The Only Necessary Identity Statements

 

The Only Necessary Identity Statements

I. Introduction

The twentieth-century Analytic philosophers, particularly the later ones, have spelled out several problems related to identity statements. It can be difficult after reading them to confidently make statements of the form “A is B”. This essay will explore some of the problems outlined and provide some additional insights to better understand the nature of identity. Specifically, it will address the notions of necessity and contingency to provide them with clearer definitions and to find the difference between “necessary identities” and “contingent identities”. What I will show by the end of this paper is that contrary to the opinions of philosophers who came before me, our understanding of this dichotomy is flawed. Then, I will demonstrate where identity statements really originate and how to tell whether they show a necessary or contingent relationship.

II. Definitions:

Kripke does a superb job of pointing out and attacking the problem. The division between necessary and contingent statements is most obvious when discussing scientific findings. When saying that one thing is another, the implication is that we do not mean the same thing by the two different components. In his words:

“a large number of other identity statements have been emphasized as examples of contingent identity statements…. One of them is, for example, the statement ‘Heat is the motion of molecules.’…they are plainly contingent identity statements, just because they were scientific discoveries. After all, heat might have turned out not to have been the motion of molecules” (Martinich and Sosa 81).

This alludes to an earlier contention made by Quine:

“The two singular terms name the same thing. But the meanings must be treated as distinct, since the identity ‘Evening Star = Morning Star’ is a statement of fact established by astronomical observation. If ‘Evening Star’ and ‘Morning Star’ were alike in meaning, the identity ‘Evening Star = Morning Star’ would be analytic” (Martinich and Sosa 519).

To both scholars, identity statements such as “heat is the motion of molecules” or “the Evening Star is the Morning Star” are contingent, seemingly because the components of the statements have different meanings. That is, when one uses the word “heat” in everyday language, they obviously have a very different conception in their mind than “the motion of molecules”. Outside of chemists, physicists, biologists, and other scientists who work very closely with molecules on a regular basis, few people would have a picture in their mind of particles speeding up rather than the image of a stove when thinking about “heat”. It is not necessarily true that heat happens to be the motion of molecules; we can easily conceive of a world in which this fact could have been otherwise. Similarly, it is not necessarily true that the Evening Star and the Morning Star are the same thing. Prior to astronomical discoveries, it would not have appeared irrational to suggest that they were two different celestial bodies. These are contingent identities, and they differ from necessary identities which cannot be otherwise (common examples of this include mathematical statements such as 2+2=4).

However, I am not satisfied with these definitions because they lack clarity. What exactly does it mean to say that a fact “could have been otherwise”? Do we mean that if we rewind the clock then there is a chance for that fact not to come true? Most people would consider the following statement to be a contingent one: Joe Biden is the forty-sixth president of the United States. I am not bothered by the suggestion that, if history were to have played out differently, then this identity statement could have turned out false. However, I do not think we can take it for granted that history could have played out differently. If we assume that all the atoms and molecules in the universe were in the exact same place moving at the exact same speed, if we assume that the same political campaigns, speeches, debates, and voters were influenced in the exact same way, then what reason could we have to think that Joe Biden would not have ended up as the forty-sixth president? At best, we could only get to a state of agnosticism due to our inability (as far as I can tell) to time travel. We cannot say that this historical fact may not have turned out true without repeated trials, and we don’t even have access to a second trial.

There is also the definition from imagination; a contingent fact is one in which we could imagine it not to be the case. While it is true that the Morning Star is the Evening Star, we can easily imagine a scenario in which they were not discovered to be the same. But what difference does this make? Are we supposed to make philosophical judgments based on what we can imagine – about what makes sense to us? By this logic, any mathematical equation that is complex enough would have to be contingent if we do not know the answer. There are proofs online showing that 0 is equal to 2, and these proofs may appear as rational and sensible to someone who is unaware of certain rules in mathematics (I will not go through the proofs here but suffice it to say that they do make sense conceptually if we ignore specific principles). This can’t be right; math supposedly contains statements that can’t be false! Why would their ability or inability to be false depend on our creativity or capacity to be wrong? We should abandon this definition if we wish for the necessary-contingent distinction to lie outside of flawed human thinking.

