Say I have an object that is currently red, but in one day will be painted green. If I put it in my formal language with a constant symbol 's' and define the colours green and red using 'g' and 'r' with the relation C that relates an object to it's primary colour.
The formula "C(s,r)" can be interpreted easily, however the truth value varies with time. Is this an acceptable case in FOL or do I need to move to a temporal logic to deal with this case?
Logic does not approach change.
Remember that Logic is the formal expression of the rules of Reason. Reason is the dynamic of thinking. When we speak of Logic as such ("Logic" in uppercase means the discipline that deals with logic) , we are referring to Propositional Logic. When we speak about Reason, we assume the Rules of Thought.
Logic is just a set of rules that apply to abstract objects, that is, metaphysical, ideal, rational objects; Logic never applies to physical objects. Physical/empirical objects are impossible to define mathematically, they only can be abstracted (e.g. you can abstracted as a dot inside a circle). Abstractions (that is, the modeling of huge amounts of atoms which have no boundaries as if they were perfectly homogeneous and bounded objects) occur during the process of reason, thinking. Reason and Logic can only address static objects.
Before and after a logical operation (e.g. If I have an apple, then, I'm rich, then, I am a capitalist), reason expects for the object, the apple, to remain similar to what was at the beginning, it expects for the object not to change too much. If, along an operation, the object has changed too much (e.g. the apple has been smashed), then, the logical operation is not valid anymore. In other words, logic (the tool, not the discipline) can't cope with change. Change is not a problem of the apple (meaning that in order to apply logic you need "apples that don't change"), this is a problem of logic. In fact, the apple is changing continuously in the physical world, although it does not change in the metaphysical space where it exists in your mind.
When you think of an apple, you think of a shape, texture, taste, smell, color, etc. You don't think in the interactions between fundamental forces that are impacting electrons in such local context, causing structural modifications amplified by brownian movement, etc. You are thinking on an object as you understand it. Not as a noumenon, not as a set of natural phenomena dependent on your subjective capabilities.
There is no way to logically address change, because change occurs in infinite dimensions. The problem is not related to logic, logic is just a set of rules. The problem is about the understanding, which takes arbitrary chunks of nature and calls them objects. You, for example, are just a set of atoms. Does it include the water you drank yesterday? The waste in your body? Your beard, if you don't like it and shave everyday? What about all the molecules you are losing in this precise moment? Mathematically, you are 1 person, but physically, it is literally impossible to define you.
Not even Mathematics is able to address change as such. Mathematically, an apple (total=1a) keeps being an apple along time, even if water evaporates and energy and molecules disperse in all directions. Eventually, when the apple rottens, and according to some metaphysical and subjective rule (e.g. apples are fruits that weigh 30 or more grams), then, suddenly, you don't have anymore total=1a, but total=0a.
Notice that in consequence, total=1+1=2 is a huge abstraction of physical facts. Physically, adding apples by means of a judgement like total=1+1 is impossible, because there are no two identical apples.
Worst even, any possible physical entity is constantly changing, there are no physical entities that do not change. Nature is pure change, there are no static objects in nature, only in our minds. Logic and mathematics address static objects, but nature is dynamic, it is permanently changing.
That's why Aristotle's Laws of Reason include the Principle of Identity: A keeps being A, remaining static and identical through a process of logical reasoning.
If you really need to assess physical change, you can use, for example, thermodynamics, that address certain specific types of change, by defining a set of rules (The Laws of Thermodynamics): for example: matter conserves (1st law); energy tends to disperse (2nd law), at 0k, energy dispersal is zero (3rd law); temperature is a transitive relationship (0th law).
However, not even thermodynamics approach real change: while thermodynamic systems are closed, meaning they are perfect and don't lose energy, in reality, all systems are open, closed systems are impossible, thermodynamic exchange systems are never perfect, etc. This discipline can only cope with one type of change: the change that occurs from a thermodynamic perspective (e.g. adiabatic, isothermal, quasi-static, etc.).
It is a fact that the logic of human reasoning allows humans to survive in a world which is in flux. It also allows human babies who come into this world knowing not one of its contingent facts to become proficient within a few years at navigating not only the pitfalls of making a living out of nature but also the pitfalls of relying on other humans to survive and prosper. Change, therefore, does not even register as a difficulty.
Logic allows humans to assume constant laws of nature out of the changing world of contingent facts, and then allows them to give up on these laws if they realise they do not work so as to be able to adopt new laws that seem to work. Change is fundamental to how human logic works.
