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I have read a number of the great questions and answers regarding syntactic consequence, semantic consequence, the material implication and the logical implication. In particular the answers to this question $\to$ vs. $\vdash$ in logic made it clear that a distinction between the object and metalogical language is important.

My question is about what it means to prove a material conditional statement by direct proof, I.e. to prove $P \rightarrow Q $ by assuming $P$. Is it the case that this method of proof is tantamount to showing the statement (in the meta-language) $P \vdash Q$ is true?

I am quite confused as to whether this is the case or not, as on the one hand from what I've read, it seems like I am confusing the proof in the object language, with a statement in the meta-language.

However at the same time, my understanding of a proof is that we have a sequence of sentences deduced from that are assumptions, axioms, or deduced statements using the rules of inference of our system. To me this means that when we assume $P$ to prove $Q$, we are ultimately trying to show that $P \vdash Q$.

Which of these two notions is correct?

As an additional question, if we have shown that $P \rightarrow Q$ is indeed true, we then eliminate the model of $P$ being true while $Q$ is false as a possibility. Does this mean that $P \vDash Q$ since the only model in which $Q$ is false while $P$ is true is known to be impossible? In other words, if I know in my object language that a particular model is impossible, does this affect the meta-language statement of entailment? (My apologies if this should be posed as a separate question; I thought it was relevant since I think my confusion is stemming in both cases from how the meta-language statements and object language statements affect each other.)

masiewpao
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    Basically YES. See Direct proof : we assume the premise $P$ and derive $Q$, i.e. we have established the relation : $P \vdash Q$. Then, we apply the Conditional proof rule (aka: Conditional introduction) and we conclude with $\vdash P \to Q$. – Mauro ALLEGRANZA Nov 25 '19 at 14:53
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    And YES: if the proof systems is sound (i.e. derives only TRUE conclusion from TRUE premises), having established the relation $P \vdash Q$, we can infer that $P \vDash Q$ also holds. And thus : if $\vdash P \to Q$, then $\vDash P \to Q$. – Mauro ALLEGRANZA Nov 25 '19 at 14:55
  • @MauroALLEGRANZA thank you very much, I think i understand the first point. If I could clarify the second point, in general, it would be incorrect to say $p \vDash q$ right? It is only when I have specific information about what $p$ and $q$ are, such that $p \rightarrow q$, that I can say $p \vDash q$? (I suppose in other words I am just checking that in general, the language pertaining to the objects actually influences the metalanguage statements?) – masiewpao Nov 25 '19 at 15:47
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    Nitpicking issue : consider $P,Q$ formulas; otherwise (i.e. if they are propositional letetrs) we cannot have neither $P \vdash Q$ nor $P \vDash Q$. – Mauro ALLEGRANZA Nov 25 '19 at 16:08
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    Not clear your question... $p \land q \vDash p$ is perfectly correct. Check it with truth table : it is not possible that $p \land q$ is TRUE and $q$ is FALSE. – Mauro ALLEGRANZA Nov 25 '19 at 16:10
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    A derivation of $p \to q$ in a specified proof system is a finite sequence of formulas such that ... ending with formula $p \to q$. The existence of the derivation is expressed through the metalinguistic statement $\vdash_{PC} p \to q$, where $\text {PC}$ identifies the (propositional) proof system used. – Mauro ALLEGRANZA Nov 25 '19 at 16:14
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    The Soundness Theorem is a meta-theorem stating that... It is a theorem about the proof system and not a derivation in the proof system. Using it, we can infer from $\vdash_{PC} p \to q$ that $\vDash p \to q$. – Mauro ALLEGRANZA Nov 25 '19 at 16:15
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    The fundamental distinction is that between e.g. $\to$ : a symbol of the language denoting a propositional connective, and $\vdash, \vDash$ that are not part of the language but are used in the meta- to express relations between formuals of the language. – Mauro ALLEGRANZA Nov 25 '19 at 16:19
  • @MauroALLEGRANZA Thank you, I think I understand now! – masiewpao Nov 25 '19 at 19:50

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