# Plausibility Theory & Paradox Resolution

The concept of plausibility is not new. The concept dates back to the Ancient Greeks; the idea can be found in Aristotle’s Topica (Rescher, 1976, Preface). Plausible reasoning is used in evaluating hypothetical reasoning, scientific reasoning and even inductive reasoning. However, it will be argued that when applied to paradox resolution, it rests upon a fundamental misunderstanding of what a paradox is. It must be granted that the plausibility theory succeeds in providing us with a method for resolving a wide variety of paradoxes. However, in resolving the paradoxes, it over-looks the essential nature of paradoxes.

Contrary to Rescher’s theory of paradox resolution, paradoxes are not mere problems to be resolved, but rather important problems which challenge our basic intuitions of the world. In failing to appreciate the fundamental nature of paradox, one fails to appreciate what can be learned from paradoxes. Moreover, it will be demonstrated that plausibility theory suffers from serious counterintuitive consequences regarding truth and usefulness.

Motivation Behind Plausibility Theory

One of the driving forces behind plausibility theory is to provide us with a systematic way of resolving inconsistent sets. Take the contradictory set: [P, ~P]. If we assume classical logic, we know it tells us that one of these propositions must be rejected. However, we do not know which one must be rejected based purely on logic; pure logic alone cannot tell us which proposition to needs to be abandoned (Rescher, 1976, p.2).

Since logic fails to give us a definitive answer in the inconsistent case, something more is needed. Ergo, while logic tells us that if we want to restore consistency, we must abandon one of the above propositions, plausibility theory tells us which one of the propositions we ought to abandon. How plausibility theory works will be the focus of the following section.

The Logic Of Plausibility

As stated earlier, the function of plausibility theory is to provide one with a method for resolving cases in which we have accepted a set of inconsistent propositions. Another way to say it is that plausibility seeks to provide us with a systematic way to resolve cognitive dissonance (Rescher, 1974, p.1). For example, take the following set of propositions: [p Ú q, ~p, ~q]. Let us suppose that the propositions have the following plausibility breakdown: p Ú q = .7, ~p = .4, ~q = .5. According to plausibility theory, in order to restore consistency, we must abandon the proposition ~p, leaving us with the following set of propositions: [p Ú q, ~q].

In order to understand how plausibility theory works, we must know its cardinal maxim: In the inconsistent case, retain those propositions with the highest level of plausibility, and abandon the least plausible proposition (Rescher, 2001, p. 32). This, however, can only make sense after we have a firm grasp on exactly what “plausibility” is suppose to mean.

Nicholas Rescher (1974) states: “…plausibility is intended to reflect an index of what reasonable people would - and should - agree on, given the relevant information” (p. 5). In his most recent book on paradoxes, Rescher (2001) claims that, “Its precedence [with regards to a proposition] and priority ranking bears upon the systematic standing or status that we take a claim to occupy in the cognitive situation at hand” (p. 46). From the above information, it would seem that the theory of plausibility is merely based upon how certain we are regarding various propositions. For example, we generally think the axioms of number theory are more certain than the law of physics. While cognitive certainly is a major theme within plausibility theory, Rescher claims that he does not ground his theory of plausibility merely on how certain a proposition is within our cognitive web of beliefs.

Rescher also believes that there are objectively more plausible propositions than others. As stated earlier, we hold axioms of mathematics to be more certain than the laws of physics. The plausibility of the proposition goes up the closer it gets to (1). Rescher (2001) gives the following breakdown:

(1) Definitions, axioms of mathematics and logic, mathematical relations, and in the case of paradoxes, principles of the story. These are 100% certain and cannot be refuted

(2) The inductive sciences.

(3) Observations and experimental observations of life.

(4) Highly probable propositions regarding contingent fact.

(5) Reasonably warranted suppositions.

(6) Provisional and tentative conjectures.

(7) Speculative suppositions (p.50).

