Logic without ex falso quod libet
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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.
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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. 0=1 $\vDash_\Bbb N$ 7, but let us say that 7 gets interpreted here as a Seven-up bottle.But then we cannot arrive at A $\land$~A $\vDash$ q vv p which means that it is outside of the definition of validity. Because validity is defined: $\Sigma \vDash B \leftrightarrow \forall I:(\forall A \in \Sigma \to A(I) = 1) \to B(I) = 1$. Is this a valid argument since „q vv p“ is not a wff? But can B be really literally „arbitrary“ and still the argument holds valid? Two examples to explore the problem. It is called ex falso quodlibet and basically says that from false premises any deduction is valid. That is a well known theorem in classical logic.