Short answers, pulled from the story.
Alfred Tarski defined logical consequence as a relation that relies strictly on the logical form of sentences and exists independently of empirical evidence or sensory experience. He identified three features required for an adequate characterization of entailment during the mid twentieth century.
The turnstile symbol originally appeared in 1879 when Gottlob Frege introduced it to mark the relationship between premises and conclusions. Its current usage dates back to Rosser and Kleene between 1934 and 1935.
Modal accounts define logical consequence by asserting that an argument holds true if it is necessary that all elements being true makes the result true. It is impossible for all premises to be true while the conclusion remains false, such as the case where frogs are green yet Kermit is not green.
Intuitionists prioritize justifiability because they argue good inferences never move from justifiably assertible premises to an untrue conclusion. This approach shifts focus away from binary outcomes toward the status of assertions themselves rather than guaranteeing absolute truth values.
Non-monotonic logic systems describe exceptions where additional facts can invalidate previous conclusions unlike standard monotonic rules which preserve results when new information is added. For instance adding the fact that Tweety is a penguin removes the conclusion that Tweety can fly despite Birds typically flying.