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The way I've always thought that philosophy worked is that philosophers have a certain set of tools (deduction, laws of thought, basic sources of knowledge) which they use to come to reasoned answers to questions. Most importantly, these tools are taken as axiomatic. That is, they are seen as starting points from which all reasoning must proceed. To question these axioms wouldn't be possible.

However, I've recently seen an attitude that has puzzled me. Many philosophers state that very rarely does reasoning in philosophy rely on axioms. Axioms are things to be avoided and go against the spirit of philosophy.

What am I misunderstanding here? If philosophers don't take their tools of reasoning as axiomatic, how do they go about doing philosophy? More importantly, if philosophical reasoning is so pervasive that it questions its own tools, from what framework does the questioning occur? What tools does the philosopher use to question their own tools?

When this question was posed to a philosophy student, they responded that:

Philosophers don't tend to think of human thought or reasoning in terms of strict "axioms". Axioms are part of a formal logical system and it's not clear that a lot of our reasoning is like that. We hold many beliefs that we might typically think of as taken for granted. Philosophy is really about trying to understand what those are, whether they really fit together properly, and what properties of those beliefs we might want to look at to determine whether we can trust them or we ought to abandon them . . . [philosphers] generally share the idea that we take seriously our basic intuitions about cases of reasoning and we determine general rules and principles from them.

Is this the case? Is this how professional philosophers typically go about doing philosophy I've simply held a naive view?

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    It is important to realize that axiomatic method only dates back a little over a century even in mathematics, and people reasoned long before that. Euclid's reasoning in the Elements is not from axioms only, he has no logical axioms or rules of inference, his geometric postulates are incomplete, and his demonstrations rely on diagrams and "common sense". Part of the task of philosophy is to reflect and balance which set of "axioms" is "right" to use, so it would be nonsensical, aside from impractical, to confine it to some fixed axioms from the start. – Conifold Jun 12 at 4:55
  • @Conifold Yeah, that makes sense. I think I had this preconceived notion of how philosophy worked and this has lead to quite a few problems. But thanks for the clarification. I'm starting to see the approach of the philosopher is much more pervasive and akin to a kind of reflective equilibrium (I use the term generally, not as a reference to a specfic kind of theory). – Christian Dean Jun 12 at 6:25
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    There might just be something of a good intuition in the way you've perceived philosophy, there exists - for every philosopher - some functional axioms, these would not maybe be technically equivalent to what axioms represent in the axiomatic-logical understanding, but they act as anchors for the reasoning, at least in some temporary capacity, until the consequent construction/result shows shortcomings or distinct mistakes. – Gloserio Jun 12 at 8:35
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    It's then more of a termonological task to qualify such "anchors" as "more certain beliefs" or "axioms" as long as you understand how the reasoning operates: make a fixed combination of propositions and see how (combined with your tools of deduction) the resulting corpus of demonstrated/argumented propositions holds (well or not). – Gloserio Jun 12 at 8:35
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    @Gloserio I think even this relaxed model is misleading. Unlike mathematics, which is narrow but tall (few axioms, long inference chains), philosophy is broad but short. The substance and the skill is in eliciting premises, from judgment, reflection, intuition, tradition, etc., the inferences that follow are mostly trivial and quickly run dry without adjoining more premises. It is not like exploring theorems of a fixed set of axioms at all. There certainly are anchor beliefs that are used throughout, but they are like Euclid's postulates, nothing of interest can be deduced from them alone. – Conifold Jun 13 at 4:51
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If you look at how philosophy is usually taught and practiced, the answer you quoted is certainly correct. At least in my experience the strict axiomatic method you took for granted is something you will not usually find in academic practice, no matter on which level. On the contrary, the one principle you will find championed most often is that everything can be questioned and that philosophy should be understood as the discipline where exactly this is practiced.

Now, obviously such a description is itself problematic, as it borders on a form of relativism that many philosophers would itself reject vigorously. Let's say most would agree that not everything can sensibly be questioned, but regarding the question what exactly basic not-to-be-questioned axioms could be, you will not find much agreement.

Take the debate about the law of non-contradiction ( ¬(p ∧ ¬p) ) as an example:
Aristotle attempted several proofs of this law and many philosophers have considered these proofs satisfactory. Some philosophers like Hegel are believed to be 'opponents' of the LNC though, while yet others consider this claim a complete misunderstanding. And if you look at contemporary philosophy, there is a full fledged philosophy and logic built around the rejection (of at least some forms) of the law of non-contradiction. See https://plato.stanford.edu/entries/dialetheism/ for more information on that.

So far, this whole answer is purely descriptive of course, so it leaves open the question how philosophy could or should be ideally.

What should minimally be clear is that even if one sees a more axiomatic approach as desirable, finding the 'correct' axioms is obviously a huge challenge. This is perhaps most obvious if you compare how much disagreement there (already) is over foundations of mathematics or logic.

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    Thank you for taking the time to answer, I appreciate it! So would you say philosophy is generally done by taking our intuitions or our seemings seriously, such as our intuition that a law like non-contradiction is correct, and work from there to correct and clarifying our intuitions? Basically, something like the process of reflective equilibrium? – Christian Dean Jun 12 at 2:18
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    Yes, I'd consider that a good description, even though the SEP reflective equilibrium entry shows how complicated such a description itself is (specifically the 'criticisms' section.) I might add that I consider the axiomatic method a good ideal for philosophy, specifically in light of somewhat relativistic trends, where the fact that "almost everything has been questioned at some point" is sometimes brought up as an indication that rational consensus was not even possible, which clearly produces something of a non-ideal. – Yven Johannes Leist Jun 12 at 2:44
  • okay cool, thank you! – Christian Dean Jun 12 at 4:53
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    By the way, I would actually be interested to hear what kind of axioms you had in mind yourself - I think this could in fact be an interesting questions in its own right, along the lines of "What do people see as the most promising axioms of philosophy? or "If you had to give the most fundamental elements (axioms) of philosophy, what would they be?" – Yven Johannes Leist Jun 12 at 17:45
  • Well, most of them are pretty common "axiomatic" truths I would say: The three laws of thought, rules of inference, induction, abduction, and basic sources of belief. I probably also had in mind certain moral facts as being fundamental truths of morality that we discover. Although the reason I held to this view is more so because a found the line of reasoning persuasive which argued for perception being akin to ascertaining moral truths. – Christian Dean Jun 12 at 19:50

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