My primary insight around epistemology, one that I appear to be the only person on the intertubes pushing, is that Deductive Arguments both aren't and shouldn't be very persuasive.
Inductive proof the 1st:
How many times have you seen a deductive argument settle a factual matter under dispute? I hear Coase managed it a time or two. I believe that James C. Maxwell had a strikiing deductive conclusion with pretty strong smackdown capabilities. But really...how many deductive arguments have *settled* a matter of fact? Without strong confidence in the conclusion, I'm inclined to suggest: 2. The two I mentioned above. Given the history of the world (2500 years), and the existence of roughly two known cases where deduction has given us a conclusion...I am inclined, by observation, toward the conclusion that deduction has *tremendously* less power than we credit it with.
What does deduction do well?
Provide hypotheses that we can take seriously enough to test them.
Inductive proof the 2nd:
When you watch deductive arguments...what results does a thorough demolishing of one of the premises currently held by one of the disputants have? Does it result in the disputant decreasing his probability that his position is correct? Or does the disputant whose premise was demolished replace his premise, or even his entire line of argument, and leave his conclusion unchanged? By observation, human behavior overwhelmingly chooses the latter option. By the basic economic methodology of revealed preferences, one should then conclude that people do not take deductive arguments as truth-bearing.
Inductive proof the 3rd:
When you watch in an argument for which you have sympathy for both sides of the argument...you can rather usually easily determnine that the arguments are not actually opposed to one another. The conventional theory of argument is that premises A, B, and C forces conclusions D, E, and F, which results in position G. The opposition asserts premises N, M and L, which force conclusions K, J, and I, which results in position H, directly opposed to G. Everything nice and linear.
A,B,C -> D,E,F -> G-> | <-H <- I,J,K <- L,M,N
In most cases, though, that's not what's being argued. In most cases, the arguments appear to be opposed, but end up being angularly opposed, as opposed to directly. For instance....the argument for determinism (or quantum indeterminacy, effectively the same line) goes something like this: We have reason to believe that the universe is composed of particles in motion which obey specific laws. You are in the universe, composed of particles in motion. Therefore your internal particles obey specific laws also. Therefore, determinism. The argument for free will goes something like this: I experience consciousness, and the experience of choosing at the same level of awareness that I experience the external world. Attempting to suggest that my strong awareness of choice is false gets pretty hinky, and should not be taken seriously. Rather than plumping for Dennett's compatibilism here...I'd like to point out that the two arguments are not discussing the same topic. Free-willies are discussing the experience of free will, whether one makes choices. Determinists are discussing the question of whether the choices could have been different (in another run through the universe). Two entirely unrelated questions...that sound similar.
If you listen to the arguments between the two sides...without they themselves noticing what's happening, you will observe that the argument comes down to a dispute over which premises, which method of analysis is more important. The Free Willies say: The direct experience of free will is more important to this analysis than the unsupported conclusion that there is nothing in the universe but particulate motion. The determinists say that the scientific edifice that analyzes everything from the point of view of particulate motion is more important than the cognitive illusion of free will.
If you watch closely enough...most (honest) arguments follow this path. Two disputants, or two sides, each arguing the importance of their method of analysis or approach. Not two disputants arguing in any real sense about which is true.
Inductive proof the 4th:
If scientist/theoretician proves that human flight is impossible, and then you see the Wright brothers fly, which do you believe? Observation trumps theory, in roughly all cases. Induction FTW. Deduction is not truth-bearing...or more accurately, Deduction's truth-capabilities are entirely subject to the prior modeling problem...of whether one has abstracted the problem sufficiently well.
Deduction should be thought of as a red-headed step-child: Wild, unruly, badly behaved, and usually wrong.