Grounding Relevance (in logic)
Assertions about relevance or irrelevance of arguments are often made as if relevance were uncontroversial, obvious, but this is not the case. Mistakes about relevance are common and result in illogical conclusions. Crucially, for the concept of relevance to be consistent with the fundamental laws of logic it must be reducible to the fundamental laws, or else it would be inconsequential to the meaning/sense of any logically valid conclusion, and thus possibly irrelevant by its own standard. I consider how relevance can be formalised, grounded in the fundamental laws and therefore provable.
A premise is relevant to a conclusion if the conclusion logically 'depends' on the premise. Logical dependence consists in each premise being 'about' a logical relation (=, and, or, <, > etc.) between terms that are common with at least one other premise, or between a common term and some auxiliary term that is not common with any other premise, where all premises are connected by the relation of identity mediated by their common terms (in A>x; x=y; y>B the common terms are x and y), and the auxiliary terms (A, B) are present in the conclusion.
Relevance is a necessary but not a sufficient condition of logical consistency and therefore of a valid argument, because the conclusion may be inconsistent with the premises even if the premises are relevant to the conclusion.
An irrelevant premise is not part of the argument if its auxiliary term is not in the conclusion, because a logically consistent argument must disregard irrelevant premises.
Relevance consists in the continuity of identity relations between all the premises. To allow non-relevance implies that anything (that does not have the identity relation with other premises, but its auxiliary term appears in the conclusion) can be a premise in an argument, therefore the opposite can also be a premise in the same argument, which implies contradiction, which proves that there is no meaningful, logically consistent conclusion without relevance.
Alternatively, it is possible to test the consistency of the argument in reverse, by analysing it for sufficiency of reasons. Premises are reasons that are logically 'sufficient' for knowing a conclusion if and only if they collectively 'prove' the conclusion. To deny this would imply that any conclusion that does not logically follow (is not proven) may be asserted, which implies that the opposite of any conclusion may be asserted, therefore contradiction.
A conclusion is proven if its negation implies contradiction.
Example (i)
1: A=B
2: B=C
C: A=C
Negating the conclusion (C) and substituting on the basis of identity: not(A=C) implies not(A=B), which implies contradiction at Premise 1, which proves the conclusion.
Example (ii)
1: A=B
2: C=D
C: A=D
Negating the conclusion and substituting on the basis of identity: not(A=D) implies not(A=C) and not(B=C), which exhausts the possibilities and does not imply contradiction, which disproves the conclusion.
Example (iii)
1: A>B
2: C>D
C: A>D
Negating the conclusion and substituting on the basis of identity: not(A>D) implies A=D or A<D, where A=D implies C>A and D>B, and where A<D implies C>A and C>B, which exhausts the possibilities and does not imply contradiction. This disproves the conclusion, therefore the argument was not consistent, but it does not prove that non-relevance is the cause of the inconsistency (the cause could be an inconsistency in logical relations). We can nevertheless determine the cause by inspection.
Non-relevance in an argument violates the law of non-contradiction, and renders the conclusion nonsensical beyond the inconsistency of its premises, which cannot be grasped as a singular thought, integrated as an idea, or intended as an action.
Example (iv)
1: I saw her crying.
C: I think she is sad.
In this final example the ‘conclusion’ includes terms that are not common with or logically equivalent with the terms in the premise, which makes the premise both irrelevant to the conclusion and not part of the argument, therefore it is not a consistent argument but two logically disconnected statement. One could nevertheless argue that crying implies sadness, in which case the argument would still require development to achieve relevance, irrespective of whether the additional premises are valid:
1: I saw her crying.
2: Seeing someone crying implies that that someone is crying.
3: Crying implies sadness.
C: She was sad.
Step 1 posits an auxiliary term 'her' in a logical relation with the subjective ascription about 'crying'. Step 2 establishes an identity relation between the subjective ascription about 'crying' and the general sense of 'crying'. The identity relation in steps 2-3 is mediated by the term 'crying', and a logical relation between 'crying' and 'sadness' (auxiliary) is posited. The conclusion captures the auxiliary terms via linguistic equivalents.
This is a work in progress and may be subject to future amendments.
IMO, the illusion of relevance (wherein the considerations are demonstrably not relevant) should be added to the list of logical fallacies. Yes, I realize that such illusions are manifest in "red herring" and "straw man" logical fallacies. In rape trials the defense frequently deliberately calls attention to the victim's sexual relations with other men who are not the defendant in the usually successful effort to sway the jury to conclude that the victim's involvement was consensual, or at least to introduce that critical element of reasonable doubt. Hence it isn't likely, to take an extreme example, that a prostitute can persuade a prosecutor to prosecute a charge of rape, (theft maybe, but not rape). It's unjust, and it's a factor in the exponential rise in unreported sexual assaults in the United States.
But more to Michael's point, there are alarming instances that involve relevant facts to establish the supporting element of a logical syllogism, that require complex justification for which the target audience lack the frame of reference, interest and attention span needed to follow sufficiently enough to be persuaded of the desired conclusion.