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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 to say "exactly/more than one" in object and subject position when translating to predicate logic

In predicate Logic, I'm trying to work out when both the subject and object are exactly one how is this translated? I've seen that to work out how to say there exists only one dog such that P(d) is ∃d: (P(d) & ∀x: P(x) ⇒ d=x but where do we add the…
user932571
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Why is Universal Generalization a necessary rule, if "for ALL" is already the definition of ∀?

I am just learning about Universal Generalization and don't understand why it is even brought up. There is nothing to prove in "∀xP(x) is true, given the premise that P(c) is true for all elements c in the domain", the ∀ already literally says that…
Rautermann
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Prove that if $ \Gamma\vdash\alpha$, then $\Gamma\vDash\alpha$ is equivalent to "Every consistent set is satisfiable"

I am trying to prove the equivalence of these two ways of stating the Soundness Theorem for FOL. I am proceeding by reductio for each direction. So on the left to right direction I reason as follows: If $ \Gamma\vdash\alpha$ then…
beachtrees
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Prove that a set of first-order formulas contains instances

A set $\Gamma$ of (first-order) formulas is said to contain instances when, for every existentially quantified formula $\exists x A$ of the language, the following holds: There is some term $t$ in the language s.t. $\Gamma \models \exists x A…
paul bernays
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Help on a parsing algorithm for FOL

In Enderton, Section 2.3, to prepare to prove Unique Readability for the wffs in FOL he proves a series of Lemmas involving a function K. On pg. 105 says the following about $K$: "Recall that the terms are built up from the variables and constant…
beachtrees
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Translating "All humans have a father" into propositional logic

Translating: All humans have a father Let $Hx$ be x is human and $Fx$ be x has a father. Then should it be $$\forall x(HxFx)$$? or $$\forall x(Hx\rightarrow Fx)$$?
Xetrez
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$Fd$ from $Gd$, $∃x¬Gx$ by natural deduction

I'm working on a bigger proof using natural deduction and I'm struggling with a final step, which is showing $Fd$ follows by natural deduction from $Gd$, $∃x¬Gx$ ($d$ is a random constant, $G,F$ are unary predicates). No matter what rule I try, it…
Othman El Hammouchi
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Predicate logic from formula to English

I study for an exam and I have this exercise that I've practiced Let: Monkey(x) = "x is a monkey" Likes(x, y) = "x likes y" ∃x (Monkey(x) ∧ ∀y(Monkey(y) ∧ x ̸= y ⇒ Likes(y, x))) My response: There exists someone to like all monkeys, but not all…
Kinfol
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What is the correct translation of this sentence in predicate logic?

Using this glossary: $Gx: x \text{ is green}$ $Hx: x \text{ is heavy}$ $Rx: x \text{ is red}$ My textbook says that the sentence "All red things heavy, but some green things aren't" is translated to: $$\forall{x} (Rx \implies Hx) \land \exists{x}…
Vivek Joshy
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How to paraphrase this sentence into quantificational notation

Sentence: If a tenor respects all sopranos who respect him, then that tenor is respected by all sopranos. Using 'Tx' to symbolize as 'x is a tenor'; 'Sx' to symbolize 'x is a soprano'; and 'Rxy' to symbolize 'x respects y' The sentence seems to be a…
beachtrees
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How to paraphrase this sentence into quantificational notation?

Sentence: Anyone who helps all those who help her is helped by all those she helps. (∀x)(x helps all those who help her ⊃ x is helped by all those she helps) I paraphrased the antecedent, by itself, as: (∀y)(y helps x ⊃ x helps y) I paraphrased the…
beachtrees
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What interpretation makes '$\exists x(Fx \iff Gx)$' true and '$\exists x(Fx) \iff \exists x(Gx)$' false--given the domain {1, 2, 3}?

To make the antecedent true at least one thing that is or is not in the extension of $F$ must also be or not be in the extension of $G$. But then this seems to automatically make at least one case of the second bi-conditional, true. And to find an…
beachtrees
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Question concerning predicate Logic

how do I say in predicate logic: $\forall x > 2 $ there exists ... Thanks for every help!
user804333
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Equivalence of contrapositive and contradiction proofs with quantifiers

I have read that contraposition proof is a special case of contradiction proof. For example, the conditional statement: $P \rightarrow Q$, both proofs suppose $\neg Q$. If we show the contradiction $P \wedge \neg P$, then both proofs are equivalent.…
Carlos Villaseñor
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Is there a language that checks the validity of a proof?

Is there a language/interpreter, that checks the validity of a proof with which you can define a model and use whole predicate logic?
David
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