Consider a universe where causality failed to hold. Would that universe be described by classical/standard logic? Or would we have to use a radically different logic where causality was not necessary? If causality is violated, would this break any laws of logic?
Classical logic can't be incompatible with an acausal universe, because it never talks about causality. It is merely rules of inference. Truth tables say nothing about something causing something else (although, of course, we can still prove the validity of a syllogism that mentions causality, or any other noun). In particular, to a logician "if p then q" just means q and/or not-p. "If Paris is the capital of France, Jupiter is the Solar System's largest planet" doesn't imply the two parts are causally related.
Ayn Rand thought that "the law of causality is the law of identity applied to change," saying that "you can't have your cake and eat it too, nor eat your cake before you have it" (paraphrase of J. Galt's absurd speech). If we think this or that causal fact is essential to some object, we might say it was analytically (meaning logical-identitywise) true that such an object causes this or that.
The issue is akin to that of "thick concepts," then. In ethics this is the sense that some concepts are more than just a bare normativity + descriptive factors. Rather, the "thick" concept's "parts" are strongly united. I personally find the idea of "thickness" here to perhaps be suspicious, an attempt to validate preconceived descriptors, and re: causality there are kindred examples (claims that something was essential to a species or subspecies, like women's blood would be damaged if they voted (or some such excuse as patriarchists used)).
Otoh, think of time as a general function system. Let the past, A, be relatively determinate, and the future B relatively indeterminate. The variables of B can only be "solved for" by the determination from A plus the general function of the system, or rather all things related under this system as such exhibit causation. However, this is more or less Kantian causality, which might be logically presented but not as a derivation from the law of identity...
If Trump and Biden both are candidate, then Biden is candidate.
If Trump and Biden both are candidate, then Biden can win.
You can reason logically from assumptions made about causality or about the causal relation of events, but the assumptions would not be logical in themselves, that is, you can reason logically without assuming causality, and so you do not necessarily assume causality just because you are reasoning logically.
Of course, maybe it is the case that our world is causal and that causality is a necessary condition of what we are. If this is true, then maybe we would not be logical without causality to make us logical, so to speak. In this case, without causality, logic would not hold. But, presumably not much else would either.
Logic can only tell us, if it does, whether it is true that if something is true, then something else is true.