For example, if "All a are b" and "All b are c" then "All a are c". To make this syllogism, does a conscious being need to exist? Or is it inferred from the sentences?
A syllogism is just that: some premises and a conclusion. For that to exist requires no human being.
Of course, we do need a human being (or at least some cognitive agent that can reason) to draw the conclusion from the premises (i.e. infer it).
But again, with a syllogism the conclusion is already there.
Don't think about a syllogism as part of the reality (concrete world of what you are perceiving or studying), but as an abstract construct, just a tool, a device of our own to reason about such reality.
We can discuss forever (pseudo-quantum-mechanics-wiseassess-writers-of-new-age-and-founders-of-sects aside) whether reality exists if no intelligent being exists there to observe it. But in this case, you are asking about a tool to reason and to understand such reality.
In that way, syllogism need an intelligent being or machine to understand and process them. Ultimately, the machine could not know, perhaps, what is it doing, and thus do not know that it is handling or computing a syllogism.
So, depending on what you'd like to know:
- Syllogism, being abstract, requires a conscience (of what that abstract thing is) to exist.
- However, it just requires a turing-complete intelligence to be processed (inferred).
- But the facts covered by the syllogism will occur regardless either being or processor exists or not.