While this is how Rescher actually categorizes the hierarchy of plausibility, a better hierarchy can be given that is more clear and without doing damage to his overall idea. For example, it remains unclear as to how (5) and (6) differ in any significant way. Furthermore, does (2) refer to just the natural sciences, or the social sciences as well? Given what we know about plausibility theory thus far, here is a proposed revised hierarchy:

(1) Definitions, axioms of mathematics and logic, mathematical relations, and in the case of paradoxes, principles of the story. These are 100% certain and cannot be refuted

(2) Physical laws

(3) Scientific theories

(4) Conjectures

(5) Speculative suppositions.

The above hierarchy ought to be clear, only a minor elucidation is needed. When we say a conjecture, we mean that which we expect to be true, but as of yet, has remained unproven. For example, take Fermat‘s Last Theorem. Before it was proven, it was considered merely a conjecture within mathematics. We can also use the example of the continuum hypothesis as an example of a conjecture. It is unclear as to whether the continuum hypothesis is true or false.

The level of plausibility given to a proposition is also based upon the Rescher’s notion of “reliability.” We would apply Rescher’s idea of reliability when we are dealing with claims people make. For example, we would say that a physics professor is more reliable when it comes to matters of physics than a non-physicist. In the case of legal testimony, take two individuals who claim to have seen person x leaving the bank after it was robbed. Let us say that one of those individuals was inebriated at the time, whereas the other individual was not. We would say the individual who was not inebriated is more reliable than the individual who was inebriated.

One must also understand that plausibility theory and probability theory are distinct. The biggest difference between plausibility theory and probability theory is how they would handle the following set: [p, ~p]. Let us assume that we know that proposition p has probability of .8. Since p has probability of .8, we know that ~p must have a probability of .2. Hence, the probability of either p or ~p will either rise or fall depending upon the probability of the other proposition: this is a basic truth of probability theory. In plausibility theory, it is within the realm of possibility that both p and ~p could have the same value; thus, p could be .8, and ~p could be .8.

A Critique Of Using Plausibility Theory To Resolve Paradoxes

As stated earlier, resolving inconsistent sets, especially paradoxical sets, is at the very bedrock of plausibility theory. It will be granted that the logic of plausibility is not only a robust theory, but also internally consistent. Moreover, it will be granted that plausibility theory succeeds in giving us a method of resolving paradoxes. However, this method fails to appreciate not only the very nature of paradoxes themselves, but also collapses into a sort of incoherence. In order to show this, two paradoxes will first be considered: The Lottery Paradox and The Liar Paradox.

The Lottery Paradox

The Lottery Paradox (LP) is a paradox one runs into when working in epistemology. The Lottery Paradox challenges the common sense belief that fallible justifiability only requires high probability. The typical lottery paradox runs as follows: Suppose there is a fair lottery in which 1,000,000 tickets are sold. Moreover, suppose that only one ticket can be the winner. If we select any one ticket, we will see that the probability of that ticket being the winning ticket is .000001%. Since it is highly probable that this ticket will not win, and by assuming the traditional fallible justifiability, we know that the ticket we choose will not win. However, since our choice was random, and since the same probability would have applied to any ticket we selected, we can use the principle of universal generalization to apply our result to every single ticket. The immediate result that we get is that we know that every single ticket will not be the winning ticket. However, as part of the game, one ticket must win. Ergo, we get a contradiction: we know that one ticket will win, but we also know that no ticket will win.

Rescher (2001) concedes that his theory of paradox resolution does not apply in this case, since all the premises of the argument are equally plausible (p. 224). While Rescher is correct, this is not the main point of the paper.

Ever since Descartes proposed that knowledge demanded absolute certainty, there have been skeptical worries. Since infallible knowledge is slim, many epistemologists, both then and now, have been defending theories of fallible knowledge. Until the Lottery Paradox was formulated, it was generally held that high probability was sufficient for justifiability. Intuitively, this seems extremely plausible (excuse the pun). However, the Lottery Paradox demonstrates that high probability is not enough to warrant fallible knowledge. Furthermore, the Lottery Paradox can be formulated using the principle of mathematical induction. Thus, the Lottery Paradox challenges not only modern Epistemology, but also a principle of mathematics.

