The Law of Excluded Middle
Ordinary mathematics usually poses some arguments that present knowledge about logical reasoning or pure logic. For this same reasoning, the Law of Excluded Middle (LEM) came into existence. This law states that for every proposition, either its negation or the proposition itself is true. The law of excluded middle is one of the three laws of thought alongside the law of identity, and that of non-contradiction. One of the prominent mathematicians who argued out the law of the excluded middle is De Morgan. According to De Morgan’s laws, LEM is logically equivalent to the law of noncontradiction. Nevertheless, none of these laws provide inference rules. LEM is usually confused with the principle of bivalence, which states that every proposition is either true or false. Philosophers have had several special attractions to mathematics. Such philosophers include Plato. Plato is well known for his various works concerning the mathematics of geometry. In geometry, the universe is developed and understood as opposed to how it appears. The same way the constructs of geometry were built paved the way for similar philosophical reasoning in developing the LEM. The most powerful argument against assuming LEM is that it can decide things that a machine or a human being cannot decide on their own.
De Morgan’s rule for deriving LEM
The law of noncontradiction says that no statement can be both true and false at the same time. LEM suggests that all statements must be on one side of the argument or the other. LEM can clearly be understood by considering what it implies for the principle of non-contradiction to be true. For the law of noncontradiction to be upheld, the proposition for the law of noncontradiction must be true, i.e. [¬(p^¬p)] must be true (Shapiro 2012, pg. 306). This implies that p^¬p must be false. But according to the true definition of a conjunction, p^¬p can only be false if at least one of the conjunctions is false (Gonthier 2013, pg. 173). Thus, either ¬p or p is false. So if p is false, then ¬p must be true, and the vice versa holds. So what is left is the disjunction p_¬p, which is the formulation for LEM. Thus LEM can be derived from the principle of non-contradiction.
The most powerful argument for not assuming LEM
Suppose two real numbers are picked at random say a and b. then it becomes obvious that a>b, a<b, or a=b. But suppose a=b, how would that be known? Therefore, the process starts by comparing the digits a and b assume on the real number line. It is difficult to tell that a and b are the same such that a= b holds for all real values of a and b. due to the infinite sequences of the digits, it is difficult to tell a=b (Gonthier 2013, pg. 178). Therefore, the program comparing the two numbers would never halt, if a and b are the same. This brings the concept of constructive mathematics. If a program cannot decide, then the statements are undecidable, and constructively not a theorem that for all real values of a and b, we have a>b, a<b, or a=b. LEM can be expressed by a propositional formula p_¬p. This means that a statement is either true or false. Because LEM excludes the middle ground between truth and falsity, human beings or machines cannot decide on where to place an argument, neither are the machines. But the truth is that such arguments exist in the philosophical world.
Constructive mathematicians believe that certain mathematical universes have sets which are like topological spaces or open sets. From this aspect of the empty set theory (ɸ), we can construct another mathematical argument on the reason why LEM should not be assumed in mathematics (Gonthier 2013, pg. 165). In the same fashion of using the random numbers of a and b in the above example, for any property of an empty set ɸ, there is an x with ɸ(x) or there is no x with ɸ(x) based on LEM. In terms of the constructive mathematics, this argument is senseless. This is because, if we say there exist, it means we can find one. In the case of the continuous function f: [a,b] →R with f(a) <0 and f(b)>0, then there exists an ϵ such that f(ϵ)=0, however, it is a very difficult task to find the exact ϵ. ϵ can only be approximated, thus the theorem of LEM can fail. But if we argue that a zero can be approximated, then LEM will work in this case.
LEM itself has certain exceptions regarding the conditionals that provide the philosophical frameworks for its operation. The statements on the truth tables are inherently configured to be true or false (Shapiro 2012, pg. 311). This might have certain exceptions regarding the conditionals. When faced with a conditional statement in which the antecedent is false, it is commonly held the tradition that the conditional is automatically true. So if we have a condition where we say if it shines tomorrow (p), then I will go for a run (q), in the event of nonoccurrence of p tomorrow, the entire condition is considered valid. However, from certain conditions, it would neither be true nor false, and its analysis can only determine the whole meaning of the statement. But since the antecedent never occurs, it is difficult to argue that such a condition, if taken as a whole is true or false. LEM is a crucial ingredient in proofs by contradiction as opposed to the contrapositive proof (Shapiro 2011, pg. 136). Certain proofs by contradiction establish the existence of mathematical objects. In constructive mathematics, proof by contradiction does not provide indications of how mathematical models can be fully described.
LEM has simple alternative inclusions into the principles of logic that informs its decision. Its bare omission without the assertion of contrary principles reduces the number of theorems that could be proved and to render certain principles vacuous (Gonthier 2013, pg. 176). LEM cannot help in deriving theories, which contradict theorems in the obtained by its aid unless some assertions that are contrary to LEM are made. By making some assertions that a proposition is neither true nor false would deny LEM yet LEM is so fundamental in making judgments that even machines cannot make (Shapiro 2012, pg. 311). On the ground of forming “tiers” as suggested by the French word used by Barzin and Errera, the assumptions about the existence of tiers and its properties produce a system of logic that is consistent with itself and LEM would be added.
