# Questions tagged [first-order-logic]

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science.

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### About the first order logic (valid, Unsatisfiable, Syntactically wrong)

I am in trouble , I searched a lot about how to solve this kind of questions but I did not get any answers. I understand how can I know when the sentences is valid and Unsatisfiable in propositional ...
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### A graph in descriptive complexity - is $x$ already a vertex?

So suppose that there is an undirected graph with edge connections known. Now in first-order logic there is quantifier $\forall x$. Then does this automatically refer to vertexes, or can we use ...
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### a program discovering himself how to solve propositional calculus

it is well-known that propositional logic problems such as $$(p\leftrightarrow q) \lor r \quad\overset{?}{\vdash}\quad (((p\lor q)\to(p\land q)) \land \lnot r)\lor r$$ can be simply solved by ...
371 views

### Relations between statements involving universal quantifier, conditional and biconditional

If we consider two predicates: $b(x)$: x is a boy $c(x)$: x is clever Then, there are four statements involving $∀, b(x), c(x), →$ and $↔$ . These are below along with my interpretation of their ...
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### Is Inductive Logic Programming approach applicable to general theories (not just sets of Horn clauses)?

Inductive Logic Programming (https://en.wikipedia.org/wiki/Inductive_logic_programming) find hypothesis theory H for background theory B and set of examples E. ILP algorithms and implementations ...
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### Does FOL extended with least-fixed points satisfy the Compactness Theorem?

I am aware that first-order logics (FOL) satisfies the compactness theorem. That is, if a FOL theory is insatisfiable, a finite subset of the axioms of such theory is insatisfiable too. Is it the case ...
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### Consistent theory based on L and not(A->A) is a theorem

I am working on this problem in which I have a theory $T$ based on language $\mathcal{L}$ and the only information we have is that T is consistent and $\vdash \lnot(A \rightarrow A)$. Given this ...
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### Situation calculus: how to find pre-conditions in 15-puzzle game?

I have been working on finding the preconditions for a situation calculus example for some time now. This example is called the game "15-puzzle" where you can find a discription here https://...
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### Show that exist a finite set of clauses F in first-order logic that Res*(F) is infinite

I'm kind of desperate at this point about this question. A predicate-logic resolution derivation of a clause $C$ from a set of clauses $F$ is a sequence of clauses $C_1,\dots,C_m$, with $C_m = C$ ...
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### Z-Specification = Routes

Im trying to make an invariant for this Z schema about routes. 1) The invariant should express that each route should contain at least 20 different places. First of all i thought of doing a universal ...
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### Abstract syntax tree: expression->operation->lhs|rhs vs expression->lhs|operation|rhs - what should I take into account in decision?

I am trying to build class hierarchy for the abstract syntax tree of First Order Logic as specified in the grammar https://github.com/antlr/grammars-v4/blob/master/fol/fol.g4 (ANTLR parser generator). ...
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### Synthesis of the crisp programs from the Bayesian (probabilistic) programs?

https://arxiv.org/abs/2001.00805 is the article which is pointing towards the direction that the optimal policy of the reinforcement learning can be expressed as the Bayesian/probabilistic program (...
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### Bounds on the size of the universe of a model for an FO-sentence

Sorry for the weird title. The Problem: Consider the first-order logic sentence φ≡∃s∃t∃u∀v∀w∀x∀yψ(s,t,u,v,w,x,y) where ψ(s,t,u,v,w,x,y,) is a quantifier-free first-order logic formula using ...
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### Can full first order knowledge base be written as the single sequent in the sequent calculus?

The knowledge base of the first order logic essentially is single formula: conjunction of individual formulas (I guess, I am right). The sequent for the sequent calculus is the formula in the special ...
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### Is it possible to write down every Prolog program+query as the sequent in the sequent calculus?

Prolog program P is set of Horn (definite) clauses, effectively it is the conjunction of implicational formulas. I guess that every Prolog program P with some query Q can be written as ...
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### Is FOL representation of probabilistic assignment statement correct?

For instance, $x = x + 1[0.3]x+2$ sets $x$ to $x + 1$ with probability $0.3$ and to $x+2$ with probability $0.7$. If I use notation used in the paper "An Analysis of First-Order Logics of Probability"...
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### How to compute (partial) consequence set for premises of the first order logic?

I am playing with the Sequent Calculus Trainer https://www.uni-kassel.de/eecs/fachgebiete/fmv/projects/sequent-calculus-trainer.html . It is game with judgments, where each judgment consists from: 1) ...
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### How do you derive a type $∃e(e)$ in terms of universally quantified types, without invoking Void initially?

I wrote a "proof" for this, and though it was enough to convince myself, there are a few things that bother me about it. Primarily I'm not sure about the loose way in which I'm swapping between first-...
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### Description logic representation

If Manufacturer manufactures electronic equipment, then manufacturer generates e-waste. I have generated the triple. However how to represent the if-then statement in DL? ...
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### Show the Invalidity of the sequence

∃x (P(x) → Q(x)), ∃x P(x) ⊨ ∃x Q(x) I am trying to find the invalidity of the following sequents. A = Set of natural number P(x) : x is odd Q(x) : x is not divisible by 2 What i don't understand ...
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### Convert conjunctive normal form to equivalent boolean formula with only NOR gates

Let $\varphi$ be a boolean formula in 3-CNF form (conjunctive normal form with three literals at most per clause). I want to convert it to an equivalent boolean formula that uses only NOR gates with ...
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### Precedence of satisfiability operator

I'm just reading a textbook in mathematical logic, as following: What is the precedence to consider the equality and satisfiability operators in the equation pinpointed in red?! In other words, which ...
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### What does an = sign with an x beneath it mean?

I am studying for a test in Logic right now, and saw the symbol $\underset{x}{=}$, which is used like this: $I \underset{x}{=} I'$. I've seen it in the solutions of questions like this one: Prove ...