only recently started learning this topic and I'm confused about these definitions: I found two definitions for each type of variable which seemed to make sense: Free variable 1: In mathematics, a free variable is a variable that specifies…
Amin Parvaresh
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correct negation of an existential quantification sentence?

I was given the sentence: There is at least one person on earth who does not know logic and was tasked to negate the sentence. I understand from class that existential quantifications become universal quantifications when negated. My…
Jessica Tiberio
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A finite language with an infinite model but no finite ones

So I have this question from Logic for Applications by Nerode and Shore. The chapter on Predicate Logic. Find a finite language $L$ and a finite set of sentences $S$ that has an infinite model but no finite ones. Proposition 7.5 says that a…
Tim
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Order of three or more quantifiers

This is my understanding of meaning of order of two quantifiers: $\forall x \forall y$ is true if the scope of quantifiers is true no matter which values from universe for x and y you choose. $\forall x \exists y$ is true if any value from universe…
Hanlon
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How does Universal Generalization agree with facts?

The law of Universal Generalization states that: P(c) (x) P(x) Now, I understand that this works only if c is any random element from the universe. Such arbitrary selection makes this rule mathematically valid. However, I do not understand how it…
PPK
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Predicate logic: Aristotle vs Frege

Consider the following argument. 1) People who write novels are more sensitive than people who play soccer. 2) Alf writes novels. 3) Brian plays soccer. Conclusion: 4) Alf is more sensitive than Brian. Here is how you formalise this argument using…
user405159
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Predicate logic: Why ∀x(Px ⊃ Px) is not a tautology?

I'm learning predicate logic and my textbook says that some logical truths expressible in the language of predicate logic are not tautologies. For example, according to my textbook, $∀x(Px ⊃ Px)$ is valid even though it doesn't instantiate a…
user405159
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What does $[ \forall x\left( P\left( x\right) \Rightarrow q\right) ] \Leftrightarrow [ \left( \exists x:P\left( x\right) \right) \Rightarrow q]$ mean?

In my lecture materials on predicate logic I encountered the following equivalence: $$[ \forall x\left( P\left( x\right) \Rightarrow q\right) ] \Leftrightarrow [ \left( \exists x:P\left( x\right) \right) \Rightarrow q]$$ However, I cannot make sense…
Zyx
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