Conceptual Modality

Penrose triangle

Epistle 19. Posted on 2019-05-27.

Conceptual modality is one of the most useful innovations in ancient logic. Of the many different kinds of modalities, we are concerned with alethic modality. “Alethic” derives from aletheia in Ancient Greek and means truth, and “modality” derives from modus in Latin and means measure, method, or way. An alethic mode, then, is a way for truth to be, or to be regarded. In logic, alethic modality distinguishes a proposition that is impossibly true from possibly true, and necessarily true from not-necessarily true.

Conceptual modality, as presented here, is my interpretation of alethic modality in Stoic logic, and so it differs substantially from modern modal logic.

The conceptual modality of a proposition is an evaluation of the truth value of the proposition, due to concepts. There are four conceptual modalities of the truth of a proposition: necessary, non-necessary, possible, and impossible. These modalities are related to truth and falsity. A necessary proposition is necessarily true due to concepts, meaning it is either always true according to its own concepts or it is forced to be true by other concepts. An impossible proposition is impossibly true due to concepts, meaning it is either never true according to its own concepts or it is forced to be false by other concepts. In addition to propositions, conceptual modality also applies to arguments.

Conceptual modality is symbolized here, but not with operators (as in modern modal logic): the box symbol □ is the necessity symbol, and the diamond symbol ◇ is the possibility symbol. While □(p) means that proposition p is necessarily true due to concepts, ◇(p) means that proposition p is possibly true due to concepts. The symbolization of conceptual modality is akin to scribbling a “T” or “F” above or below a true or false proposition; it is simply a way of showing that truth has been evaluated.

The following list presents two symbolizations for each of the four conceptual modalities, depending on whether you prefer the necessity or possibility symbol:

Conceptual Modality

Necessary (Greek: anankaion): □(p) or ~◇(~p)
Non-necessary (Greek: ouk anankaion): ~□(p) or ◇(~p)
Possible (Greek: dunaton): ~□(~p) or ◇(p)
Impossible (Greek: adunaton): □(~p) or ~◇(p)

These symbolic definitions also imply:

A proposition is necessary, □(p), if it is either analytic and true or synthetic and hindered externally from being false.

A proposition is non-necessary, ~□(p), if it is either analytic and false or synthetic and not hindered externally from being false.

A proposition is possible, ◇(p), if it is either analytic and true or synthetic and is not hindered externally from being true.

A proposition is impossible, ~◇(p), if it is either analytic and false or synthetic and hindered externally from being true.

To understand the above definitions, analyticity must also be understood. Although analyticity is a modern term, the Stoics used it to distinguish modes of truth. There are two categories of propositions: analytic and synthetic.

Analytic Proposition

The truth value of an analytic proposition can be determined by examining the definitions of its relevant concepts. To determine if a proposition is analytic is to evaluate the proposition only with respect to the definitions of its relevant concepts. If the truth value may be determined in this way, it is an analytic proposition; otherwise, it is a synthetic proposition. An analytic proposition that is true is conceptually necessary, and an analytic proposition that is false is conceptually impossible. If the proposition is analytic, not only can the truth value be determined without reference to other propositions, but other propositions cannot alter its truth value.

For example, if a human is defined as a rational animal, consider the following proposition: “Humans are rational.” This proposition is indefinite and is translated into the following non-simple proposition: “If x is human, x is rational.” Since a human is defined as a rational animal, by substitution this proposition is equivalent to “If Both x is an animal and x is rational, x is rational”. This clearly reduces to “If p,p,” and is conceptually necessary by definition. Therefore, this is an analytic proposition.

Likewise, the simple proposition that “Dion is rational” is analytic if the concept of Dion is that he is human, and that a human is defined as a rational animal. However, if Dion is unknown to the agent evaluating this truth value, this is a synthetic proposition because its truth value cannot be determined solely by the concepts of Dion and rationality.

Examples of conceptually impossible non-simple propositions include “Both p and Not p” and “If p, Not p.” Examples of conceptually possible non-simple propositions include “Both p and q.” Examples of conceptually necessary non-simple propositions include “Not Both p and Not p,” “Either p or Not p,” “If p, p,” “If p, Not Not p,” and “If Not Not p, p”.

