Solving First-Order Logic Problem with Valuation Existence

In summary, to show that a closed well formed formula B is true in I, you need to find a valuation in I that satisfies B. This can be done by using the fact that every closed, well-formed formula B is satisfied by some valuation in I and the property of the valuation which says that a valuation v satisfies (\forall x_i)\mathcal{B} if every valuation v' which is i-equivalent to v satisfies \mathcal{B}. With this approach, you can show that for any given B, there is at least one valuation v in I that is i-equivalent to v and also satisfies B, thus proving the desired result.
  • #1
D_Miller
18
0
I have a problem I can't quite figure out:

I have a first order system [itex]S[/itex], and an interpretation [itex]I[/itex] of [itex]S[/itex]. I have to show that a closed well formed formula [itex]B[/itex] is true in [itex]I[/itex] if and only if there exists a valuation in [itex]I[/itex] which satisfies [itex]B[/itex].

I've done one of the two implications, but I still have problems with the part in which I have to show the existence of the valuation. I'm thinking that perhaps the wff being closed along with the property of the valuation which says that the valuation [itex]v[/itex] satisfies [itex](\forall x_i)\mathcal{B}[/itex] if every valuation [itex]v'[/itex] which is [itex]i[/itex]-equivalent to [itex]v[/itex] satisfies [itex]\mathcal{B}[/itex]. Is this idea way off? I can't seem to get started on the proof.
 
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  • #2
Any help would be much appreciated. The idea you have is on the right track. To show that there exists a valuation in I which satisfies B, you need to use the fact that every closed, well-formed formula B is satisfied by some valuation in I. This means that for any such B, there is always a valuation v in I that makes B true. Since B is closed, we can use the property of the valuation which says that a valuation v satisfies (\forall x_i)\mathcal{B} if every valuation v' which is i-equivalent to v satisfies \mathcal{B}. Now, you just need to show that by finding a valuation v that is i-equivalent to v, you can guarantee that B is true for that same valuation v. To do this, you will need to show that for any given B, there is at least one valuation v in I that is i-equivalent to v and also satisfies B. Once you have found this valuation, you can conclude that B is true for that valuation, and thus true for I.
 

1. What is First-Order Logic?

First-Order Logic (FOL) is a formal system of symbolic logic that allows for the representation and manipulation of statements about objects and their properties. It is commonly used in mathematics, computer science, and philosophy.

2. What is a Valuation in First-Order Logic?

A valuation in FOL is a function that assigns truth values (true or false) to atomic statements, which are basic statements that cannot be broken down any further. This allows for the evaluation of more complex statements that are built from these atomic statements.

3. How does Existence play a role in Solving First-Order Logic Problems?

Existence is an important concept in FOL, as it allows for the representation of statements about objects that may or may not exist. In solving FOL problems with valuation existence, we consider whether or not an object exists in the domain of discourse and assign a truth value accordingly.

4. What are some common strategies for Solving First-Order Logic Problems with Valuation Existence?

One common strategy is to use quantifiers, such as the universal quantifier (∀) and the existential quantifier (∃), to express statements about all or some objects in the domain of discourse. Another strategy is to use logical equivalences and inference rules to manipulate complex statements into simpler, equivalent forms.

5. Can First-Order Logic Problems with Valuation Existence be solved using computer programs?

Yes, computer programs can be used to solve FOL problems with valuation existence. There are various software tools and programming languages that have built-in support for FOL, making it easier to represent and manipulate statements using valuations and quantifiers.

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