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Let $φ_1,φ_2,\ldots$ and $ψ$ be sentences. Let $T$ be set of sentences $$\{φ_2\rightarrow φ_1,φ_3\rightarrow(φ_1∧φ_2),φ_4\rightarrow(φ_1∧φ_2∧φ_3),\ldots\}$$ Is it necessary that there exists $n$ such that for each $ψ$, $T⊨ψ$ implies $φ_n⊨ψ$?

Can you give me some hint ? How to generally begin such tasks ?

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    Maybe you wanted to ask "Is it necessary that $\forall\Psi$ we have $T⊨\Psi\rightarrow\exists n \phi_n⊨\Phi$?" Isn't it ? – 2017-02-20
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    No, I correctly rewrote it from book – 2017-02-20
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    Is this predicate calculus or propositional calculus? – 2017-02-20
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    I dont understand your question – 2017-02-20
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    Your sentences are first order sentences or propositional calculus sentences? – 2017-02-20
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    What happens if $\psi$ is $\varphi_2\to\varphi_1$? – 2017-02-20
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    it is not first order – 2017-02-20
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    @LuizCordeiro can you be more precisely ? – 2017-02-20
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    Maybe I rephrase as follows: "Are $\phi_n$" formulas or boolean variables? – 2017-02-20
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    these are sentences, so formulas (without quantifers) – 2017-02-20

1 Answers 1

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Remark that if for some fixed $n$, $\phi_n$ is true, then all $\phi_i$ with $i\leq n$ are true. Now, assume that all $\phi_n$ are either all true or all false.

If they are all false, then your implication is trivial.

If they are all true, then your implication is true if and only if $\Psi$ is a tautology.

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    Why can't there be an $N\ge 1$ such that $\phi_n$ is true for $n \le N$ and false for $n > N$? – 2017-02-20
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    Because I went too fast ;-) -> going to fix. However, the answer does not change. It depends on the sequence $\phi_n$. – 2017-02-20
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    Sure: the bottom line is much the same! – 2017-02-20