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

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How to use a predicate function when formalizing predicate propositions?

I've tried the following question from my maths assesment but not sure if I'm using the predicate, fool(p,t) correctly. Can anyone advise on if I'm using it ...
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1answer
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If there are Predicates before Predicate Calculus, why is it called such?

In my understanding, predicates are synonyms of relations: mappings of an ordered set (a,b) to the set of values "True" and "False" Well, propositional calculus comes before predicate calculus, and ...
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69 views

How show $ S \models \forall x ( \alpha \Leftrightarrow \beta)$?

I read some notes on Logic Course. I read that we can conclude: $$ S \models \forall x ( \alpha \Leftrightarrow \beta)$$ if and only if $ S \models \forall\, x\, \alpha$ has conclusion $ S ...
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Why $ M\models \forall x ( \alpha \to \beta)$ Is False? [on hold]

if M be a model and $\alpha$ and $ \beta$ be two formula the following is False: $ M \models \forall x ( \alpha \to \beta)$ if and only if $ M \models \forall x \alpha$ has conclusion $ M \models ...
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1answer
37 views

Equivalence of $\forall x (L(p,x) \implies \neg L(s,x))$

"Everyone who Patricia likes, Sue doesn't like" Let $L(x,y)$ stand for "$x$ likes $y$" and $p,s$ for Patricia and Sue, respectively. Then the statement in logic is: $\forall x (L(p,x) ...
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1answer
34 views

How do logicians notate a proposition that posits the instantiation of a property?

In The Oxford Companion to Philosophy, the entry on existence includes this paragraph. It is often held that ‘exist’ is not a firstlevel predicate. What this means is that ‘exist’ does not ...
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Logical Equivalence of Wffs in Sentence, Predicate Logic using Tables, Interpretations Resp.

just curious if there is a formal name for the results that: a) Two wffs in Sentence Logic are equivalent iff their truth tables are equal , as binary functions of {T,F} b) Two wffs A,B in ...
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1answer
111 views

I need help understanding Frege's definition of number

I have really been trying to understand Frege's definition of a number or at least gain a strong intuition of it. However, my attempts have not been fruitful. If someone could help me it would be much ...
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1answer
43 views

How to prove $(\forall x,y\in\mathbb{Z})(5\nmid xy\to(5\nmid x\land 5\nmid y))$

Question: Prove $x,y\in\mathbb{Z},\Bigl((5\nmid xy)\to(5\nmid x\land 5\nmid y)\Bigr)$ where $\forall a,b\in\mathbb{Z},\bigl((a\nmid b)\leftrightarrow(\forall k\in\mathbb{Z},b\neq ak)\bigr)$ and ...
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Predicate Logic and Calculus

Question of the week came up in my schools logic club but there is not much information to it. Here is the question: Show that $$ \exists x\,[R(x)\wedge \lnot Q(x)],\ \forall x\,[P(x)\to Q(x)],\, ...
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Predicate Calculus - Resolution

A question came up at the our schools logic club this week which involves using resolution to prove an argument in predicate calculus. I am slightly aware of how to find prenex normal forms but to ...
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1answer
59 views

Counterexample to Fraissé's Theorem for infinite signature

Let S be a finite signature and $\mathfrak{A}, \mathfrak{B}$ S-structures. Fraissé's Theorem states: $$\mathfrak{A} \equiv \mathfrak{B} \Leftrightarrow\mathfrak{A} \cong_f \mathfrak{B}$$ Where ...
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2answers
81 views

Predicate calculus (formal deduction vs resolution) [closed]

I am part of the logic club at my school and the question of the week was; Use formal deduction for predicate calculus to show that the following argument is valid. State each rule you use. Premise ...
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1answer
23 views

Constructing Logical Derivation

All Texans speak to anyone whom they know intimately. No Texan speaks to anyone who is not a Southerner. Therefore, Texans know only southerners intimately. (We have to use These predicates : $Tx, ...
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1answer
58 views

Prove $\exists x(\varphi\wedge\psi)$ is $\underline{not}$ logically equivalent to $\exists x\varphi\wedge\exists x\psi$

Question: Prove for all $\mathcal{L}$-formulae $\varphi,\psi:$ $\exists x(\varphi\vee\psi)$ is logically equivalent to $\exists x\varphi\vee\exists x\psi$ Show that it is $\underline{not}$ the ...
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1answer
81 views

direct hint to showing a formula is valid?

we know A formula is logically valid (or simply valid) if it is true in every interpretation. These formulas play a role similar to tautologies in propositional logic. which one could direct me to ...
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how we can prove that argument $P_1,P_2,…,P_n $?

