logic in the presence of doubt, uncertainty, lies
There isn't really a single formalization of the kind of thing you are asking. There are many, many aspects to truth, trust, lies, and fallible reasoning, and this leads to an enormous variety of logical formalisms, each handling different aspects of this problem. 


If you want to account for uncertainty about your hypotheses, the traditional route is via Bayesian probability theory. See E.T. Jaynes' (sadly incomplete) Probability Theory: the Logic of Science for a good exposition of this point of view.
One difficulty with probabilistic methods is that it is difficult to interpret quantifiers, essentially because limits do not necessarily exist. That is, a proposition may hold for all finite approximations, but fail to hold in the infinite limit.

Taking this into account leads you to view propositions topologically, which (a) leads you the Beth semantics of intuitionistic logic, and (b) also leads you to geometric logic. See Steve Vickers' Topology via Logic for an introductory exposition, and Peter Johnstone's Stone Spaces to jump into the deep end of the pool. 

(However, to my knowledge there is not yet any satisfactory constructive account of probability theory.) 
If you are interested in the difference between truth and belief, then you need to look at epistemic logic. The idea is to extend traditional logic with a modality $B_X(A)$, which is intended to read "$X$ believes $A$". See the SEP entry on epistemic logic for an introduction, and follow the bibliographic links in the article for more details. 
However, when you have multiple agents, you also have to think about the difference between truth and assertion. This is particularly important in applications like authorization (e.g., I can look at a file, because the owner of the file has the right to delegate the right to look at it, and says that I can look at it). There is a huge amount of work on this; a good entry point into this literature is in Deepak Garg's PhD thesis, Proof Theory for Authorization Logic and its Application to a Practical File System. 
Next, one of the traditional principles of logic is ex falso quodlibet: that is, $\bot \to A$ holds for all propositions $A$ — if you assume false, everything follows. If you are thinking about the possibility of believing false things (for example, someone has lied to you, or you have a database with errors in it), this is potentially disastrous. In practical reasoning, you don't want to derive a contradiction and then happily believe everything — you want to find a contradiction, and deduce that you have made an error. 

The study of what happens to logic when you drop ex falso is called relevance logic, so named because the idea is that you should only make inferences from hypotheses that are relevant to the conclusion. Again, see the SEP article on relevance logic for more. Also, you might wish to tolerate contradictions in your logic system. In this case, the thing to look at is paraconsistent logic. 
Next, you mentioned a worry that individual inferential steps might not be completely reliable. In traditional logic we accept the unrestricted use of modus ponens. That is, inside a proof, if we know $A \to B$ and $A$ hold, we can conclude $B$ holds, any number of times. If you start to consider the idea that individual inferential steps might not be completely reliable, then you might think that proofs depending on long chains of inferential steps are "less reliable" than short ones. 

It's a bit tricky to formalize this, but it is one of the (several) motivations for ultrafinitism. See Mannucci and Cherubin's draft Model Theory of Ultrafinitism I: Fuzzy Initial Segments of Arithmetic, for an exploration of this idea (and some explanation of its connection to fuzzy logic). 


Finally, note that that none of these approaches actually talk about Frankfurt's idea of bullshit as an assertion which is made with an indifference to its truth value. You probably want to look at J.L. Austin's theory of speech acts (e.g., his book How to Do Things With Words) to help organize your thinking on this, and if you try to formalize it, you will probably find Per Martin-Löf's judgemental methodology (see for example his Siena lectures) helpful.