A theory is unsatisfiable if no structure makes all sentences from the theory true. Any theory that contains a formula and its negation is unsatisfiable.

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- November 12th 2011, 08:38 AM #1

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## Fol 9

Enderton 2.6.2

Let and be theories (in the same language) such that

(i) ,

(ii) is complete, and

(iii) is satisfiable. Show that .

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Before I get started, I need to understand some definitions related to this problem.

(*) A*theory*T is said to be*complete*iff for every sentence , either or .

I am confused with an unsatisfiable theory, where a theory is defined to be a set of sentences closed under logical implication.

The textbook gives an example of an unsatisfiable theory, which is the theory consisting of all sentences of the language. Why is this unsatisfiable theory? Any description or example?

Thank you.

- November 12th 2011, 09:48 AM #2

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- November 12th 2011, 09:28 PM #3

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## Re: Fol 9

A

*theory*is defined to be a set of sentences closed under logical implications.

As you said, if I am not mistaken, for an unsatisfiable theory T, there exists a sentence such that

and , where and .

I was having hard time finding an example of T and . Is there any concrete example of an unsatisfiable theory T and for the above?

Additionally, is there any example of an inconsistent theory T and a sentence such that

and ?

- November 13th 2011, 12:40 AM #4

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- November 13th 2011, 04:25 AM #5

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## Re: Fol 9

What is the completion of the set under logical implication?

As you know, T is a*theory*iff T is a set of sentences such that for any sentence of the language,

.

If it logically implies anything, is it the theory consisting of all the sentences of the language? If the theory consists of all the sentences of the language, is the empty structure the only model for it?

Thanks.

- November 13th 2011, 02:58 PM #6

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## Re: Fol 9

It's the set of all sentences.

Yes.

No; rather, no structure is a model of it. Also, the definition of structure in first-order logic usually has a requirement that the universe is nonempty. Otherwise, the formula is not a tautology.