How much do you need to know about science to begin learning about the philosophy of science? Some but not a great deal if you are interested in very general metaphysical questions about e.g. the nature of explanation, laws and causation, and in very general methodological questions about how scientific theories are confirmed and refuted. You'll need to know quite a lot more if you are interested in understanding more specific foundational questions about the interpretation of quantum mechanics or are worrying about the nature of natural selection. Similarly, how much do you need to know about mathematics to begin learning about the philosophy of mathematics? Some but not a great deal if you are interested in very general metaphysical and epistemological questions about e.g. the nature of numbers and the nature of our knowledge of such things (if they are "things"). Quite a lot more if you are bugged by more specific questions about how we are to settle axioms for set theory or to decide...

Any university library will have dozens of elementary textbooks in its collection with titles like "Logic", "Formal Logic", "Elementary Logic", "Beginning Logic", "The Logic Book", etc. etc. The best advice is probably just to quickly browse through the opening chapter or two of a whole pile of them, till you find one that works for you. But here are three specific suggestions. I often recommend Samuel Guttenplan's The Languages of Logic for beginners who are struggling. Another quite excellent resource -- and freely downloadable -- is Paul Teller's A Modern Formal Logic Primer , available at http://tellerprimer.ucdavis.edu/ And then I have to confess I do still rather like P*t*r Sm*th's An Introduction to Formal Logic (though use the heavily corrected 2009 reprint).

Any good textbook that covers second-order logic should in fact clearly answer your question. Here, though, is a summary answer. An inference in the formal language L from the set of premisses A to the conclusion C is valid if every interpretation of L (that respects the meaning of the logical operators) which makes all the members of A true makes C true too. A deductive proof system S for sentences in the formal language L is complete if, for every valid inference from a set of premisses A to the conclusion C there is deduction in the system from (some of) the premisses in A to the conclusion C. If L is a first-order language, then there is a deductive system S1 which is complete in the sense defined ("first-order logic", meaning any standard deductive system for first-order logic, is complete). If L is a second-order language, with the second-order quantifiers constrained to run over all the subsets of the domain, then there is no deductive system S2 which is complete in the same sense (any...

It sounds as if you have relatively little background in philosophy. So I would suggest that, after doing an MA in the History of Philosophy, it would be wise to do another one in contemporary philosophy before doing a PhD, both for intellectual and for career purposes. Intellectually, because a lot of the best work in history of philosophy involves a kind of conversation between the Great Dead Philosophers and contemporary philosophy -- you need to appreciate both sides of the conversation. For career purposes, as many departments are not minded to appoint those they see as narrow specialists in the history of philosophy.

You certainly don't need to be able to do original research in maths to be able to work on the philosophy of maths. But you will need to be able to follow whatever maths is particularly relevant to your philosophical interests. How much maths that is, which topics at which levels, will depend on your philosophical projects. For example, compare and contrast the following questions (not exactly a random sample -- they all happen to interest me!): "Is our basic arithmetical knowledge in any sense grounded in intuition?" Evidently, you don't need any special mathematical knowledge to tackle that . "Can a fictionalist about mathematics explain its applicability?" Again, I guess that acquaintance with the sort of high school mathematics that indeed gets applied is probably all you really need to know to discuss this too. "Just what infinitary assumptions are we committed to if we accept applicable mathematics as true?" Here you do need to get more into the maths, and know quite a lot about...

It doesn't take much science-fiction imagination to conceive of creatures -- Klingons, or whatever! -- who work differently. When their equanimity is disturbed, e.g. by relationships falling apart, then they naturally recover their emotional balance after a while, so long as they don't keep dwelling on things. Rather as we heal broken skin so long as we don't keep picking at the scabs, so for our Klingons talking about their problems is like picking at scabs. For them, it is better to "let nature take its course" and for their emotional system to recalibrate itself to the new situation without paying too much conscious attention to the processes. They don't suffer from effects of "repressing" bad experiences, etc.: in fact they function better if they do "repress". Now, if we are very different from our imagined Klingons (as modern therapeutic "it's good to talk" theories suppose we are), then that's an empirical fact about us. And if it is a contingent empirical matter, as it...

This does sound a bit like a question asking for help with a student paper, which isn't really the role of this site: and certainly this sort of techie question doesn't lend itself to a snappy answer here. So just two comments. First, paraconsistent logics don't have to attack the law of non-contradiction -- i.e. paraconsistent logics don't have to say there are true contradictions (dialetheias): see here for more explanation. Second, for some defences of the law of non-contradiction, see the papers collected in Part V of Priest, Beall and Armour-Garb (eds) The Law of Non-Contradiction .

I'm not quite sure what is meant by "sublime insight"! But anyway, serious philosophy involves negotiating your way around thickets of argument . Philosophical originality is a matter of finding new moves to make (or breathing new live into old moves) in argued debates that have usually been developing and deepening for many years, in some cases for thousands(!) of years. Generations of philosophers have explored the options on (as it might be) the liar paradox, or the free-will problem, or the nature of consciousness, with ever-more sophistication, piling distinction upon distinction, argument upon argument. And yes, of course, lots of progress is made -- refining the options, working out their costs and benefits, and often engaging with the relevant science (or work in logic, etc.) as that develops. Now, it is hard enough for graduate students who've devoted five, six, or more years studying philosophy to start making much progress -- they have to get to grips with so much first, in order...

Well, there is a logical truth in the vicinity of 1 + 1 = 2. Or perhaps better, a whole family of logical truths. Fix on a pair of properties F and G . Then it is a theorem of first-order logic that if exactly one thing is F and one thing is G and nothing is both F and G , then are exactly two things are either- F -or- G . Here the numerical quantifiers 'exactly one thing is' and 'exactly two things are' can be defined in standard ways from the ordinary first-order quantifiers and identity. And the theorem holds whatever pair of properties we choose. This elementary logical result probably captures what is driving your intuition that in some sense 1 + 1 = 2 is "reducible to formal logic". (For a bit more on this sort of thing, see my Intro to Formal Logic §33.3 -- or any other standard logic text!) But all that is quite compatible with Gödel's first incompleteness theorem. For Gödel's theorem isn't about some limitation or incompleteness in our ability to prove ...

I'd say: if you've done a maths course or two already, then you should have learnt some lessons about arguing rigorously and giving absolutely clear gap-free proofs. Doing further courses won't teach you any more about that. So if you are not going to specialize in the philosophy of mathematics, a little maths in addition to some logic is already enough. If you want to work eventually in the philosophy of mind and language, then much better to do some courses on scientific psychology, neuro-biology, and linguistics.