The Liar Paradox

Ever since the time of the Greeks, the Liar Paradox has been a proverbial thorn in the side of philosophy. In the Liar Paradox, we are asked to considered the following sentence L - L: “this sentence is false.” The question is posed, is (L) true? Assume that (L) is true. If (L) is true, then what it asserts is the case. (L) asserts of itself that it is false. Therefore, (L) is false. Assume that (L) is false. (L) says of itself that it is false. Therefore, (L) is true. From this we get the following biconditional: T(L) Û F(L).

The question naturally arises, how does plausibility theory handle the Liar Paradox? Rescher (2001) declares that: “…‘This sentence is false’ is semantically meaningless: neither truth nor falsity can be ascribed to it” (p. 43). Beyond this sentence, Rescher gives us no further analysis of why (L) is meaningless. To be fair, however, here are the following Liar presuppositions:

(1) (L) is a meaningful declarative sentence.

(2) (L) says of itself only that it is false.

(3) If (L) is meaningful declarative sentence, then it must be either true or false.

(4) Meaningful sentences can be self-referential.

Given what we know about plausibility theory, it is highly likely that Rescher would give the above propositions the following ranking (If he had chosen to rank them: (3) > [(4) (2)] > (1)#. Proposition (3) is a truth of classical logic. Since we can think of countless self-referential sentences that do no lead to absurdities, it seems plausible that (4) is not wrongheaded. In addition, since (2) seems equally as plausible as (4), it receives the same priority ranking as (4). We end up with (1) being the weakest link, for it is a mere supposition.

Regardless of which proposition one chooses to reject, there is an important insight this paradox gives us: that ether the nature of a proposition is not all that manifest, or bivalence is false. If Saul Kripke is correct, then propositions do not necessarily have to be either true or false; they can fall into the “gaps” of truth and falsity. If Graham Priest is correct, then we know that while the law of non-contradiction may be applied in most cases, it is not a necessary truth of logic. If Kripke is correct, we have learned something significant about the nature of truth and of propositions. If Priest is correct, then we have also learned something significant about the nature of truth and propositions. If Bertrand Russell and Alfred Tarski are correct, then we have learned that contrary to intuition, propositions cannot refer to themselves. Furthermore, if Tarski is correct, then we know that any artificial language which has it’s own truth predicate will be inconsistent. Once again, if they are correct we will have learned something significant about the nature of propositions and truth.

Critique of Rescher’s Hierarchy

Given Rescher’s hierarchy of plausibility, we can see that the following relation holds: (1) > (2) > (3) > (4) > (5). Starting with (5) we can see that a proposition gains plausibility the higher it is on the hierarchy. Thus, if (5) was given a plausibility ranking of .2, then it is a basic truth within plausibility theory that (4) must receive a minimum of a .21 ranking. When a proposition receives the ranking of (1) it is given a plausibility ranking of 1.0. Intuitively, the hierarchy seems reasonable. However, what justifies the hierarchy? The hierarchy can be justified in one of two ways: objective evidence external to the mind, or subjective evidence internal to the mind.

Assume for the sake of argument that the Rescherian Hierarchy can be justified in virtue of objective external evidence. If the hierarchy is justified in virtue of external objective evidence, then the propositions within logic and mathematics cannot be refuted. If they are given a ranking of 1.0, then it is logically impossible for the propositions within logic and mathematics to be false. Examining both modern logic and mathematics shows, however, that this is dubious. In order to show it is dubious, we must consider naïve set theory.

In the late 1800’s and early 1900’s, any collection of objects was thought to comprise a set. Before Russell’s paradox, naïve set theory was considered to be one of the most intuitively obvious theories within mathematics. Given the interpretation given to the Rescherian Hierarchy, naïve set theory receives a plausibility ranking of 1.0. However, we need only consult Russell’s paradox to see that naïve set theory is contradictory. Given the interpretation given, a counterintuitive result is that a proposition within mathematics or logic could be 100% plausible, but yet be demonstrably false. An even better example to illustrat this point is the Rule of Necessitation within Modal Logic.

The Rule of Necessitation (RN) within Modal Logic states that if one is able to prove a proposition from the axioms of Modal Logic, then one has also proven that the said proposition p is necessarily true. Hence, one can say that P is a necessary truth, if one can prove P from the axioms. However, this rule is only justified if the axioms of Modal Logic are true. However, if the axioms of Modal Logic are dubious, then saying that a proposition is necessarily true could very well only be a possible or contingent truth.

