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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 can I prove it by predicate logic?

For every natural number there exists another natural number which is greater than it. A greatest natural number has no natural number greater than it. Therefore, there exists no greatest natural number. Use N(x) to denote “x is a natural…
John.K
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Quantifying statements w.r.t variables

Suppose we have a predicate symbol of one variable i.e $P(x)$, where the individial $x$ are in some fixed set $X$. Consider the formula $$\exists yP(y)\rightarrow P(x) =: \mathcal F $$ It has one free variable, so one could say $\mathcal F =…
AlvinL
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Conservative extension of theory $T =\{c_1\neq c_2 \}$ and $T = \{(\forall x) (x=c_1 \lor x=c_2 ) \}$

(1) $T = \{(\forall x) (x=c_1 \lor x=c_2 ) \}$ (2) $T =\{c_1\neq c_2 \}$, $T' = \{(\exists x)(\exists y)((x \neq y) \land (\forall z)(z=y \lor z=x))\}$ I am not sure, how to determine that T' is or isn't an extension/conservative extension of T. In…
Mafi
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Defining addition function through predicate logic

Let universe be natural numbers: $\mathbb{N}=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...\}$ Let this be Addition set whose elements are 3-element arrays, such that $\langle x, y, x+y \rangle$. $\mathbb{A}=\{\langle 1, 1, 2 \rangle\,…
Hanlon
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A set of sentences is satisfiable iff it has an infinite model

Let $L$ be any language for predicate logic and $S$ be any set of sentences in $L$. Prove that $S$ is satisfiable iff it has an infinite model. So I know that a set of sentences is only satisfiable if every finite subset is also satisfiable. I…
Tim
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Determining truth value of statement with more than one quantifier

Domain: $\{1, 2, 3, 4\}$ Referents: $a: 3, b: 4$ Extensions: $M:\{1, 2\}, F:\{3, 4\}, J:\{1, 3\}, S:\{2, 4\}$ How can I determine the truth value if there is more than one quantifier applied to the same expression? For example, how to determine the…
Hanlon
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Real numbers field with ternary predicates instead of binary functions symbols

I know that in first order logic one can use (n+1)-ary predicates instead of n-ary function symbols to "simulate" a partial function. So, I'm loking for the axioms for real numbers field expressed using ternary predicates (es. sum(x, y, z)) instead…
Tony
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Is this equivalent?

Im having a dillema , is ∀x(p(x) -> ∃y n(y)) and ∃x p(x) -> ∃y n(y) equivalent? it seems at all cases its the same, is it really equivalent?
Xiang Ramadhan
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Predicate Logic Family Members

I am having a bit of trouble with predicate logic involving family members. This is a question from an assignment I am doing. Let the domain be a group of three dogs, Tiger, Ashes and Smokey. Consider the following premises: • ∀x∀y(Puppy(x) ∧…
asdfjkdsg
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How to solve this logics problem?

question is that \begin{align} 1.∀ x (P(x)⟹Q(x)∨R(x))\\ 2.∀ x (P(x)∧ㄱR(x))\end{align} hence $∃xQ(x)$ i know that $$∀x(p(x)∨q(x))≢∀x(p(x))∨∀x(q(x))$$ then $$∀xP(x)≢∀xQ(x)∨ ∀xR(x)$$ right? Hmm I don't understand how to solve this problem.
kim
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How to proof that the predicates before and after the loop holds

So this is the loop: predicate $ m + n = odd$ $while (m>=0 $ and $m <=100)$ $m = m +4$ $n = n -2$ $end while$ So this loop is supposed to be true before and after the loop, but i dont know why. Before the loop we have: $m +n $ = $m+4 +n -2$ if we…
bassie
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How do you determine whether a logical statement with nested quantifiers is true or not?

I have the following statement $$\neg\exists a\forall b \exists c (ab = b^2c + c)$$ where $a$,$b$,$c$ are all real numbers. How do I determine whether its true or not, what is a good approach with these type of problems?
Arnold Doveman
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Need clarification regarding **free variables** and **ground sentences**

I have been reading Logical Fundations of Artificial Intelligence by Michael R. Genesereth, and have questions regarding some of his paragraphs on Page 20. 1. A variable can also occur as a term in a sentence without an enclosing quantifier. When…
Yan Yang
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Strong induction in logical symbols

I found the following definition of strong induction in Analysis 1 (Amann/Escher, third print). Let $n_0\in\mathbb{N}$ and $\mathcal{A}$ is predicate defined over all integers $n\geq n_0$. Suppose the following two statements are…
Hölderlin
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Verb is and predicate logic

I am a bit confused when translating categorical sentences in language of predicate logic. I am trying to be as exacted I can be. That means capital letters for verbs only, the rest are individual constants. Example: All Greeks are men maybe could…
Slit
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