Finally, there is the definition regarding “all possible worlds”. To use Kripke’s examples, we can imagine a world in which Benjamin Franklin was not the inventor of bifocals, but we cannot imagine a world in which the square root of twenty-five is not five. The former contains what Kripke calls a nonrigid designator, while the latter contains a rigid designator. In any world where the former exists, the statement “Benjamin Franklin was the inventor of bifocals” is not necessarily true, but in any world where the latter exists, the statement “5 is the square root of 25” is necessarily true. In one case, the existence of each component does not require the relationship between them, whereas the other one does require it. To put it another way, a rigid designator denotes the same thing in all possible worlds. It is unclear how this distinction differs significantly from the previous case. Even if we assume that there are other “possible worlds”, surely, we can only deal with the ones which we can imagine? What is the difference between conceiving of a world in which an identity statement is true there but not in this world and simply being wrong? Or, alternatively, flip the script. What if there is a possible world that I cannot imagine because my ability to think is constrained by the world in which I live? What if, instead of Benjamin Franklin, another person living in North America around the same time as him, whose name has unfortunately been long forgotten throughout history, invented bifocals? Can I imagine such a world if I do not know who the person is? Once again, it appears that the necessary-contingent distinction rests on our ability to think rather than the truth of the identity itself.

III. Conjunctions:

Perhaps we should start over. Let’s restate our original definitions. A necessary statement is one that could not have been otherwise, whereas a contingent statement could have. This sort of language is reminiscent of deductive arguments, in which a conclusion must logically follow from the premises. Maybe the answer to this quandary lies somewhere in there.

Let’s start with G.E. Moore’s proof of an external world as an example:

“[I]f I can prove that there are a pair of things, one of which is of one of these kinds and another of another, or a pair both of which are of one of them, then I shall have proved ipso facto that there are at least two ‘things outside of us’” (Martinich and Sosa 172-173).

To put the argument in the form of a logical syllogism:

Here’s a hand. (H)

Here’s another. (A)

If there are two spaces external to one another, then the external world exists. (If T, then E)

If there is a hand and another hand, then there are two spaces. (If H and A, then T)

Therefore, the external world exists. (E)

Notice, however, that this proof contains if-then statements, which imply a necessary connection between the two clauses. Since we are trying to figure out the meaning of “necessary”, we must better understand how if-then statements are formed. But that is what we are trying to do right now, which means we are going in circles. The presence of an if-then statement in this argument only pushes the question further back; where do necessary statements come from originally?

Let’s try a simpler argument – a simple conjunction.

Arthur is bald. (A)

Bethany has brown hair (B).

Therefore, Arthur is bald, and Bethany has brown hair (A and B).

This shouldn’t be controversial; surely the conjunction of the premises will always be a conclusion that follows in a valid deductive argument. If we assume that each component of the conjunction is true, then the entire conjunction must be true as well. So, if each premise is true individually, then a single statement combining them must also be true. Notice the subtle use of the word “must” here. I think we are there. Necessary statements seem to come from deductive arguments. If all the premises are true, then the conclusion must also be true. If premises, then conclusion. So, we can define “necessary” as there being an argument that can be made showing that the declaration in question is the logical endpoint of known facts. If you know that a given set of statements are true, then anything that logically follows from them, such as a conjunction of those statements, is “necessarily” true.

What is an “identity” statement? It is one in which one component is the conclusion of an argument, while the other component is the conjunction of the premises. This is how math works; when we say that 2 + 5 = 7, aren’t we saying that the “sum” of all the numbers on one side of the equation are equal to the number on the other side? When we say that “a bachelor is an unmarried man”, aren’t we saying that a “bachelor” contains the properties and only the properties of “unmarried” and “man”? The truth value of the conclusion is not isolated to any individual premise but is revealed because of all the premises put together.

Although, this is only true some of the time. Let’s return to Moore’s example: the result of the argument, that the external world exists, is true if all the steps leading to it are true. That is, the statement “the external world exists” is true if the following conjunction is true: “‘Here’s a hand’ and ‘Here’s another’ and ‘If there are two spaces external to one another, then the external world exists’ and ‘If there is a hand and another hand, then there are two spaces’”. However, it seems possible to maintain the conclusion that “the external world exists” while assuming that all the premises are false (or at least any one of them, since that would be enough to disprove the conjunction). Additionally, we can imagine a scenario in which Socrates is mortal, but either not all men are mortal, or Socrates is not a man (perhaps he is a cat). It seems that necessity is not a two-way street. If we know that the premises are all true, then the conclusion must follow; it cannot be false. On the other hand, if we know that the conclusion is true ahead of time, we cannot infer from that alone that the premises are true. We cannot infer from “the external world exists” the statement “Here is a hand”. Nor can we infer from “Socrates is mortal” that “All men are mortal, and Socrates is a man”.