Consider a proposition such as “Either p or q.” This proposition is defined by conflicts between the disjuncts, and by the lack of self-conflict of each disjunct, and by a conflict between the conjunction of all contradictory disjuncts. If a conflict occurs when it should not or does not when it should, this disjunction is conceptually non-necessary. Otherwise, it is conceptually necessary. The disjuncts are uninstantiated inside the disjunction, and hence the conceptual modality of the disjuncts is not evaluated within the disjunction.

A proposition is conceptually necessary when it is is capable internally of being only true, and its truth value cannot be hindered externally. A proposition is conceptually impossible when it is capable internally of being only false, and its truth value cannot be hindered externally. A proposition that is capable internally of being either true or false is a proposition that has a truth value that may be hindered externally, and when unhindered externally the conceptual modality is either non-necessary or possible, depending on the context. For example, proposition p is capable internally of being either true or false and is not hindered externally:

Either p or Not p
But ◇(p)
Therefore ◇(Not p).

Proposition “Either p or Not p” cannot be false, and is conceptually necessary.

Synthetic Proposition

The conceptual modality of a synthetic proposition can be changed by an external proposition. To determine whether or not a truth value is hindered externally, two conditionals are evaluated. In each conditional, the antecedent is the external proposition. For the first conditional, the consequent is the proposition of interest. For the second conditional, the consequent is the contradictory of the proposition of interest. The conditional and a contradictory proposition are presented elsewhere. The following are evaluated:

If p, q.
If p, Not q.

Each conditional is evaluated for a conflict between the antecedent and the contradictory of the consequent. If a conflict occurs in the first evaluation, the truth value of the proposition of interest has been hindered externally to truth, which results in conceptual necessity. If a conflict occurs in the second evaluation, the truth value of the proposition of interest has been hindered externally to falsity, which results in conceptual impossibility. If a conflict does not occur in either conditional, the truth value of the proposition of interest is not hindered externally by this external proposition.

For example, the external proposition is “Dion is sitting” and the proposition of interest is “Dion is standing.” In the evaluation of the first conditional, “Both Dion is sitting and Not Dion is standing” does not result in a conflict. Dion may be both sitting and not standing. In the evaluation of the second conditional, “Both Dion is sitting and Dion is standing” results in a conflict. Dion cannot be both sitting and standing. Due to the conflict in the evaluation of the second conditional, if “Dion is sitting” is true, the truth value of “Dion is standing” is hindered externally to falsity, and so “Dion is standing” is conceptually impossible in that case, and its contradictory, “Not Dion is standing” is conceptually necessary.

Now consider if the external proposition is “Dion is talking” and the proposition of interest is “Dion is standing.” In the first conditional, “Dion is talking” does not conflict with “Not Dion is standing.” Dion may be both talking and not standing. In the second conditional, “Dion is talking” does not conflict with “Dion is standing.” Dion may be both talking and standing. Due to the lack of conflict in both evaluations, if “Dion is talking” is true, the truth value of “Dion is standing” is not hindered externally and so “Dion is standing” remains conceptually contingent in that case:

□(Either p or Not p)
But ◇(p)
Therefore ◇(Not p).

Finally, consider if the external proposition is “Dion is walking” and the proposition of interest is “Dion is standing.” In the evaluation of the first conditional, “Both Dion is walking and Not Dion is standing” results in a conflict. Dion cannot be both walking and not standing. In the evaluation of the second conditional, “Both Dion is walking and Dion is standing” does not result in a conflict. Dion may be both walking and standing. Due to the conflict resulting from the evaluation of the first conditional, if “Dion is walking” is true, the truth value of “Dion is standing” is hindered externally to truth, and so “Dion is standing” is conceptually necessary in that case, and its contradictory, “Not Dion is standing“ is conceptually impossible.

In notation, when proposition q has a truth value that was hindered by proposition p, it is symbolized in a conditional as:

□(If p, q)
But □(p)
Therefore □(q).