I ran into a one claims on LOGIC. how can add more direction or hint to me? if we have an argument $P_1,P_2,...,P_n $ such that $ n>3$ ($p_i$ is premise) why $P_1,P_2,....,P_n,P_1$ is ...
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24 views

Show that there are only finitely many $\mathcal{L}$-structures with domain $D$

Question: Let $D$ be finite and non-empty, let $\mathcal{L}$ be finite. Show that there are only finitely many $\mathcal{L}$-structures with domain $D$ Answer: (1) Let $k$ be the cardinality of ...
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2answers
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Maximal consistency proof for set of propositional logic with specific restriction?

I ran into struggle when I comes to one sentence on logic. Why the set of all propositional that under any valuation has value 1 is not maximal consistent ? ...
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Interpretation and truth table is enough to showing validity or a better way?

I'm so glad that find this useful site. anyway, I ran into some challenging ways to find a formula is valid. Here is two example in my note that called valid. I ran into such a problem with making ...
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1answer
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RAA elimination and inference a theory ?!

Can somebody explain the why if we eliminate RAA rule in natural deduction system on propositional logic, why ~$(p \wedge $~$p)$ is not inference from the ...
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38 views

Valid Formula in First Order Logic

I am a little confused about the validity of first order logic formulas. How we can using formal notation to prove the following is VALID? $ \exists x \exists ...
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28 views

Is an infinite, recursive predicate valid?

Consider the following predicate: $$P(x) =\ "True\ and\ P(x)"$$ Does it make sense to claim that $P(x)$ is true $\forall x$? Specifically, Consider the case where, instead of a boring $True$, you ...
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1answer
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how would you convert the following to CNF [closed]

How would you convert the following to CNF? ∃y.(g(y) ∧ ∀z.(r(z) ⇒ f(y,z)))
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Proving $\square(\forall v_1\neg\psi(v_1))\rightarrow\forall v_1\neg\psi(v_1)$ for a particular $\psi$.

I have a formula $\psi(v_1)$ that is equivalent in $\mathrm{PA}$ to $$\exists a\exists b\exists c\left[\neg\exists ...
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1answer
54 views

Construct a calculus which produces exactly all pairs $(S,t)$, such that $free(t)=S$.

Construct a calculus which produces exactly all pairs $(S,t)$, such that $free(t)=S$. This calculus will operate on pairs $(S,t)$, where $S$ is a set of variables and t is a term. I've got an ...
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Please help me translate these english sentence into Predicate calculus. I have hard time doing it.

Everything is greater than or equa to itself; For evert number n, the resut of adding n to 9 is greater than or equal to 9; Everything has something greater than it. This is for my semantics ...
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1answer
102 views

Unable to prove this simple inference using sequent calculus

In a recent question, I asked if the folowing inference is valid provided that the variable $z$ does not occur free in $\Gamma$ (Note: No restriction regarding whether or not $z$ occurs free in $\phi$ ...
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Is this inference valid?

Is the following inference valid provided that the variable $z$ does not occur free in $\Gamma$ (Note: No restriction regarding whether or not $z$ occurs free in $\phi$ is assumed) ? $${ \Gamma \vdash ...
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Validity of a few FOL formulas

How we can obtain that the following examples (in my book wrote) is logically valid, I) $ \exists y \forall x p(x,y) \to \forall x \exists y p(x,y) $ II) $ \exists x \exists y p(x,y) \to \neg ...
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translating one sentence to FOL , an interview question is wrong !?

We ran into an Interview question, writing part 3 days ago. one of the question is as follows: (definition of A(x) and B(x)‌ is not given by OP) what is the logical interpretation of following ...
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Do all logic problems have one solution? [closed]

Analyze the logical forms of the following statements: x and y are natural numbers, and exactly one of them is prime. Below are the two answers that I got. The first one is the one the author ...
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1answer
42 views

What does it mean to axiomatize a logic?

