I believe our disagreement can be reduced to three problematic premises; first the premise that knowledge must be verified by observation; second, that true statements must be precise; and third, that a statement cannot be true unless it is also known. To thoroughly refute each of these premises is no simple task, so I have to ask that you bear with me while we examine each.
From the position that all knowledge must be verified by observation there follow some interesting conclusions:
(1) It is impossible for me to know whether the statement "Toonia is feeling calm," is true or false. I can only verify statements such as "Toonia has a slow heart rate," "Toonia has low levels of cortisol in her bloodstream," or "Toonia is slouched in her chair." I cannot verify the statement "Toonia is feeling calm," because Toonia's subjective experience is unverifiable to all but Toonia herself.
Therefore, only Toonia can really know whether the statement "Toonia is feeling calm" is true, such a statement is only subjectively verifiable.
(2) The kind of statements which can be inter-subjectively known are those of science, such as the statement "the electromagnetic waves arriving from the direction of the sky and interacting with the photosensitive proteins in my retina, have a wavelength between 440 and 490 nanometres." This statement can be verified by both of us.
Therefore, the statement in (2) can be known by Toonia and anyone else, or may be called objectively verifiable.
That difficulties may arise with such a pressuposition that knowledge must be verified by observation can be easily shown. For example, we can make the following deductions:
(3) The kind of statements commonly found in science, such as "every object in the universe attracts every other object of the universe with a force proportional to the objects' mass and distance" (i.e. universal quantifiers) are unverifiable, and therefore, unknowable
In fact, every scientific law is unverifiable, since a law is a prohibition of a particular set of occurences, but since we can never scour the entire universe to verify the law, such laws cannot, according the presupposition that knowledge is verified by observation, be considered knowledge.
(4) The future cannot be known or predicted, since statements about what has not yet occured cannot be verified, and thus cannot be considered knowledge.
This problem, in conjunction with (4) places us in a position where only the past and that which has been observed so far can be known.
(5) The statement "memory is a collection of previously verified experience," cannot be verified by experience, and so cannot be known to anyone.
This places us in the sorry position that memory cannot be verified as a source of past verifications, and cannot be considered knowledge.
(6) It is impossible for me to know whether the statement "Toonia has subjective experience," is true or false. I can only verify statements such as "Toonia has a slow heart rate," "Toonia has low levels of cortisol in her bloodstream," or "Toonia is slouched in her chair." I cannot know that Toonia has any subjective experience at all.
So far this principle leads us to the conclusion that the past is unknowable, the future is unknowable, science is impossible and that other minds are unknowable. If this was not enough to demonstrate the poverty of such a principle, then the following may perhaps suffice.
(7) The statement "there is a reality beyond our sensory perceptions," is univerifiable, since all verification of the external world consists of sensory perceptions, we can never know there is anything but those sensory perceptions.
(edit: I forgot here to also mention that ethical knowledge suffers the same fate)
These problems would seem enough, but lurking behind all this is a deeper error; the presuppostion that observation is an epistemic authority, or that observation can in some way justify our statements. That we can only really know something if we can justifiably know it, and to that end we must justify our premises with reference to an authority, such as sense observation, The Bible, clear and distinct ideas, subjective experience, etc. Indeed, it can be easily shown that any such principle leads to an infinite regress, and even if we could somehow stem that regress, would be a ciruclar argument.
For example, let us take the proposition "R" and try to justify it. You'll notice that to justify "R" we must be able to derive "R" from a justifier, so we invoke "Q" as our justifier:
If Q then R
However, this may seem to justify "R", but now "Q" and "If Q then R" are unjustified. Now it is evident that "R" is not really justified at all if "Q" and "If Q then R" are unjustifed, so we may invoke "P":
If P then (If Q then R)
But this obviously does not solve our problem, since "P" and "If P then (If Q then R)" are unjustified, then it follows that neither are "If Q then R," "Q" or "R." In fact, even should we somehow stem this regress, perhaps by dogmatic assertion ("P is true and that's just the end of it!"), the argument would still be circular. This is because if "If Q then R," "Q" and "R" are to be deduced (i.e. deducted) from P, then P already asserts them to be true. The argument is circular!
(edit: It can also be shown that the requirement of justification (or verification, confirmation, good reasons etc.), is itself unjustifiable, not only any proposed justifiers.)
This brings me on to the premise that I cannot assert statements to be true unless they are also known to be true. Firstly, if a statement is true then it is true irrespective of whether we have satsified some supposed epistemic authority; secondly, if such epistmic standards lead to inconsistency, as argued above, then the very demand should be dropped. This leads me on to pointing out that true belief, not justifiably true belief should be our goal. Indeed in practical matters the two are indistinguishable.
I will get to approximate truth and precise truth later, now I need a break.