Although conditionals are introduced later, this argument is read as: “Necessarily If p, q, but necessarily p, therefore necessarily q.” This implies that p hindered the truth value of q from being false: whenever p is true, q is true. Although there may be numerous propositions that hinder externally the truth value of another proposition, only the minimum number of propositions should be noted that is required to hinder the truth value. Although it is desirable to be thorough and include premises in an argument that list or show, visually, which propositions hindered externally the truth value of another proposition, this is not always possible to do in a valid argument, because additional premises may be inessential premises that are irrelevant to the conclusion. In this case, an additional and independent argument may be made for the proposition in question, and attached as an appendix of sorts.

Examples

“Dion is walking” occurs in an argument regarding the present tense. This proposition is synthetic, and there is not currently another proposition to hinder externally its truth value:

It is not necessary, because even if true, it is synthetic and is not hindered externally from being false.

It is non-necessary, because it is synthetic and is not hindered externally from being false.

It is possible, because it is synthetic and is not hindered externally from being true.

It is not impossible, because even if false, it is synthetic and is not hindered externally from being true.

“Dion is walking” occurs in an argument regarding the present tense, along with the true proposition “Dion’s legs are tied to a chair.” The truth value of “Dion is walking” is hindered externally from being true by the true proposition “Dion’s legs are tied to a chair.” The first proposition is synthetic, and its truth value is hindered externally from being true. The first proposition is conceptually impossible, and the evaluated truth value must be false, due to the second proposition.

It is not necessary, because even if true, it is synthetic and is not hindered externally from being false.

It is non-necessary, because it is synthetic and is not hindered externally from being false.

It is impossible, because even though it is synthetic, it is also hindered externally from being true.

It is not impossible, because even if false, it is synthetic and is not hindered externally from being true.

Practicality

This system of conceptual modality is essentially Chrysippean modality symbolized with modern modal symbols and including analytic and synthetic as simplifying terminology. Chrysippean modality has been criticized as odd for allowing a possible proposition to become an impossible proposition, or for a proposition to be regarded as possible, though the related fact never occurs. According to Gould (1970, p. 147) in The Philosophy of Chrysippus, Chrysippus and the Stoics interpreted their word for “possibility” in relation to human ignorance, not with respect to “an objective contingency in the nature of things.” Therefore, something that seemed possible can turn out to be impossible, always to be false. The only absolute modalities are necessity and impossibility. This is actually desirable because Chrysippean modality is conceptual, and so it accommodates both necessitarianism and uncertainty; the evaluated modality is limited intentionally by human experience. Given enough experiences, conceptual modality should agree with logical and factual modality. But, when less information is available such as in an argument with few premises, the conceptual modality of a proposition that is synthetic tends to be contingent, driven by the uncertainty in the argument.

For example, the proposition “Dion decided consciously to wear blue rather than red socks” is conceptually impossible according to neuroscience, because that decision was made unconsciously, and shortly thereafter his consciousness became aware of his unconscious decision. Therefore, it is analytic and contradictory because “decided consciously” is conceptually impossible. This predicate verb and adverb cannot be meaningfully combined. If all causes were known, any event must be either necessary or impossible.

A simple proposition such as “It is day” is synthetic, but it is fully determined, necessitated, or caused by other external factors. If it is true that it is day, it is not merely possible because it is capable internally of either truth value, but it is hindered externally by the position of the sun in relation to the earth, which results in believing it is a conceptual necessity. And, once a proposition is regarded as conceptually necessary, its contradictory proposition becomes conceptually impossible, such as “Not it is day.”

A proposition that is synthetic has a truth value this undetermined until it is believed to be either true or false. For example, a proposition about next week is conceptually possible, and so is its contradictory proposition, until next week occurs. The conceptual modality can be evaluated at any time, and is updated as more information is learned and more premises are included accordingly.

There are other differences to consider between a conceptual modality and absolute modality. Conceptual impossibility does not imply that the agent regards something as absolutely impossible, but instead implies only that there is a conflict between a concept in the proposition and another concept, or between concepts in the proposition. Likewise, a student may believe an analytic proposition is necessarily true because it was just explained to be a tautology, but the student may still have their doubts. On the other hand, the student may believe it is only possibly true, because the student may have a concept that it is best to delay assenting to new scholastic material, which in turn may relate to a concept of the incompetence of a professor to explain clearly.

