I'm not sure logic necessarily is the key to philosophising. For starters, without meaning to be hideously pedantic, I'd really want a detailed definition of what was meant by logic and come to think of it philosphising.
Moving on from there (which otherwise could hold us up for an awfully long time) and just considering the question using my own personal undisclosed conceptions of what these words might mean I'm thinking, no not really. Certainly I think it has an important role (in what respect?) and is an essential component of philosophical argument (really?) but then also so is imagination, perception, vision, intellectual courage, the willingness to question the seemingly unquestionable, the refusal to just accept what we're told without thinking. Logic may be many things but philosophy without imagination, vision and courage would be of little worth.
Just my thoughts. Compare and contrast.
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Logic is crucial in philosophy because it helps to structure arguments and evaluate ideas. By applying logical reasoning, philosophers can develop coherent and sound arguments, analyze concepts and propositions, and critically assess the validity of their conclusions. In essence, logic serves as a foundational tool for clear thinking and effective communication in philosophical inquiry.
Some methods of philosophizing include critical thinking, analyzing arguments and evidence, questioning assumptions, exploring different perspectives, and engaging in thought experiments and hypothetical scenarios. Philosophers often use logic, reason, and ethical principles to derive insights and draw conclusions about complex issues and concepts.
Philosophizing involves reflecting on complex questions about existence, morality, knowledge, and reality. Insight is gained through this process by critically analyzing these questions to deepen understanding and uncover new perspectives. Philosophizing allows individuals to explore and contemplate profound ideas, leading to personal growth and enhanced critical thinking skills.
The Barcan formula in modal logic is significant because it helps to understand the relationship between necessity and possibility. It states that if something is necessarily true in all possible worlds, then it is also true in the actual world. This formula plays a key role in reasoning about the nature of possibility and necessity in modal logic.
Some of the best logic textbooks for studying logic include "Introduction to Logic" by Irving M. Copi, "Logic: The Laws of Truth" by Nicholas J.J. Smith, and "A Concise Introduction to Logic" by Patrick J. Hurley.
David Hume proposed that relations of ideas are based on logic and reasoning, and that they are necessary truths that can be known through reason alone. He argued that these relations are based on the principles of mathematics and logic, and do not depend on experience or observation. Hume's key idea is that relations of ideas are certain and unchanging, and can be known with certainty through rational thought.