Previously I made the claim that nothing about the world can be known a priori (in the absence of experience), and that the only things that can be known a priori are facts about constructed systems, leaving the question as to what in the world corresponds to the system, if anything, to be an a posteriori one. Of course I am sure that not everyone is happy with this claim, especially other philosophers. Let me assume that my positive claim from last time, that nothing about the world can be known to be the case a priori, is accepted. Even so, some would still insist that some things can be known a priori to be impossible. For example, consider the circular square. Some would say that nothing could possibly be a circle and a square at the same time, and thus that we know it can’t exist in the real world a priori.
In actuality it is very easy to show that circular squares are in fact possible in some worlds (although not in this world I would hope). Assume that we define a square as a shape with four sides and four right angles, and assume we define a circle as a shape in which all points are equidistant from some other point. Now assume that the geometry of the world is such that the distance between two points A and B is defined as: max(|Ax – Bx|, |Ay – By|), meaning that the distance is whichever is greater: the difference in the x components or the difference in the y components. And if you trace out a circle using this definition of distance you will find out that it is in fact a square. Of course you might insist that we can know a priori that the distance between two points is defined in the standard way. But this is something you can’t know a priori because distance is not in fact defined in the standard Euclidean way in the real world, one of the advances of special relativity was to show that the geometry of the world was not Euclidean. Thus, as history shows us, the definition of distance is something that must be determined experimentally.
Of course perhaps this is simply a bad example, because the definitions didn’t truly conflict, in the sense that it was possible for me to construct a model in which they could both hold of the same shape. Let us consider instead the claim that nothing is both A and ~A, where A is some property. Of course one part of the object could be A and a different part could obviously be ~A, but let us assume that we have ruled that out (and ruled out the possibility that A can hold to different degrees). Even so, to assume that such objects are impossible is to assume that an object cannot be two different things at once (like a standard baseball and a standard car at the same time). Of course it probably is impossible for objects in our world to be two things at the same time (although this might be a very real possibility in the quantum world, but let’s not go there). But we can’t know that it is impossible for objects to be two things at the same time without experiencing the world, to see that things don’t behave in that way.
Another way to look at this problem is to return to the distinction between “real” properties and constructed properties, which I developed when talking about inter-world relations. Real properties were those for which we had some way of determining, in an objective manner, when they apply (for example, a test for height by comparing an object to markings on a surface), versus constructed properties, which apply only to objects in abstract systems (such as the lines and points in geometry) or to the real world by fiat. We might claim a priori knowledge about constructed properties, but such properties tell us nothing about the world. And we can’t claim a priori knowledge about real properties, not even regarding the assertion that objects will never both have a real property and not have a real property, as we cannot rule out a priori the possibility that the test by which we tell if the property applies will either both yield a positive and negative result simultaneously (some kind of superposition perhaps?) or that the result will be indeterminate, or in some other way unknowable. Anyone who claims otherwise simply lacks imagination about how the world might work (or, perhaps more charitably, are simply unable to separate their reasoning in these matters from their deeply ingrained intuitions about the way the world works that are themselves a posteriori).