Assume for the sake of argument that the Rescherian Hierarchy is justified in virtue of its place within our cognitive web of beliefs. If this is the case, the hierarchy will differ from person to person; the hierarchy will even differ among logicians given the soundness of various logics. Given that Graham Priest is a Dialetheist, we can say with confidence that he would not put the Law of Non-Contradiction as having a cognitive status of 1.0. A logician who subscribes to 3-valued logic will explicitly reject the principle of bivalence, while a logician who accepts 2-valued logic will hold the principle of bivalence to be of 1.0. If this is what justifies the hierarchy, then it seems that the hierarchy collapses into mere subjective judgments.

Any way of justifying the hierarchy seems problematic as of now. However, the hierarchy is at the very bedrock of plausibility theory. Without the hierarchy as the foundation for plausibility theory, there is no coherent way of assigning plausibility rankings to propositions. If one cannot assign plausibility rankings to propositions in a coherent way, then plausibility theory collapses.

This criticism of plausibility theory is not an refutation of plausibility theory per say, but rather an informal demonstration that plausibility theory is problematic. Until these problems can be resolved, we must look upon any results plausibility theory produces with great dubiety.

What Paradoxes are “About”

On Rescher’s account of paradoxes, they are mere puzzles of the mind that cause cognitive dissonance. This can be seen most evidently with how Rescher defines what a paradox is. Rescher (2001) describes paradoxes as follows: “a paradox arises when a set of individually plausible propositions in collectively inconsistent” (p. 6). The only problem with this line of reasoning is that it implicitly assumes that plausibility is at the heart of paradoxes. However, this mistakes the effects a paradox might have on ones cognitive apparatus for the paradox itself.

In general, when we say a proposition is more plausible than another proposition we are making a judgment. We are saying “according to me, this proposition seems more reasonable.” Thus, according to Rescher’s definition of what a paradox is, it implicitly assumes that a sentient being is a necessary condition for paradoxes to arise. However, this is not the case.

One can easily imagine a possible world where no sentient beings exist, but the Lottery Paradox propositions are collected together in an inconsistent set. Hence, Rescher’s definition of what a paradox is needs a slight revision. A paradox is a set of propositions that seem to be individually plausible, but are nevertheless inconsistent. On the revised definition, a sentient mind is not necessary condition for a paradox to arise.

Once we eliminate the “human element” from paradoxes, the question remains, what are paradoxes about? The answer to this question will depend upon the type of paradox we are dealing with. For example, in the case of Russell’s Paradox, the paradox is about the flagrant usage of Frege’s Unrestricted Comprehension Principle. We can safely assume that Russell’s Paradox is not about cognitive dissonance. While it may have caused set theorists cognitive dissonance, Russell’s Paradox is not paradoxical in virtue of the cognitive dissonance the set theorists felt; even if the set theorists felt no cognitive dissonance, Russell’s Paradox would still be a paradox. It would be incoherent to think that the truth or falsity of a set of propositions is made true or false merely by a sentient mind cognizing it.

As stated earlier, when trying to resolve paradoxes, Rescher focuses on the cognitive status of the propositions of the paradoxes, rather than whether these propositions are actually fallacious. Thus, according to Rescher, the essential piece of paradox resolution is not in finding actually fallacious principles, but of merely trying to resolve cognitive dissonance. This approach should strike most people as very unsatisfying. It seems, at least prima facia, that when we try to resolve paradoxes we are not merely resolving cognitive dissonance, but that we have come to understand the fabric of reality in a much better way. Moreover, it seems that what a given paradox demonstrates is that something is fundamentally flawed with the way we understand the universe. This can be seen most evidently with the alleged paradoxes of infinity that plagued philosophers and mathematicians since antiquity.