Assuming LEM in mathematics would be disastrous to the mathematics of infinite numbers and thus hinder the decisions that might have been based on the “tiers.” The law of the extended middle is extended into the infinite. Acceding to Church, the assertions that there are finitely numerous prime numbers or they are infinite in number hinders the decision by the intuitionists who believe in LEM when it’s only confounded on sets (Church 1928, pg. 79). When used in the discourse of the infinite sets, LEM tends to be disallowed by the intuitionist. The intuitionists also find LEM a very important law when discussing the mathematics of sets (Shapiro 2011, pg. 149). They disallow the blanket assertion that the for all propositions p, concerning infinite sets D. thus LEM must not be assumed in mathematics because of the role it plays in the discussion of the set theory. In this regard, LEM operates in such a way that it can decide certain integers in a set operation but only when the numbers are finite or in a closed set (Church 1928, pg. 78). Barzin and Errera argue in their paper that the systems of logic proposed by Brouwer are contradictory. Nevertheless, this is an erroneous conclusion because the method of argument assumed LEM. Rejecting the law of the excluded middle always is a pre-requisite to any anti-realism.
LEM implies realism for a domain. The realism is about the argument presented by the mathematical Platonism and intuitionism. The logical basis of metaphysics asserts that most of the metaphysical arguments take a top-down approach starting with the metaphysical positions and applying them in the domain in the question. Such problems should be approached from bottom up to make the good use of LEM by not making any assumptions regarding the domains in question (Church 1928, pg. 75). The realism of a domain encompasses the entities of that particular domain being independent. That is, if realism is true or false independently, then a well-formulated claim on that domain will be true or false even if we cannot establish their truth values.
Using the truth tables, the disjunctive statement p_¬p can always be shown to be true using tautology. P_¬p is a tautology regardless of the truth value of p. So in this way, LEM is a tautology no matter what statement p represents. The famous example of Bertrand Russell: “The current king of France is bald” gives more overview why LEM should not be assumed or at worse, rejected (Church 1928, pg. 76). Since LEM says every statement is either true or false, the statement by Russell must also bear the same properties. But in the contemporary world, there is no present king of France. Thus the claim is quite unusual. But if we accept the claim based on LEM, we would only have one option left- to claim that is false. At this point, we may choose to reject LEM or contend that it doesn’t hold (Church 1928, pg. 77). But because the decision, in this case, is to be made by humans or programs, LEM will find a place if not assumed. The truth tables can be constructed to imply LEM holds when it doesn’t. These truth tables area is also forming conceptual frameworks for the development of mathematical programs.
LEM cannot be approved or disapproved based on mathematical reasoning, it must be assumed when using it in mathematical operations, but the law itself cannot be assumed to exist. Denying LEM is self-contradictory (Shapiro 2011, pg. 143). The other laws in this category include the Law of identity and the law of noncontradiction as mentioned before. The principle of realism holds that these laws apply to everything since they are the most general truth of reality. LEM not only applies to what we think and say but also apply to what we think and talk about. Every person who contends with LEM must be forced into a two-valued orientation of thoughts. Then LEM is only valid in cases where one restricts himself to ambiguous statements that are precise. Therefore, real paradox is not possible under LEM; thus it cannot be assumed.
In conclusion, LEM must not be assumed based on its importance in offering solutions where human beings and machines can fail in offering solutions. Several philosophers such Bertrand Russell and Aristotle also argued that LEM could be construed on the truth tables and used in explaining facts that are evident to the natural eye. The concept of constructive mathematics itself makes several assumptions on LEM and thus crucial in constructive reasoning as presented by the constructive mathematicians. LEM excludes the middle ground between truth and falsity and offers an avenue for the “tiers” in the world of philosophy. The concepts of the set theory with regards to the empty sets originate from LEM. If a point zero can be approximated on a continuous interval f(x), then LEM can work. This means that the tier is located on that interval by merely approximating the value of x to be zero. The philosophical frameworks for the operations LEM have assumptions. Therefore, LEM must not be assumed. Besides, LEM has alternatives to the inclusions in the field of logic that inform its use as a tier in some cases, and always imply realism in specific domains. In so doing, the law of excluded middle will always work in such a way that it has some theoretical understandings that must be well explained for clear understanding. Concerning its technicality, LEM should be assumed because it can solutions based on mathematical reasoning that cannot be provided by the machines.
References
Church, A., 1928. On the law of excluded middle. Bulletin of the American Mathematical Society, 34(1), pp.75-78.
Gonthier, G., Asperti, A., Avigad, J., Bertot, Y., Cohen, C., Garillot, F., Le Roux, S., Mahboubi, A., O’Connor, R., Biha, S.O. and Pasca, I., 2013, July. A machine-checked proof of the odd order theorem. In International Conference on Interactive Theorem Proving (pp. 163- 179). Springer, Berlin, Heidelberg.
Shapiro, S., 2011. Epistemology of mathematics: What are the questions? What count as answers?. The Philosophical Quarterly, 61(242), pp.130-150.
Shapiro, S., 2012. Higher-order logic or set theory: A false dilemma. Philosophia Mathematica, 20(3), pp.305-323.