The phrase "different logics" is correct and can be used in written English.
You can use it when referring to a plurality of distinct logical systems or approaches. For example, "We must consider different logics when analyzing the data."
Biconditionals of different logics may have different logical properties, and it surely matters what kind of the equivalence connective is used for defining truth values.
Applications treat each schema differently based upon different logics using advanced programming, such as object oriented principles in the form of polymorphism, classes inheritance or object arrays etc. Hence developers eagerly define and model the schemas that yield robust characteristics for their problem domain in general.
The second kind of tolerance is a tolerance for different logics, something that is naturally construed as a kind of logical pluralism.
Logic has close historical links to the study of reasoning and so the idea of logical pluralism can quickly bring to mind the idea that different logics might be correct for different groups of people to use in reasoning, that there might be, for example, different logics for different cultural and biological groups an African logic, a female logic, a middle-class logic etc.
Nowadays, a general theory of the algebraization of logics around the concept of logical consequence has grown from the different algebraic treatments of the different logics obtained during the last century.
It seems clear that he intended his logical pluralism to be both 'horizontal' that is, to allow for different logics at the same level, such as classical and intuitionist sentential logics—as well as 'vertical' allowing for logics for new kinds of expression, such as intensional logics and second-order logic (the terminology is from Eklund 2012).
