Tagged Questions

For questions about formal deduction of first-order logic formula or metamathematical properties of first-order logic.

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The axiom systems of predicate logic

I'm writing an article about logic for absolute dummies, so I want to make everything crystal clear; now I'm going to discuss predicate logic. After Googling, I found there are mainly 2 slightly ...
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Confused by a step in the 'Rule C' proof in Mendelson's Logic Textbook

I've been working through 'Introduction to Mathematical Logic, 5th Ed' by Mendelson, and I've found a step in the proof of proposition 2.10 (Rule C) that I cannot understand. In the proof of this ...
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What is it that we are missing in defining the following in the following manner?

Recently one of my friend and I are working on a project on a certain generalization of set theory. This is the same friend of mine of whom I talked in this post. However, the basic outline of his (...
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Is there an error in this textbook about Peano Arithmetic?

I encountered this doubt in an online intro-logic open course offered by Stanford Uni. Under the section 9.4 of this textbook here: http://logic.stanford.edu/intrologic/secondary/notes/chapter_09....
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Resolution Algorithms and one Example Problems?

We have a problem in one Resolution question. There is $5$ clauses, and want to prove the $6$th clause is true. which of the following clause is need more than one times to prove $6$th clause? $t$ ...
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CNF Conversion and one $2015$ exam questions?!

if $\text{likes}(x,t)$ means that person $t$ likes food $x$, and $\text{food}(x)$ means $x$ is a food, $\text{CNF}$ of sentence "No food is liked by all person", and $F$ is Skolem function. The ...
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Proving a formula is valid

Let a formula $A$, and a term $t$ such that $x\in FV(t)$. Show that $\varphi = A\{t/x\}\to \exists x (x=t\to A)$ is valid. So let's assume by contradiction that the formula isn't valid. Therefore ...
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Consider a signature $\Sigma = \{ P^1, R^1, c\}$. Where $P^1, R^1$ are unary predicates, and $c$ is a constant. Let A be a formula in FOL over $\Sigma$. Prove/Disprove: If A is satisfiable ...
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understanding a proof which uses induction on the length of a formula

This comes from Shoenfield's textbook Mathematical Logic. Here is the theorem and its proof: If $u_1,\dots,u_n, u_1',\dots,u_n'$ are designators and $u_1\dots u_n$ and $u_1'\dots u_n'$ are ...
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Contrapositive of the statement involving “for every” and “there exists”

I have a statement (∃x.(P(x) -> (∀y.P(y)))) I am trying to formulate and understand the contrapositive of the formula. ...
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Software for solving first-order logic

Is there any class of software that can help me with the following problem in first order logic: given $\phi$ a formula with a "hole" in it (a subformula which is undetermined) and a particular set of ...
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can you help me to transform ∀x FO logical formula to it equivalent ¬∃ formula?

i have this formula ∀x ∀y (A(x,y) V A(y,x) → B(x,c1) ∧ B(y,c2) ∧ c1≠c2) to the equivalent formula that start by ¬∃x ¬∃ y ? you will find the question here ...
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What are the different ways to get a first-order formula that express the statement“$P$ is the $n$-th prime”

I know that such a $2$-predicate formula exists since Enderton's have already constructed such a formula in his text on mathematical logic but it was not easy to remember so I wonder if there is other ...
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Precise definition of Σ00 in the arithmetical hierarchy

I encountered several different definitions for Σ00 = Π00 = Δ00 of the arithmetical hierarchy. Following are two definitions which seem to me different but I'm not sure: All first-...
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First-Order Logic ternary predicate

I want to express the meaning of the following sentence in FOL: All the GermanAthlete are exactly those Persons who play a Sport for a Country where that Country is Germany. We have a ternary ...
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Source proof of equivalence between relation algebra and three variable first order logic

I am reading up on relation algebra and a lot about first order logic. Being a bit of a heavy torsion on the latter one as I am not used to it. Either way I read on wikipedia that up to 3 variable ...
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Optimize nonlinear model over time period (w/ Kuhn-Tucker Conditions)

I'm working on a nonlinear model that includes a time interval of 12 months. The goal is to maximize the total net benefit (NB) over the entire time period given the constraints listed below. I've ...
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Unification of $P(x,x)$ and $P(a,b)$ [closed]

Why can we not unify $P(x,x)$ and $P(a,b)$? Don't the substitutions $a/x$ and $b/x$ provide unification?
I'm having trouble figuring out my error in this exercise: the task is to prove by resolution refutation that $p \rightarrow (q \land r)$ is a consequence of the set \$\{ p \rightarrow r, (p \land r)...