1) Three conceptions of logical validity
Let me start with the three major options available, which can be traced back to the medieval logicians, at least. King has a nice review about the treatment of consequence by Aristotelian logicians of 14th century (it may come as a surprise that logic then was far more advanced than commonly thought today).
Conceptual containment
The first is what is often called conceptual containment, the conclusion follows due to the "meanings" of the premises and the conclusion. It fits well with Aristotle's idea of concepts as quivers of attributes, later adopted by Kant. As articulated by Lavenham around 1370:
"A consequence is formal when the consequent necessarily belongs to the understanding of the antecedent (necessario est de intellectu antecedentis), as it is in the case of syllogistic consequence, and in many enthymematic consequences."
This was a minority opinion, however, although it regained prominence after Kant's endorsement in the Critique. As he would express it, logical inference is not ampliative or synthetic. In the modern formalizations it is approximated by the relevance conditional.
Modal necessity
As King remarks:
"The second and most common account of inferential validity among our authors is modal: the consequence A ⊢ B is legitimate when it is impossible for A to be true and B false. More precisely, the modal criterion spells out at least a necessary condition for consequences in general to be legitimate. The intuition at work here is familiar. Modal accounts of logical consequence date back to Aristotle, and live on today in Tarskian model-theoretic explanations of logical consequence that take possible worlds to be the models in which an interpretation is evaluated".
However, this modal account is a big tent indeed. For one, it is not clear what the "possible world" represents. Is it something like a metaphysical realm, a la Molina and Leibniz, or is it purely linguistic, just an assignment of meanings or referents to the non-logical terms in the sentences?
Linguistic formality
The consequence based on assignment of interpretations to the non-logical terms is the most "formal", and the closest to the modern semantic consequence. Buridan, and arguably Ockham, already express it:
"Uniform subsitutivity, of the sort proposed by Buridan, is the third account of inferential validity. He is clear that (3) goes beyond the modal account in at least two ways. First, it applies equally to material (non-formal) consequences; thus the formality of an inference is a feature that goes beyond its necessity, neither explained by nor reducible to it. Second, it takes legitimacy to be a function of truth-value relative to a set of terms, namely the nonlogical vocabulary, rather than appealing to possibilities."
Shapiro in Logical consequence: Models and modality characterizes the metaphysical and the linguistic views as the two leading modern contenders as well, but shows that both are flawed, and some blendings of them work better.
2) Aristotle and the "causing" of conclusions
As for Aristotle, it is hard to pin him down on either the modal or the conceptual view, although the majority of scholars seem to lean towards the latter. Andres in Aristotle and the Conventional Logicians on the Fourth Syllogism Figure argues that Aristotle's rejection of the fourth syllogism figure (which weakens "all" to "some" in the conclusions) is motivated by his insistence on the understanding/relevance, and that Aquinas agreed with him:
"First, let us recall Aristotle's definition of the syllogism. It is speech in which, certain things being given, something else necessarily follows, these being so. Then Aristotle explains, "I mean by the last phrase that they produce the consequence". A more literal translation of the same passage is "I mean by 'these being so' that it follows because of these". Both translations, however, convey that the premisses are related to the conclusion as a cause is related to its effect. Aristotle's definition of the syllogism implies an order of causality between the premisses and the conclusion. St. Thomas makes this same point more explicitly. He writes:"The principles are in a certain way the efficient cause of the conclusion, as we say that demonstration is a syllogism making someone know..." If Aristotle and St. Thomas are right in their understanding of what a syllogism is, then Aristotle is right in defining the terms of the syllogism as he does and thus in rejecting the fourth figure".
He further suggests that the acceptance of the fourth figure flows from the Lockian attitude towards syllogisms as useless for "causing" knowledge (and hence pushed things in the linguistic direction):"it is of no great use, since the mind can perceive such connection, where it really is, as easily, nay, perhaps better, without it... and when they see that, they see whether the inference be good or no; and so syllogism comes too late to settle it".
3) Aristotle and relevance logic
Steinkrüger in Aristotle’s Assertoric Syllogistic and Modern Relevance Logic takes a broader view:
"This notion is characterized by two conditions imposed on the concept of validity: first, that some meaning content is shared between the premises and the conclusion, and second, that the premises of a proof are actually used to derive the conclusion. Turning to Aristotle’s Prior Analytics, I argue that there is evidence that Aristotle’s Assertoric Syllogistic satisfies both conditions. Moreover, Aristotle at one point explicitly addresses the potential harmfulness of syllogisms with unused premises. Here, I argue that Aristotle’s analysis allows for a rejection of such syllogisms on formal grounds established in the foregoing parts of the Prior Analytics."
He also considers the possibility that Aristotle works with two different notions of validity, but "the reasons to reject this view are more compelling than the reasons to accept it" and so "we can, cautiously, uphold the result that Aristotle’s logic is a relevance logic". The relevance logic does reject weakening rules as valid inferences. This is also corroborated by Felt in Impossible Worlds, who argues that Aristotle's view of potentiality is incompatible with either linguistic or metaphysical interpretations of the possible worlds semantics:
"For whether with Lewis one takes possible worlds to be as real as the actual, or one tries to replace them solely by the actual, the upshot seems the same: all is reduced to a planar understanding of what it means to be. In these controversies the anti-Parmenidean (Aristotelian) notion of potentiality, as an intrinsic character of the actual, has tended to be supplanted by possibilities (in the plural), Lewis’s “ways things could have been,” purely formal and discrete patterns. The dynamism of potentiality has been exchanged for a dust of homeless forms."
