Mathematics Stack Exchange
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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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Predicate into clause form

How would one convert this into clause form? (1) ∀x[Dog(x) ⇒ Animal(x)] Some hits, or literally anything is appreciated..! Thanks in regards, O. Dripp
O. Dripp
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What is the easiest way to define the substitution of a term to a variable in a formula?

In the first order logic, I would like to know what is the simplest way of defining the substitution of a term to a variable in a formula. As it is defined obviously by induction on terms then on formulas, the only difficulty is for quantifiers. Is…
fyusuf-a
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How can this be broken up into a math statement?

I'm trying to solve this question: "Prove or disprove that the product of two irrational numbers is irrational" but I am having trouble breaking this question up into a math statement. Is an appropriate statement "If a and b are irrational, then ab…
Jason
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Does an empty string fulfill the condition 'the string can contain alphabet and number only'?

On one hand, an empty string doesn't contain any alphabet and number, which looks violate the condition 'contain alphabet and number', on other hand, the opposite of the condition is 'contain non-alphabet and non-numeric characters', which an empty…
Gstestso
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Help with some predicate Calculus arguments

I just need a little help finishing off this proof, not sure where to go from here with introducing (∀x) (∀x) F(x) ⇒ G(x), (∃x) F(x) ⊢ (∀x) G(x) 1 (1) (∀x) F(x) ⇒ G(x) A 2 (2) (∃x) F(x) A 3 (3) F(a) A 1 (4) F(a) ⇒ G(a) 1 ∀ E 1, 3 (5)…
feeblefeeble
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Determine whether $∀x∈ℝ,∃y∈ℝ$ such that $x+y=0$ & $∃x∈ℝ,∀y∈ℝ$ such that $x+y=0$ is true or false.

Please help me out of this I am easily confuse by this kind of question. Determine whether $∀x∈ℝ,∃y∈ℝ$ such that $x+y=0$ is true. Logical thinking I know that it is true because for all $x$, I can choose a corresponding $y$ that will satisfy…
user292965
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Answering questions with existential and/or universal quantifiers

Let P(n): 2n = n² What is P(2) as a statement? 2(2) = 2² What is P(3) as a statement? 2(3) = 3² Based off of this, would it be correct to say ∀n P(n), ∃n P(n), both, or neither? Explain. Answer: Only the existential quantification of the…
Sage Hopkins
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Assessing Truth Value of Predicate Logic Statements

Consider the following interpretation: Domain = {1, 2} Assignment of constants: a = 1 and b = 2 Assignment of functions: f(1) = 2 and f(2) = 1 Assignment for predicate P: P(1, 1) = T; P(1, 2) = T; P(2, 1) = F; P(2, 2) = F Evaluate the truth value of…
dibdub
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Converting English to predicate logic

Given the following: The domain is X, the set of people. And the functions S(x), meaning that “x has been a student of Course101,” A(x), meaning that “x has gotten an ‘A’ in Course101,” T (x), meaning that “x is a TA of Course101,” and E(x, y),…
user1658296
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find out if formula is logically true

I have an formula , and i am to find if it is logically true. ((∃x)A ∧ (∃x)B) ⇒ (∃x)(A ∧ B) By definition , to check if it is true , i should neg it and create a semantic tree. But with that it results in this formula to be true , while it isnt ,…
trolkura
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How to prove $(\forall x)(A\to B)\to(\exists x)A\to(\exists x)B$

I am stuck on this one, I know I have to use Ax3 which is $$(\forall x)(A\to B)\to(\exists x)A\to(\exists x)B$$ and convert the existential quantifier to universal, but I have problem making it to become just as Ax3 The original was just like the…
swordgit
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logic formula: transform to cnf / dnfb

I try to transform some formulas into CNF and DNF but I am not sure how to use distributivity Law here. Given: $$ ((C \lor D) \land (A\rightarrow D)) <-> (C \rightarrow A ) $$ I applied the rules for Implication and Biimplication and transformed it…
Nemos
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Writing Quantified Statements with Predicate Logic

I have a few examples I need help working out. I feel pretty comfortable with most English to Predicate logic statement problems but there were a few I was unable to figure out on my own and I could use some guidance. This is what we were…
mm19
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Finding an interpretation that makes a formula true

I understand the basic rules of propositional and predicate logic, and what makes formulas true in a general sense, but I am stuck on how to answer specific questions that ask of "each of the formulas below, state an interpretation that makes it…
Rory
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