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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.
    – Caleb Stanford
    Commented Jul 13, 2016 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.
    – Caleb Stanford
    Commented Jul 13, 2016 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
    Commented Jul 16, 2016 at 0:02
  • Feel free to post an answer even if you're not certain so I can accept it.
    – James P.
    Commented Sep 9, 2016 at 16:47
  • This is an old question, but you need to understand that different people use the word "logic" to mean different things. To a professional logician, "logic" means mathematical logic, including the proper study of logic itself, and the proper use of classical logic in deducing truths about various structures from axioms about them. To a philosopher, "logic" often means hand-waving or completely useless things such as paraconsistent logics, and it is nearly impossible to find any of them that truly understand even simple theorems and proofs about logic (e.g. incompleteness theorems).
    – user21820
    Commented Apr 9, 2023 at 4:34

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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
    Commented Jun 4, 2017 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
    Commented Jun 5, 2017 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
    Commented Jun 5, 2017 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
    Commented Jun 5, 2017 at 19:12

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