Example

Suppose an agent endeavors to evaluate the proposition that “Dion is tall” within an argument. First, the agent evaluates the conceptual modality of the proposition, isolated from other propositions. On its own, without the agent having a concept of “a dion,” this proposition is synthetic. Second, the agent must complete the evaluation of conceptual modality by evaluating whether or not this proposition is hindered externally from truth or falsity by other concepts or evaluated propositions. Before external hindrances are considered, this proposition and its contradictory both begin by being evaluated as conceptually possible: Dion may or may not be tall.

If Dion is unknown to the agent, empirical investigation is required to develop a concept of “a dion” and evaluate external hindrances to truth and falsity. The agent finds Dion and sees that Dion is tall. This information (that Dion was seen to be tall) is external to the information in the investigated proposition (that Dion is tall). The new information becomes a new proposition that is relevant to the investigated proposition. The new proposition externally hinders the truth value of the investigated proposition from being false, at least for the agent who experienced the observation. Since the investigated proposition was synthetic, and it is hindered externally from falsity, the conceptual modality has changed from possible to necessary. Given the observation of the agent, it is now conceptually necessary that Dion is tall, for that agent. The only truth value compatible with necessity is “true.” This proposition is now supposed by the agent to be in agreement or consistent with the cosmos.

If Dion is unknown to the agent and the agent cannot empirically investigate this proposition, or chooses not to, then both this proposition and its contradictory remain regarded as possible, and either truth value is compatible with this conceptual modality.

Higher-Order Conceptual Modality

Since a conceptual modality is an evaluation of truth, higher-order conceptual modality is not allowed. For example, a proposition cannot be necessarily impossible or possibly necessary. The application of a modality to a modality is itself impossible. A modal proposition may be reevaluated, resulting again in only one modality. Modalities operate on propositions, not on other modalities.

Consider a simple and non-modal logic with only two possible truth values: true and false. An evaluation of the truth value of a proposition does not affect the proposition, and it does not affect other propositions in the argument. Likewise with conceptual modality here.

Consider converting ~◇(p) to □(~◇(p)). Let p be “I was born in 1776,” which is conceptually impossible due to concepts in other propositions such as q, which is “I have seen my birth certificate.” To further declare □(~◇(p)) is to declare that another proposition, r, hinders not p, but ~◇(p). Although q makes p impossible, it does not make it necessarily impossible. If proposition r is “I looked in a mirror and do not look hundreds of years old,” then r may hinder p, but not ~◇(p). Why? Because whatever proposition r could be, proposition r hinders the truth value of a proposition such as p, not the truth value of the evaluation of p. For example, who would wonder if □(p) is capable internally of being true? That has already been answered.

Contingency

Two additional alethic modalities are contingecy (both possible and non-necessary) and non-contingency (either necessary or impossible). Consider that there is an order of evaluation of the conceptual modalities in any judgment. Briefly, each simple proposition is first evaluated independently of other propositions as either conceptually impossible or possible, depending on whether or not a conflict occurs. If it is conceptually impossible, its contradictory is conceptually necessary. If it is conceptually possible, it may be conjoined with other propositions, and may again be either possible or impossible. To test for necessity or non-necessity requires going further and using a conditional. As an example, Dion evaluates the definite proposition “This marble is blue” as conceptually possible. Next, he wonders if blueness is a quality of a marble, and considers “If x is a marble, x is blue.” Dion tests this conditional as “Both x is a marble and Not x is blue,” and a conflict does not result. Therefore, blueness is not a quality of a marble, and more importantly, “x is blue” is conceptually non-necessary, given that “x is a marble.” It is possible this marble is blue, and non-necessary that it is blue because it is a marble. Either contingency is not needed, or it is understood that a proposition is contingent when it is self-possible and non-necessary from a conditional. Non-contingency was most likely not expressed in Stoic logic.

Vale (pronounced WAH-lay is Latin for “Farewell”),

Ron Hall

Ron Hall



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