Owen Griffiths and A.C. Paseau. — Implication vs Inference

Griffiths, Owen and A.C. Paseau. One True Logic: A Monist Manifesto. Oxford: Oxford University Press, 2022.

 

"As just hinted, we take capturing implication and capturing reasoning as distinct applications. The implications of some premises are their logical consequences; they follow from them, whether or not one can deduce them from the premises. In contrast, an inference is what an agent does when she deduces a conclusion from some premises. Reasoning or inference tries to respect implication, though is distinct from it. Thus we write ‘implicational’ rather than ‘inferential’ whenever we are interested in what follows from what—as we typically will be—rather than what can be deduced from what. All this applies even to idealized notions of reasoning (which for example prescind from human subjects’ errors in reasoning). [Fn6: So long as the sense of reasoning/inference is not so idealized that it means nothing other than the ability to accurately reflect implicational facts.]

     It follows in particular that there is no reason to suppose at the outset that the correct foundational logic is completable by a (sound and effective) deductive system . . . Perhaps no deductive system can capture all logical entailments. (Incidentally, we use the words ‘implication’, ‘consequence’, and ‘entailment’ interchangeably, with the epithet ‘logical’ understood when omitted.) [Fn7: With the usual act/outcome ambiguity, resolvable by context; e.g. ‘entailment’ can mean the relation that holds between some premises and a conclusion, or the conclusion itself.] Implication is modelled by model-theoretic consequence (⊨) and derivability by deductive consequence (⊢)." (Prologue xx-xxi.)