# Tagged Questions

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

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### Reasoning informally about $\{x \in B \mid x \notin C\} \in \mathscr P(A)$

Attempting to apply more flexible, informal reasoning to predicate logic as demonstrated helpfully to me by another user in answer to my last question. $\{x \in B \mid x \notin C\} \in \mathscr P(A)$ ...
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### Is this conclusion via rules of inference correct?

Use rules of inference to show: ∀x(P(x) → Q(x)) premise ∀x(Q(x) → R(x)) premise ¬R(a) premise ¬P(a) conclusion I have a lot of trouble with these sort of questions and was wondering if I did this ...
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### Presenting well a sentence with quantifiers

What is the syntax rule to present a syntax with quantifiers ? Should we rather write : $$\forall x\in \mathbb N\quad \exists y\in\mathbb N \quad x<y\qquad (1)$$ or \...
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### Symmetry and transitivity with the existential quantifier

I can't find any resource that would indicate whether symmetry and transitivity relations are possible or not with the existential quantifier or when quantifiers are combined. I'm interested in the ...
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### Predicate logic: Symbolize a sentence using a dictionary and two-place predicates

Given the following dictionary, how would the sentences below be translated in to a language using quantifiers? My attempts are shown as well: Dictionary: $L$: a two place predicate which means -...
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### Alternatives to pure quantifier logic

Are there some alternatives for pure quantifier logic? Pure quantifier logic is axioms and rules of inference added to proposition logic to result first order logic. Are there other axioms that ...
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### Simple proof involving quantifiers

Task: Prove this theorem: $\exists x (P(x) \Rightarrow \forall yP(y) )$. I got this far: I figured out this is equivalent to $\exists x (\neg P(x) \lor \forall yP(y) )$. I don't understand ...
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### How do I translate 'no philosopher student admires any rotten lecturer' into quantificational logic formula?

Let's assume that $Fx=x$ is a philosophy student, $Rx=x$ is a rotten lecturer, and $Mxy=x$ admires $y$. My translation of the sentence was $\forall x(Fx\supset\neg\forall y(Ry\supset Mxy))$, but my ...
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### B is an element of some power set of A such that A is an element of F.

$$B \in \{\,\mathcal P(A) \mid A \in F\,\}$$ I can't quite figure this out. My textbook says this statement is equivalent to \exists A \in F\ \forall x\ (x \in B \leftrightarrow \forall y\ (y \in ...
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### correct typesetting for quantifiers

For years I have been typing and writing quantifiers in a certain way. Now that I am writing my thesis, my adviser is taking issue with some of these things. Since he is my adviser I'm going to do ...
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### How can I interpret a multiply-quantified statement?

∃ x ∈ R such that ∀ y ∈ R, x + y = 0. Can anyone help me rewrite this statement in plain english without symbols or variables? So far I have "There exists a real number whose number and other number ...
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### “If everyone in front of you is bald, then you're bald.” Does this logically mean that the first person is bald?

Suppose we have a line of people that starts with person #1 and goes for a (finite or infinite) number of people behind him/her, and this property holds for every person in the line: If everyone ...
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### Translating this nested quantifier to english (negation of nested quantifiers)

So i'm a total newb at this so I need help on one the questions for my assignment: Let S(x) = “x is a student at Bronx Community College”; F(x) = “x is a faculty member at Bronx Community College”, ...
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### Universal Quantification of a Predicate P(x)

Consider the following statement: Every number is less than its square. Write the statement “Every number is less that its square” symbolically by defining a predicate and using a quantifier. Answer: ...
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### How should I prove $\forall x \in \mathbb{Q}$ where $x > \sqrt 2$ , $\exists y \in \mathbb{Q}$ where $\sqrt{2} < y < x$

How should I prove the below statement? $\forall x \in \mathbb{Q}$ where $x > \sqrt 2$ , $\exists y \in > \mathbb{Q}$ where $\sqrt{2} < y < x$ I tried to prove it by contradiction ...
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### Universal Quantifiers [duplicate]

What is the difference between ∀x∈U : ~p(x) and ~∀x∈U : p(x) ?? Could anybody give any English sentences explaining both of them?...
There is an exercise in my textbook. Suppose ‘$m$’ denotes Myfanwy, ‘$n$’ denotes Ninian, ‘$o$’ denotes Olwen, ‘$Fx$’ means x is a philosopher, ‘$Gx$’ means x speaks Welsh, ‘$Lxy$’ means $x$ loves ...