![]() This is completely independent of interpretations, which only enter once we have syntactically well-defined input to begin with. $7$ is not a formula, but a term, so $0=1 \vDash 7$ is ungrammatical and not meaningful.īoth examples are excluded for purely syntactic reasons because they do not meet the definition of "well-formed formula". $q \lor \lor p$ is not a well-formed formula, so $p \land \neg p \vDash q \lor \lor p$ is ungrammatical and not meaningful. Generally only the well-formed formulas are ever of interest in logic, malformed examples like $q \lor \lor p$ are never considered once the language is staked out, "formula" always implicitly means "well-formed formula". This is because the $\vDash$ relation is defined to hold between sentences (well-formed formulas), putting anything else around it is not a proper usage of the symbol. The conclusion can be an arbitrary sentence, but it must be a sentence. Is this whole line of thinking correct in your opinion? Remarks? We can also never have a language for arbitrary structures and interpretations since we first need to define some structure and that means someone could use this very necessity to create some X that violates the structure and all of a sudden our language would not be arbitrary-powerful anymore because X would be outside of it. So in conclusion it seems imprecise to say that from false premises follow arbitrary sentences, we mean: arbitrary sentences within the structure and interpretation of the language we use. So again, we cannot arrive where we want. Now, we cannot substitute our 7 (with the Seven-up interpretation) into B which only holds as a placeholder for wff‘s of a certain interpretation. This time we begin from a slightly modified definition due to the situation: $\Sigma \vDash_\Bbb NB \leftrightarrow \forall I_\Bbb N:(\forall A \in \Sigma \to A(I _\Bbb N) = 1) \to B(I _\Bbb N) = 1$. Is this a valid argument despite that „7“ is not interpreted in the way we interpret in the realm of natural numbers?Īlso no.
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