I'm sorry if this question is not clearly formulated: An axiomatization, or an axiomatic system, usually means a set of axioms (i.e. a theory). A formal theory is such a set of formulas in some formal ...
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Is there a way that I can contribute to this site? [migrated]

Hi guys I am studying how to prove it by Velleman. its a great book and I feel that it can really help people out with logic. However, the one thing that this book lacks are the answers to all the ...
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2answers
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Predicate Logic Practice [duplicate]

There are 5 predicate calculus questions I've been working on, I think I've correctly solved the first four, except the last one I have no idea where I'm going with. I'll present my work and the ...
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1answer
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Predicate Calculus Challenge Practice Question [closed]

Me and a couple group members from our Computer Science club have recently started predicate logic and we are given three practice challenge questions. We're quite sure we have the first one correct ...
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Prenex normal form of $\neg \big(\forall x \ P(x) \vee \forall x \ Q(x) \big)$

I have the statement $\neg \big(\forall x \ P(x) \vee \forall x \ Q(x) \big)$ and I have to write it in prenex normal form. First I use the second De Morgan law $\neg \big(\forall x \ P(x) \vee ...
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Equivalent categories are elementarily equivalent: Formalization?

Equivalent categories should be elementarily equivalent in the sense of mathematical logic. How to make this precise? Here is an attempt: Let $F : \mathcal{C} \to \mathcal{D}$ be an equivalence of ...
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61 views

What is the truth value of $\exists ! x \ (x=x+1)$? [closed]

What is the truth value of this statement? $\exists ! x \ (x=x+1)$
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Predicate Calculus Practice

I've been doing some practice questions in the textbook for an upcoming predicate calculus lecture and I think I've managed to get A and B (possibly C) correct but I am clueless on how I can manage D ...
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Predicate Calculus

(a) All students who like computer science like mathematics.(S(x), C(x), M(x)) (b) There are some students who like Socrates but do not like Aristotle. (S(x), L(x, y), S, A) (c) No student who likes ...
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Determine the correctness of that formal proof. [duplicate]

1) $a+c<b+c$……………………Hypothesis 2) $\neg(a < b)$..........................Hypothesis 3) $\forall A \forall B[\neg(A<B) ⇒(A=B ∨B<A)]$......Trichotomy law 4) $\neg(a < ...
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Statements in prenex normal form.

Put these statements in prenex normal form. a) $\exists x \ P(x) \vee \exists x \ Q(x) \vee A$, $\textit{where A is a proposition not involving any quantifiers.}$ b) $\neg (\forall x \ P(x) \vee ...
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1answer
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What is the truth value of the statement $\exists ! x \ (x>1)$?

$\textit{If the domain consists of all integers, what are the truth values of these statements?}$ a) $\exists ! x \ (x>1)$ Because the domain consists of all integers then we can loop through all ...
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51 views

Determining the correctness of a formal proof

Is the following formal proof, proving $\forall A\forall B \forall C[A+C=B+C\Longrightarrow A=B]$ correct?? Proof 1) $a+c=b+c$.............................................................Hypothesis ...
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The propositional logic expression for ∃x∀yP(x,y)

Where u.d. of x is {1,2,3} and y is {a,b} The given answer is ((1,a)Λ(1,b)) V ((2,a)Λ(2,b)) V ((3,a)Λ(3,b)) But I get the expression ((1,a)V(2,a)V(3,a)) Λ ((1,b)V(2,b)V(3,b)) Why is my one wrong ...
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How to determine if a predicate statement is true or false while giving reasons?

I am currently studying discrete maths at university. However the text book doesnt seem to explain predicate statements in much detail (only one chapter on them). In the assignment which i am ...
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Logical Structure of a Proposition

I'm having a hard time figuring out the logical structure of the following theorem : I'm not interested in proving it, for now, i'm just trying to understand its logical structure. I don't know ...
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Translation of positive integer domain?

in Translation of positive integer domain. and $G(x,y)$ with $( x < y)$ tranlation how we can find the value of $A, B$? A= $ (\forall x)(\exists y) G(x,y) \to (\exists y)(\forall x) G(x,y)$ ...
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Show that the following formulas are consistent

How do I prove if the following formulas are consistent? ∀x$\neg$S(x,x) ∃x P(x) ∀x∃y S(x,y) ∀x(P(x)$\to$∃y S(y,x)) I think I proved part of it... There is at least one value for x such that P(x) ...