Well let’s find out if it makes sense to say that they are. Let’s imagine that Adam’s culture defines “space pixie” as “one of the living creatures that has wings, and whose species is solely responsible for bringing water to Earth.”
If definitions are arbitrary matters of social convention, then Adam can reason like this:
- Premise 1: If there is water on Earth, it was brought by space pixies.
- Premise 2: There is water on Earth.
- Conclusion 1: Therefore, (only) space pixies brought water to Earth.
- Premise 3: All things that bring water must exist at the time it is brought.
- Conclusion 2: Therefore, space pixies must have existed.
This argument is deductively valid: If the premises are true, then the conclusions must also be true.
Furthermore, this argument is sound (has true premises and is valid–irrefutably correct) given the stated definition of “space pixie.” If definitions are arbitrary cultural inventions, then Premise 1 is “analytic”–that is, it is true purely by definition: One need only examine the definition of “space pixie” to find that Premise 1 is true. Premise 2 is an empirical truth, obvious to anyone who observes Earth and holds the common definition of “water.” Premise 3 simply states an indubitable fact: that things that act in reality must also exist.
Now, let’s imagine that Beth’s culture defines “space dwarf” as “one of the living creatures that has no wings, and whose species is solely responsible for bringing water to Earth.”
If definitions are arbitrary matters of social convention, then Beth can reason like this:
- Premise 1: If there is water on Earth, it was brought by space dwarves.
- Premise 2: There is water on Earth.
- Conclusion 1: Therefore, (only) space dwarves brought water to Earth.
- Premise 3: All things that bring water must exist at the time it is brought.
- Conclusion 2: Therefore, space dwarves must have existed.
This argument is also deductively valid, just like Adam’s. Given Beth’s definition of “space dwarf,” it is sound and irrefutable, as well. Yet Beth’s Conclusion 1 flatly contradicts Adam’s Conclusion 1, and both of them have apparently sound arguments that use the presence of water on Earth as the sole evidence “proving” the existence of different sets of creatures.
Indeed, the view that definitions are arbitrary means that Adam could alter the physical reality of what brought water to Earth, (billions of years before he was born) merely by altering the definitions he accepts in his mind. He could alter physical facts on his whim, implying that things have no identity and are not what they are, independent of consciousness.
How do we escape these absurdities, contradictions and clearly unjustified leaps in inference? If we say that logic itself is an arbitrary convention and that contradictions can exist in reality, then claims to knowledge of any sort are destroyed, and both philosophy and science are dead and done with. Everything we utter is self-refutation. (See: The Axioms of Objectivism.) This is self-evidently the wrong approach.
To eliminate the contradiction, we must acknowledge that the definitions we use must be justified by evidence. We must understand that definitions are not matters of arbitrary social convention: they can be right or wrong, according to whether or not they fit the available evidence. When they are held to be defining for things in objective reality, (as opposed to fantasy) they can be held to be improper, due to arbitrariness. (By “arbitrariness,” I mean the state of not having any evidence to justify the definition.)
In the case of “space pixies” and “space dwarves,” we can rule these definitions out as unjustified.
But this solution leaves many pressing questions: When and how does evidence justify a definition? If evidence is ultimately sensory, how can it justify definitions for very abstract concepts, such as “justice”? If concepts are defined in terms of other concepts, how can we avoid an infinite regress or circularity of definitions? Etc.
The broad strokes of answers to these questions can be found in Introduction to Objectivist Epistemology. It elaborates on the nature of concepts as mental universals, how they are formed from sense experience in a hierarchical fashion, and how definitions fit into the process of concept formation. Also recommended is Harry Binswanger’s How We Know and Concepts and Their Role in Knowledge: Reflections on Objectivist Epistemology by various authors. (Details on the rich theory of sensory perception that underlies Rand’s theory of concepts can be found in The Evidence of the Senses: A Realist Theory of Perception by David Kelley.)
P.S. Epistemology on an Objectivist base is fertile new territory for philosophical development and discovery: There are many philosophical questions left to be answered and technical nuances to be explored and related to other ideas. The basic philosophical framework that led to the above insight about definitions, presents a tremendous opportunity for new academic exploration. For example: What are the full implications of this theory of concepts for: inductive reasoning? various kinds of propositions? deductive logic? probability? sense vs. reference in meaning? mathematical concepts? Etc.
Addendum: After posting the above essay on reddit, a user, /u/identicalParticle, gave a couple of interesting and challenging responses to the essay, in which he made an intelligent attempt at refuting the reductio ad absurdum in the essay. I have transcribed this conversation here:
identicalParticle: I think I can refute it.
A definition might refer to nothing. If you’re arguing from a definition, you need to be sure that your argument is valid in both the case where your definition refers to something, AND in the case where it refers to nothing.
The argument (premise 1) “If there is water on Earth, it was brought by space pixies” is invalid if the definition “space pixie” doesn’t refer to anything. Since anything follows from a falsehood, the author can draw any conclusion they want.
If your argument is only valid when a definition is refering to something, you have to include another premise in your argument. (premise 0) “There is at least one thing that the term ‘space pixie’ refers to”.
