I am not a philosopher but I would like to ask this question. I know that there are philosophers like Hobbes, Locke or Foucault, who excel in the areas of ethics, political philosophy and aesthetics. However they rather use informal logic than formal/symbolic logic in dealing with practical problems posed by the respective field because informal logic is rather concerned with substance than with structure. I suppose that other, not this well known philosophers, are good in symbolic aspects, too. So can one be a philosopher without much knowledge of formal logic?
It is almost impossible to have a career in philosophy in the modern anglophone world without being a very sound logician. It is not, however, strictly necessary for the pursuit of strong academic philosopher. A lack of a strong grounding in logic will exclude you from much of the contemporary conversation, plus a strong formal education will sharpen your analytical capacity.
I know well-studied and competent phenomenologists who have never studied formal logic and are not interested in logic or mathematics in the least and have never shown any competency in them. It is definitely possible, although inadvisable and a poor professional decision.
Then again, I'm not sure there's any domain of knowledge that will not make for a more sophisticated and capable philosopher.
You must at least understand logic, especially symbolic logic because it underpins many areas of philosophy, not to mention mathematics. For example, Kant would not have written such great books if he was not interested in logic. He even studied mathematics in his spare time. He also played around with many natural sciences like physics. I doubt he ever published anything, but I know he actually explained why galaxies and solar systems lie in planes using conservation of angular momentum. Some people like Russell and Godel start by studying philosophy but then realize that they are much more interested in logic, paradoxes, proofs, etc. You must also understand that there is not just propositional logic, there are things like modal logic for example. Overall, many great mathematicians were also philosophers, and vice versa. For instance, Leibniz and Descartes are among many.
I don't think it all necessary. It's reasoning, judgement & imagination thats important. Of course it does depend on what area of philosophy you are interested in.
There have been philosophers who have been good mathematicians & philosophers. Descarte & Pascal comes to mind. But its reason that grounds both. You appear to slight the authors you mention because they're using 'informal logic'. Would could it mean to formalise their logic? Would it make it more comprehensible?
Although Plato idolised geometry, I doubt you'll find it (in a formal sense) in any of his works.
Hegel wrote 'The Science of Logic', and even that doesn't treat formal symbolic logic. Since for Hegel "the underlying structure of all of reality is ultimately rational, logic is not merely about reasoning or argument but rather is also the rational, structural core of all of reality and every dimension of it. Thus Hegel's Science of Logic includes among other things analyses of being, nothingness, becoming, existence, reality, essence, reflection, concept, and method".
But, if you're interested in Badious work, you'll need a heavy dose of Set Theory.
To understand Nietsche you need a literary imagination, since his strategies are either poetical or rhetorical, even if his concerns are not.
Spinoza wrote in a proposition - theorem style as homage to Euclid. But I don't think his concerns are formal either. Wittgenstein wrote in a similar style. But he was also interested in formal logic (an inheritence from Bertrand Russell perhaps), but he denounced mathematics as merely syntax & tautology, which as a lapsed mathematician, hurts.
If you're interested in metaphysics & mysticism, you might find that the philosophy encoded in metaphysical/mystical poetry more accessible.
Lastly if you don't like formal symbolic logic, avoid it :), but if you do go for it!