This leads to a realization that stunned me upon first discovering it: the conclusion of an argument is not always logically equivalent to a conjunction of the premises. If it is possible for one statement to be true while the other is false, then they cannot mean the same thing! This seems unintuitive; after all, don’t the truth of the premises dictate the truth of the conclusion? Yes, but this is not so in the other direction. The premises can all be true at once only if the conclusion is true, but the conclusion may still be true while any one of the premises are false. The only exception to this rule seems to be the case with which we started: conjunctions. Conjunction statements (and any logically equivalent statements) are the only statements that are necessarily true if the premises are true and necessarily imply that each premise is true. If we know that “Arthur is bald, and Bethany has brown hair”, then we also know that “Arthur is bald”. But, if we know that “there is an external world”, we do not automatically know that “there is a hand”. As far as I can tell, conjunctions in reference to their components are the only examples of “necessary identities”. Every other conclusion in reference to its premises is a “contingent identity” because the truth values of each are not logically equivalent. For example, when we say “the Morning Star is the Evening Star” is contingent, we mean that one does not imply the sum of properties for the other. The “Morning Star” does not imply the property of being seen in the evening, and vice versa.

IV. Gettier Cases:

This understanding about logical deductive arguments may help explain at least some Gettier cases. Take Frank Gettier’s own famous case about Smith and Jones, in which Smith deductively concludes that the promotion will go to the person with ten coins in his pocket. If the boss told Smith that the job will go to Jones, and Jones has ten coins in his pocket, then the conclusion that logically follows is that the promotion will go to the person with ten coins in his pocket. In syllogistic form:

Jones is the person who will get the job. (J = P).

Jones has ten coins in his pocket (or, perhaps, Jones has the “property” of having ten coins in his pocket). (F(J) = T)

Therefore, the person who will get the job has ten coins in his pocket. (F(P) = T)

“In this case, Smith is clearly justified in believing that (e) is true….But it is equally clear that Smith does not know that (e) is true; for (e) is true in virtue of the number of coins in Smith’s pocket, while Smith does not know how many coins are in Smith’s pocket, and bases his beliefs in (e) on a count of the coins in Jones’s pocket, whom he falsely believes to be the man who will get the job” (Martinich and Sosa 206).

We must ask ourselves if the conclusion (F(P) = T) is logically equivalent to the conjunction of the premises ((J = P) and (F(J) = T)). Is it possible for one to be true while the other is false?

Let’s assume that the negation of the first premise was true. That is, Jones is not the person who will get the job (J != P). The conjunction of the premises automatically becomes false because the first premise is false. Is it still possible for the conclusion to be true? Gettier has demonstrated that it is; it turns out that Smith himself has ten coins in his pocket when he is given the promotion, so the person who gets the job still has ten coins in his pocket (F(P) = T). Therefore, the truth value of the conclusion is not logically equivalent to the conjunction of the premises, because it is possible for them to differ. So, it is a contingent identity, not a necessary one, that the person who gets the job is the person who has ten coins in his pocket. Gettier has pointed out a significant problem regarding our understanding of knowledge, but he neglects to follow through with a new theory like the one I am proposing here - although, I should note that Gettier’s article is about knowledge and epistemology, whereas I am talking about the actual truth of the matter.

V. Conclusion:

To summarize, I have explained why the traditional distinction between necessity and contingency is flawed; the definitions for these words are generally unclear and misunderstood. I have then given my own theory for the meanings of these words, in which a “necessary” fact is one that logically follows from a given set of statements, and a necessary identity depends on a relationship of logical equivalence between the conjunction of premises and the conclusion of an argument. If it is possible for the truth values of the conjunction of premises and conclusion to differ, then there is only a “contingent” identity; they happen to be true simultaneously, but they do not have to be.

I do think that this leads to another problem for which I currently do not have an answer. Is there any necessary identity that can be proven besides the conjunction of the premises? As far as I can tell, conjunctions are the only statements that provide their own criteria to be proven true. Math is filled with conjunctions. Logical operators such as pluses and minuses connect numbers together; they are not "disjunctive". Is it the case that we hold math to such a high standard because it contains only conjunctive arguments? This I do not know, but I think this paper provides the groundwork for one to make that case.

Works Cited 

Martinich, A.P. and David Sosa. Analytic Philosophy An Anthology Second Edition. Wiley-Blackwell, n.d.