Questions tagged [quantifiers]

The quantifiers $\forall$ ("for all") and $\exists$ ("there exists") distinguish predicate calculus from propositional logic.

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Express the following statement without using quantifiers [closed]

Assume that the domain is {−4, −3, −1, 1, 3, 4}, where P (x) is a propositional function ∃x(¬P (x)) ∧ ∀ x ((x < 0) ⇒ P (x))
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Formal Logic/Metalogic: Defining Wff Substitution while Forbidding Double Quantification

This question is closely related to the answer here. I'm working in a system of logic which has the usual semantics for building Wffs out of terms and atomic formulas. However, in the interest of '...
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A model in which the formula holds

I'm supposed to find a model in which the formula $$\exists x(\forall y(P(x) \land Q(x,y)))$$ holds. Does the model below work?: $M$: $U = \{a,b\}$, $P=\{a,b\}$, $Q=\{(a,b), (b,a)\}$.
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First Order Logic formula transformation

I have the following predicates: Doctor(D) Hospital(H) WorksIn(D, H) HasShift(D, IdS, H) Is this FOL formula transformation correct? I'm a little unsure. Every doctor that works in hospitals has at ...
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Is $\forall x,\exists y (x \ge y^2 \Rightarrow x>0)$ same as $\forall x (\exists y: x\ge y^2\Rightarrow x > 0)$

The first statement: $\forall x \exists y (x\geq y^2 \Rightarrow x > 0)$ The Second statement: $\forall x (\exists y: x\geq y^2 \Rightarrow x > 0)$ Actually, my question is this: In the first ...
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Expressing "at most" in predicate logic

I need to translate to predicate logic: "Every natural number is the sum of at most four squared natural numbers". The word that causes problems for me is the word "at most". Here ...
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Determining the truth value of $∃x∀y (y=x^2+2x+1)$

The domain for $x$ and $y$ are all integers. $∃x∀y (y=x^2+2x+1)$ can be interpreted as: There is an $x$ for all $y$ such that it satisfies $y=x^2+2x+1$. There is a single $x$ value which results in ...
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What is the difference between ∃x∀y and ∀y∃x? [duplicate]

What is the exact difference between ∃x∀y(condition) and ∀y∃x(condition)? Translating these into English, ∃x∀y(condition) = There is an x for all y such that (condition) is satisfied. ∀y∃x(condition) ...