[Slider thankfully has pointed out some flaws in parts of my argumentation and I recommend you to also read Slider's comment.]
Everybody who wants to have agency over one's live should believe two things. Firstly, there is an objective truth about the world in which we live. Secondly, we can partially discover this world: There are ways in which we can get an approximate understanding of what's true.
However, questions remain even if these things are taken for granted. My which methods can we get to this truth, and how precise can our approximation get?
It's quite clear for me (and I did not think of these issues myself but believe that they are well-covered in classical philosophy) that any method that uses all available information in the most effective and sound way should have logic as one of its tools (without the advantages of logical inference, the method is not effective; if it does not avoid contradictions, it is not sound). Furthermore, assuming that we don't think we already know the truth (which seems not plausible), we should look out for hints from the environment, evidence, which help us to get better insights into the real world.
There are some more things which basically all human beings who are able to survive assume: that we can partially trust our senses, except when they contradict each other or we have other reasons to doubt. These are basically axioms; just as the mathematician has to assume some axioms before beginning to work, nothing about the real world can be derived without axioms (but as soon as we have those axioms, together with our sensory experiences, we can infer much more conclusions about the real world using our brains). Having nothing but logic or mathematics, there are still many possible ways the world might interact with us. So we need these axioms, or if you wish, priors to trust our sensations.
However, choosing those axioms can be tricky. To get information about the whole of our planetary society, our senses are not enough. We need to extend their range using communication technology and media: phones, newspapers, the internet. To be able to talk about what happens on the other side of the earth, we have to extend our trust in our personal sensations to trust in some newspapers or that telecommunication cables faithfully transport information between different places.
So as soon as people started to globalize their lives, and therefore increase the size of their world relative to their immediate environment, the had to add another prior to their repertoire: What people (or media) tell us about distant happenings should be considered true if there is no reason to think otherwise.
Of course "no reason to think otherwise" is not well-defined, but there are some hints. If the reporting of media about supposed facts is contradictory between different journalists, bloggers, and media enterprises, one should lower one's confidence in the overall system and find out which sources are the reliable ones. If the messages from different sources are not contradictory, this is usually a good sign, except when there's a climate of fear or conformity among these sources.
Today, we seem to be in a difficult situation. There are lots of contradictory voices, at least in the internet, and it is very hard to identify reliable information. It seems impossible to DESTROY some position with FACTS and LOGIC because for every claim, the internet provides people that have objections that can't be dismissed out of hand. At the same time, established media outlets lose trust because they sometimes consider pieces of evidence as soldiers and not as gifts and seem to fall prey to the same group dynamics most human beings do when living in a polarized society. On the other hand, there are lots of people everywhere in the political spectrum that are happy to uses these problems to erode trust and deconstruct these institutions in and of itself, until we can't trust anyone anymore.
Maybe it's all become too complex? What if our extended sensations, namely internet and media, are so indirect and have so many middlemen and the topics they try to report on are so complicated that what remains on our end has a low signal-to-noise ratio? What if the resolution of our extended sensations is too low to reliably see the information that lets our beliefs converge against the truths we need so urgently to solve the most pressing problems in a globalized world?
I really, really don't know. I just feel so empty of orientation at the moment, though full of information. I feel like my maths degree does not help at all to approach truth. Maybe in these times, what we need in addition is just wisdom? But how can we find it (in a rational, not esoteric, sense)?
It often seems to me that these are really frustrating times for people honest enough to admit that they don't understand, while having been educated about, virtually anything in this world.
But given the success of Enlightenment, well-documented by Steven Pinker, I suspect that there have to be people with a clear-enough view of what's happening in the world to be able to improve standards of living for large parts of humanity, while managing nuclear deterrence and other dangerous constellations, it's probably just me who doesn't understand how all of this works.