If you’re careful this way, you can see that the argument reduces to “if there is at least one space pixie, then space pixies exist”. Which is not a useful argument.
This type of argument is used very often, usually to demonstrate that something doesn’t exist. For example, definition: X is the number that when multiplied by 0 gives the result 1.
Premise 0: X refers to something (exists)
Premise 1: 0 * 2 = 0
Premise 2: X * 0 * 2 = 1 * 2
Conclusion: 1 = 2
Now we realize there’s a contradiction. That means one of our premises must be wrong and we can conclude it must be premise 0. In other words, X does not exist. There is no need to say that the definition is wrong, we only need to say that premise 0 is wrong.
If you’re arguing from a definition, you need to be sure that your argument is valid in both the case where your definition refers to something, AND in the case where it refers to nothing.
In conventional deductive logic, (without any modifications Objectivism would make on the basis of its theory of concepts) validity is purely a function of the form of the argument, and not content or reference. There is no difference made to the validity of an argument by whether the terms refer to anything real or not, and the extra premise you added is not needed to ensure validity.
As far as judging the truth of one of the premises of an argument by the use of a definition, I think there’s hardly an academic philosopher in the world who would, in practice, disapprove of reasoning that “All bachelors are men,” purely on the basis of the definition of “bachelor” as “an unmarried man.” Yet I would venture to say that a great many of those same philosophers consider definitions to be matters of culturally relative semantics–effectively arbitrary groupings of particulars, made out of convenience or economy of communication.
Since anything follows from a falsehood, the author can draw any conclusion they want.
In conventional logic, the “falsehood” from which anything follows is specifically a contradiction, not a false premise, nor a reference failure. There are no contradictions stated in the definition, nor in the premises of the modus ponens argument.
As far as I can tell, the only things left to question are the given definitions of “space pixie” and “space dwarf.” They are arbitrary definitions, unjustified by (ultimately perceptual) evidence.
identicalParticle: I obviously don’t have as much background in logic as you do, but it seems to me that that statement
“If there is water on Earth, it was brought by nothing”
is a contradiction. The two statements a) “everthing that is brought it is brought by something” and b) “the water which was brought to earth was brought by nothing” contradict with one another. Statement a) wasn’t written explicitly in the argument, but I think it is implicit by the fact that “brought” is a transitive verb.
So if “space pixie” refers to nothing, there is a contradiction in the argument.
Is there a way for something to be false without being in contradiction with something else? Usually if a second statement is in contradiction with a first statement, and we if beleive the first statement, then we would use the convention that the second statement is called false and the first statement is caled true.
There is no difference made to the validity of an argument by whether the terms refer to anything real or not, and the extra premise you added is not needed to ensure validity
I’m not suggesting the issue is that a definition may refer to something real, or something not real. I think the issue is that a definition may refer to nothing at all (real or otherwise). The set of things referred to by the definition “space pixie” could contain real things, or imaginary things, OR it may be the empty set.
In my comment I gave two examples where a definition referring to something was necessary for the argument to be valid. Does this mean “conventional deductive logic” is limited in the types of things it can describe? I’m not really sure precisely what conventional deductive logic is or what it is intended to describe.
Sword_of_Apollo: Thanks, /u/identicalParticle, I appreciate your insight. You have shown me that the reductio I proposed will apparently fail, if one grants a premise that I reject: Namely, that “nothing” is a something in external reality to which terms can refer.
Have you ever seen anything that you can call an absolute “nothing?” I haven’t. Everything I see is some thing. Everything in the universe is a something, not a “nothing.” In fact, it is a self-contradiction to say that a literal “Nothing is a something.” By calling it “nothing,” someone is saying that it is not a thing. By saying that it is “something,” that person is saying that it is a thing. The person who holds the premise is thus contradicting himself.
So, to what do we refer when we talk of “nothing”? We refer to an expectation or imagination in our minds that does not match what we actually see. If I say, “There is nothing in my pocket,” I don’t (and can’t) mean that there is an utter void with no identity in my pocket. I mean that the tangible things that I might expect or imagine to be in my pocket contrast with what is actually there: air, cosmic rays, neutrinos, and perhaps some sort of medium that transmits matter waves and/or forces.
If I say that there are 0 [zero] baseballs on the field, I am saying that, of the totality of things that are on the field, all of them are disqualified from falling under the concept of “baseball.” I might imagine that a baseball would be on the field, but that imagination does not match what I see.
So one cannot literally say that a concept can ever refer to some particular type of thing out there, called a “nothing.” Thus, “Refers to nothing,” really means “Does not refer to anything external.” And there is no need to ensure logical validity in that case, because then it is not really a concept of reality, but, at best, a pseudo-concept of the imagination. (And contradictions in fantasy pose no problem.)
So, if we have a definition that is alleged to refer to something in extramental reality, but we cannot give any evidence whatsoever that it may, then it should be thrown out as something unrelated to the process of gaining knowledge of reality: there is no “nothing” to refer to in reality, and so the definition is the material of fantasy.
Again, your thoughtful discussion is appreciated. I think I’ll add this discussion as an addendum to the linked essay.