I'm not clear on where logic in the broad sense of the word stands with respect to philosophy.

I do know there is mathematical logic which would be a subset of something.

If philosophy of logic and logic differ then what distinguishes the two and where does one begin and the other end?

As for philosophical logic, would I be right in thinking this is a tool used within philosophy itself?

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    "Philosophy of X" is always somewhat different than "X" just in the sense that it's the philosophical study of X, or maybe meta-X, rather than X itself. Logic is no exception. "Philosophical Logic" as far as I can tell just means the use of logic to solve philosophical problems. – 6005 Jul 13 '16 at 1:10
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    Practically speaking, logic is studied in philosophy departments because that's the field that generally has the most use for it and interest in it. On the other hand, math departments (at least in the US) don't generally hire logicians these days. – 6005 Jul 13 '16 at 1:11
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    I would say that philosophy of logic is the philosophical study of logic while philosophical logic is the "logic of philosophy", i.e. the formal methods used when one is engaged in philosophy. – George Law Jul 16 '16 at 0:02
  • Feel free to post an answer even if you're not certain so I can accept it. – James P. Sep 9 '16 at 16:47

Topics relating to the question are not neatly partitioned due to the involvement of complex fields and perspectives (e.g., contribution from mathematics and computer science to logic). Risking over-generalization, this is how differences among the three fields can be understood. Logic is like the tools in the shed; philosophical logic is using the tools to better understand problems arising in philosophy; and philosophy of logic is to ask how the tools became tools proper. Let me elaborate.

Logic: We want to understand how our ideas relate to each other. Natural languages are defective for this purpose. 'Morning star' and 'evening star,' e.g., have different ideas, although they refer to the same thing. So invented are propositional logic, first-order logic, and second order logic as tools to inquire relationships among ideas.

Philosophical logic: Some philosophers realize that logic can help sharpen important concepts in their fields. For example, philosophers of mind or of knowledge or of language ask "What does it mean to know or to believe something?"; "What does it mean for something to be possible or necessary?"; and "How to change our minds rationally (belief revision)?" So invented are epistemic logic, modal logic, and defeasible logic, respectively.

Philosophy of logic: Logicians wonder how the tools in the shed became tools. Surely, in a Flatland, a shovel would not have been a tool. Thus, the tools themselves became the object of study. For example, logicians ask "How this object came to exist as a tool?; "What logical approach would have been possible, had we understood logical particles (e.g., logical connectives and quantifiers) differently?: "What do we mean by the truth of a logical statement? Studies or topics relevant to answer these questions, respectively, are history of logic; Hilbert's axiomatic approach vs. Gentzen's natural deduction approach; correspondence theory of truth vs. deflationary theory of truth.

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    +1, I like these analogies. That said, I am a little skeptical about characterizing logic as inquiring into relationships among ideas. First, logic is not about ideas anymore so much as sentences. Second, I worry that this characterization casts too wide a net. Logic does not study any old relationships among ideas; rather, it just studies the logical ones (e.g. consequence, satisfiability, provability, and so on). – possibleWorld Jun 4 '17 at 23:56
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    I expected this excellent criticism. 1. By ideas, i did not mean the Lockean notion (the object of reflection and impression), but the object of our thought (concept, propositions, etc). 2. By the relationships of ideas, thus, I mean something relating to the properties of logical arguments. I would put satisfiability and provability under phil of logic. Under logic as a tool, I will just put proof, truth table for propositional logic, and modeling for FOL. (i.e., stuff taught in intro to logic courses) – Nanhee Byrnes PhD Jun 5 '17 at 14:15
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    Thanks for the replies. I think contemporary logicians would deny that logic even studies concepts or propositions. Rather, they study relationships between sentences in some formal language. (This isn't just border-policing, either! Substantive questions about, e.g., logical truth turn on the question of what logic studies. But maybe this is getting too deep into the philosophy of logic to pursue her). – possibleWorld Jun 5 '17 at 18:49
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    A question raised by professional logicians got to belong to philosophy of logic: at least, by my way of constellating ideas. My way is just heuristic, and open to an objection for being wrong in detail. For example, Aristotle would not think that formal logic is needed for the reason of the defect of the natural language, the way Frege would think. – Nanhee Byrnes PhD Jun 5 '17 at 19:12

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