Before Georg Cantor, the concept of “infinity” was generally thought of as a concept riddled with paradoxes. Galileo came up with a famous “paradox” that attempted to demonstrate the absurdity of the concept of infinity. Galileo observed that any infinite set could be put into a one-to-one correspondence with any one of its proper subsets. Hence, the even numbers could be put into a one-to-one correspondence with the natural numbers. It was precisely because of Galileo’s Paradox that lead Cantor to develop the theory of transfinite numbers. Thus, Galileo’s Paradox demonstrated a fundamental misunderstanding we had of infinity: contrary to intuition, there are different sizes’ of infinity. Galileo’s Paradox also demonstrated that we cannot understand the cardinality of infinite sets in the same way we understand the cardinality of finite sets.

If we had applied Rescher’s theory of paradox resolution, one of the greatest discoveries of mathematics might never have been discovered. A given proposition within Galileo’s Paradox would merely have been abandoned as “least plausible.” One of the implicit assumptions that gives Galileo’s Paradox it’s punch is that there is only one size of infinity. However, we know that not only is the above assumption is more than merely the least plausible of the paradox, but demonstrably false.

Truth and Plausibility

At the very bedrock of using plausibility theory to resolve paradoxes is that we are not making claims about what is true or false. Hence, when we abandon a proposition within the context of a paradox, we are not claiming that the said proposition is false; rather, we are making the claim that within the context of a paradox, a given proposition is “least plausible.” If by employing plausibility theory one is not making a claim about the truth or falsity of a proposition, then what are we really accomplishing? To see that plausibility theory is absurd, consider mathematical models that mathematicians construct. The combined efforts of Kurt Gödel and Paul Cohen demonstrated that within Zermelo-Fraenkel Set Theory (ZFC) the continuum hypothesis (CH) was independent of its axioms. Therefore, it would be consistent to either accept or deny the continuum hypothesis.

Given the results of Gödel and Cohen, set theorists have developed mathematical models within set theory based on either the acceptance or denial of the (CH). If we accept (ZFC), then one of the models developed will accurately map the mathematical universe. It is debatable as to which model actually represents the way the mathematical universe is. However, there can be little doubt that set theorists working on the (CH) are trying to figure out which model is true. While the models do not make any claims about what is true, set theorists constructed them with the hope of one day discovering which one is in fact true. The goal of set theorists is to discover what is true, even if they must construct useless models.

Unlike mathematical models, the goal of plausibility theory is not an attempt to get at what is true. While many mathematical models are inevitably useless, the goal is clear: while many of these models cannot be true, we hope that one of them is true. If truth is not the goal of plausibility theory, then what is?

Usefulness and Plausibility

If plausibility theory is not a method for accessing what is true, then what is its usefulness? This question can best be answered by comparing the method of plausibility theory to the method of science. The methodology of science is useful in virtue of its providing us with a way of knowing the empirical world. Hence, we know the scientific method is useful because it simply is the best method one has for knowing the truth or falsity of the propositions about the empirical world. Plausibility theory is cannot be useful in the same way that the scientific method is useful; science claims that certain propositions are true, while others false, whereas plausibility theory makes no such claim. If plausibility theory is not useful in virtue of it providing us with a method of knowing a given set of propositions, then in virtue of what is it useful?

Plausibility theory is useful in virtue of it’s ability to resolve cognitive dissonance, as stated earlier. It must be conceded that plausibility theory succeeds in resolving cognitive dissonance (to a degree). If eliminating cognitive dissonance is the way to resolve paradoxes, a much simpler idea can be given to serve this function: ignore the paradox. When a paradox arises, one can simply choose not to think about it; one can simply kick the paradox under the rug, so to speak. There seems to be no reason why one should favor Rescher’s method over simply ignoring the paradox.

Conclusion

While plausibility theory provides one with a method for resolving many paradoxes, it has been shown to rest upon a faulty diagnosis of what a paradox is. Moreover, how it resolves a paradoxes leaves one feeling unsatisfied. In addition, the way propositions are ranked suffers from serious problems. Perhaps plausibility theory’s greatest drawback is its claim to not deal with what is true or false. Given that plausibility theory is problematic, one should reject it as a viable theory of paradox resolution.

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Work Cited

Rescher, Nicholas (2001) Paradoxes. Illinois: Open Court.

Rescher, Nicholas (1961) “Plausible Implication”, Analysis 21(6): 128-135.

Rescher, Nicholas (1976) Plausible Reasoning. Amsterdam: Van Gorcum, Assen

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