“There is no cat on the mat.” Most philosophers consider truth-makers for negative facts to be problematic. In 1919, Bertrand Russell wrote:

Indeed, our truthopathy is equally strong for both sides of that medal. But it is not caused by problems concerning negative truths in particular. It is caused by conflating material reality and logic.

Intuitively, looking at our ‘empty’ mat, we have a truth-maker entailing our proposition that “There is no cat on the mat”, but the mat being ‘empty’ does not entail *that *proposition because the mat’s emptiness doesn’t refer to cats. It entails that there is no dinosaur on the mat either. Calling the mat empty makes all propositions true that say “There is no *x* on the mat.” So, ‘negative’ states of affairs seem to have the nasty habit to entail an infinite number of negative propositions.

Or maybe not, since first, the statement that on the mat there is no cat, dinosaur, or stupendously beautiful, sexually available person should rather be phrased as: “There’s nothing on the mat.” But ever since Parmenides, philosophers find the concept of ‘nothing’ very tedious. So far, ‘nothing’ has no ontology of its own. It has no properties by which it can be known and therefore exists only as a concept. And this is exactly why Parmenides denied that a ‘nothing’ *existed*:

Consequently, all negative truths are concepts and reality offers no truth-makers for them. It looks trivial to say that non-existents cannot be proven to exist, but the genuine problem is that they can neither be proven to ** not **exist. In Truth-makers, truth-bearers, truth-breakers I wrote that truth is self-service dining from the menu of reality. Reality is real, but what we make of it is human. This is even more true for negative truths, since these ask for something that isn’t even on reality’s menu.

For some elementary logic, let us call the proposition “There is no *x* on the mat” – x and the proposition that “There is no cat on the mat” – c. Now we look at – c -> – x: “If there’s no cat on the mat, it follows that there’s no x on the mat.” This is obviously wrong, since even if there’s no cat on the mat, there may still be a stupendously beautiful, sexually available person on it. So, let’s turn the premise around: – x -> – c: “If there’s no x on the mat, it follows that there’s no cat on the mat.” Dismissing technicalities such as dust, cat’s hairs, or simply air being on the mat, this proposition looks plausible. Using the transposition rule, – x -> – c is equal to – – c -> – – x. Removing the double negations we get c -> x: “If there’s a cat on the mat, there’s an x on the mat.” This premise is utterly trivial because it is only true *either if x is a cat *(by *modus ponens*: c -> x, c, therefore x) *or if there’s nothing* on the mat (by *modus tollens*: c -> x, – x, therefore – c). “There is no *x* on the mat” cannot be made true by a random value of x…

… as was to be expected since all truth-makers are necessarily definite. “Ravens are black” is made true by all individual, definite cases of black ravens. In fact, we are confronted here with an often-overlooked difference between positive and negative truths. Of truth-makers for positive truths, we may have verifiable knowledge. With truth-makers for negative truths we have two options: *either* they are acquired by some kind of *negation* – not-x, x-less –, which is deductive and therefore independent of the outside world, *or* we need to accept *negative properties* – empty, bare, absent – that are verifiable. I shall argue that both are dead-end.

The very same Russell cornered us by showing that propositions are never meaningless: they can have only two truth values: true or false. This step enabled logical positivists to grasp reality with the help of logic, but as soon as a state of affairs is said to entail a corresponding proposition, we are making what David Armstrong called a “category mistake”, mixing up logic with what exists in reality. To describe material reality, we need to express states of affairs in some language, either natural, logical or mathematical. But even if I saw my cat sitting on the windowsill, that would only materially* and not logically* entail it not being on the mat. An existent – e.g. a certain cat – can have infinitely many properties, such as ‘red’, ‘black’, ‘on the mat’ and ‘on the windowsill’. The question is: what makes it *logically* true that these colors or places are incompatible?

We already searched for the truth value of the proposition that “There is no *x* on the mat.” Splitting it up will give “There are cats”, “There is the mat” – a definite mat, our mat –, and there is some kind of relation between the two. Starting from the perspective of the cats, we get: “Of all cats, no one is such, that it is on our mat.” An important note here is that *every cat* is in *some* place. Materially, we cannot locate them all, but logically there is no problem: “All cats are in random places Y_{1}…Y_{n} that are not M (on our mat).” Thus, we turned our negative proposition into a positive one, that can *in theory *be tested for truth value: its truth-makers are the individual locations of all cats. Starting from the perspective of the mat, stating “The mat is empty”, we get “All x (things) are in a place Y that is not M (on our mat).” But as said above, *materially*, this is never the case. There is always *something* on the mat: dust, cat’s hairs, air. Obviously, it is *not the negativity* *of the proposition* that leads to over-generating truth-makers, but *the necessary incompleteness of our knowledge*. We just don’t know where all of those cats are, let alone where all *things* are.

Yet, defining negativity as incompatibility is problematic. In the example “All cats are in random places Y_{1}…Y_{n} that are not on our mat (M)”, we assume that Y_{1}…Y_{n} and M are incompatible members of the set of all places, where a place Y can be any member that is not M. A similar example: “All roses are of a random color (Z) that is not blue (B)”, where a random color Z and a definite color B belong to the set of all colors, which are considered incompatible.

Assuming the existence of these sets for the purpose of truth-making is a case of metaphysical overkill. First, they are abstract objects, so not the states of affairs most truth-maker adepts are after. Second, they are infinite: their members can be split up endlessly. Place, time, speed, color, radiation, energy, mass, and many other physical phenomena are continuous, not discrete. For example, in “Milk is not transparent”, the issue under investigation is the scattering of light in a liquid. Liquids can be opaque, translucent, or transparent, in innumerable gradations. The above sets cannot be defined by what their members have in common, but only by what their whole is about. And there we encounter new difficulties since many of these so-called sets are strongly intertwined, such as place, time and speed, color and radiation, energy and mass, and so on. Their ‘aboutnesses’ are ontologically indiscernible. We may conclude that incompatibility is not useful to underpin negative truths.

The second possible approach to negative truth is to accept *negative properties*. In “The mat is empty”, ‘empty’ could be an accidental property of mats. One may object to calling ‘empty’ a component of a state of affairs. But consider its opposites like ‘full’, ‘occupied’, ‘in use’ and so on. What might be truth-makers for these expressions? If in my tank is there is just air, is it full or empty? That would depend very much on whether I am driving or diving. If a country is being occupied, does that mean that before it was unoccupied? If the bathroom is in use, does that mean that there’s somebody in there or only that it is available? Expressions like the above get their meaning from their context. Negative properties are purely semantic and do nothing to connect language to reality.

Another case is with expressions such as colorless, odorless, whateverless. What could make such an expression true? Let us concentrate on ‘odorless’. Some philosophers, diligently mimicking each other, say that there may be “something to a liquid that excludes it from being odorless”, which serves as a truth-maker for “This liquid is odorless.” Consider the opposite: odorous. It is true under two conditions: a sufficiently vaporizing liquid and a damp that is noticeable by the human olfactory organ. If one of these conditions is not fulfilled, the liquid will be odorless *to us humans*. Since most liquids vaporize to at least some extent, the truth-maker for “This liquid is odorless” is not just ‘something to a liquid’. It only shows the limited capacity of the human sense of smell, compared to that of, for instance, dogs. By analogy, calling something colorless only shows the limited capacity of humans to see colors, compared to that of bees, and so on.