The law of non-contradiction (i.e., the LNC) can be expressed in numerous ways, including, "one cannot say of something that it is and that it is not in the same respect and at the same time," and, "contradictory propositions cannot both be true at the same time and in the same sense." Regardless of its expression, however, the LNC has been interpreted as not possible to demonstrate. Furthermore, in part due to lack of its demonstration, the LNC has in recent years become challenged by the position of dialetheism, which affirms that some statements can be simultaneously both true and false.
This chapter, then, attempts to demonstrate that contradictions cannot be allowed within our reasoning or as outcomes of it if reasoning is to in any way retain its effectiveness for us. If this demonstration is successful, it will provide an epistemic certainty for at least one version of the LNC. In so doing, it will safeguard those epistemic certainties that require an absence of contradiction.
If one has no qualms about maintaining the law of non-contradiction throughout this work, the contents of this chapter may be overlooked.
3.1. Rearticulating Contradictions
In keeping with this treatise’s initial lack of reliance upon the epistemic criterion of truth, contradictions shall be here defined in manners independent of truth-values.
Let mutual exclusivity be understood as the incapacity of givens to co-occur. A perfect silence and a loud noise can exemplify mutually exclusive givens.
Then, let a contradiction be defined as the particular result of discernment in which two or more givens that are discerned to be mutually exclusive are likewise discerned to at the same time co-occur. More succinctly, a contradiction shall within this work be defined as the discerned co-occurrence of mutually exclusive givens.
To illustrate this via example, consider a table that is stated to have both three legs and four legs at the same time and in the same respect. Such a table will be here deemed to be contradictory for the following reasons: The table’s stated two properties—that of having three legs in the same respect as it having four legs—will be discerned to be mutually exclusive, for it will be deemed that these two properties cannot be held by the table at the same time. Notwithstanding, the table’s same two properties are stated to occur at the same time. Then, because the addressed properties of the table are both discerned to be mutually exclusive and to co-occur, the stated table shall be contradictory to those here concerned.
Contradictions are thus defined, again, in attempts to circumvent any requirement to define contradictions via the criterion of truth. Instead, contradictions are specified via the psychological faculties of discernment. Nevertheless, any proposition which is specified as both true and false at the same time shall qualify the proposition as being contradictory whenever its truth and falsity are discerned to be mutually exclusive—for, here, the discernment of givens being mutually exclusive becomes superimposed with the discernment of the same givens co-occurring. Furthermore, this chapter’s offered definition of contradictions is deemed to encapsulate all instantiations of something being discerned to both be and not be, else both occur and not occur, at the same time and in the same respect—for this latter property will be likewise discerned by all those here concerned as consisting of mutually exclusive givens that nevertheless co-occur.
By comparison, let a non-contradiction be understood to be any discernment in which the co-occurrence of mutually exclusive givens does not obtain.
3.2. The Incomprehensibility of Contradictions
Relative to all those here concerned, it will be experientially evidenced that we are incapable of comprehending how two or more mutually exclusive givens co-occur.
The only means of obtaining a justifiable alternative to the just mentioned conclusion will be by acquiring a direct experience wherein the co-occurrence of mutually exclusive givens is comprehended—such that it becomes understood how givens co-occur while their incapacity of co-occurring is preserved. Respective to all those here concerned, no such direct experience has so far been obtained. Because of this, that everyone here concerned will find contradictions incomprehensible is upheld to be an unfalsified certainty.
3.3. Two Requirements for Reasoning’s Effectiveness
For the purposes of this chapter, let reasoning be minimally understood as the general conscious process a) through which beliefs are substantiated, b) that allows for analysis to occur, and c) via which conclusions are derived from premises or evidence. Hence, reasoning as it is here minimally understood shall always consist of the means by which results are obtained.
Then, let reasoning’s effectiveness (i.e., efficacy) be understood as the power of reasoning to produce desired results as its effect.
It will be presented that we can only consider reasoning effective when 1) we can rationally discern its capacity to produce correct outcomes and 2) when both itself as means and its outcomes as ends are intelligible to us.
As regards (1), it will first be upheld that, relative to those here concerned, if our wants or needs are the obtainment of A, we then use reasoning for this purpose with the intention of obtaining A—with A here being deemed the reasoning’s correct outcome for us. We therefore never want or need the obtainment of A while using reasoning for this purpose with the intention of not obtaining A—with not-A here being deemed the reasoning’s erroneous outcome for us. Reasoning’s efficacy is thereby contingent on its capacity to obtain outcomes that are either wanted or needed and, therefore, on its capacity of producing correct outcomes relative to our wants or needs. As an example, if our wants or needs are the obtainment of accurate results, then the reasoning we use must hold the capacity of producing accurate results in order for it to be deemed effective by us; alternatively, were wants or needs to be those of successful deceptions, then the reasoning used must hold the capacity of producing successful deceptions in order for it to be deemed effective in respect to the wants or needs for which it is employed. In summation, reasoning’s efficacy will be contingent on its capacity of obtaining correct outcomes relative to our wants or needs. Until anyone here concerned holds an experience contrary to that just specified, what has just been specified will be upheld to be of unfalsified certainty—for no justifiable alternative to it is currently apprehended.
Given this unfalsified certainty, the following conclusion becomes entailed: Were it to hypothetically become impossible for us to rationally discern which instantiations of reasoning can produce correct outcomes relative to our wants or needs and which cannot—because here any trust in reasoned outcomes being correct would literally be impossible to substantiate—reasoning in general would then become ineffective for us. Until anyone can find a means to substantiate any instantiation of trust given a hypothetical world wherein no instantiation of reasoning can be rationally discerned to produce correct outcomes, this entailed conclusion will be upheld as unfalsified certainty—for no justifiable alternative for it is currently apprehended. This unfalsified certainty can be reworded as follows: relative to those concerned, for any instantiation of reasoning to be effective its capacity to produce correct outcomes must be rationally discernable.
As regards (2), relative to all those here concerned, if we are incapable of comprehending the reasoning used to obtain some result, the reasoning’s incomprehensibility will in turn incapacitate us from rationally discerning whether or not its reasoned outcomes are correct relative to our wants or needs. For example, if we are in pursuit of accurate results, and if the reasoning used for this purpose consists of, “Gibberish plus zebra is greater than running and zebras are native to Africa, therefore Sam is a better runner than Adam,” we will not be capable of discerning strictly via this reasoning whether or not the inferred conclusion is correct due to our incapacity to comprehend the reasoning by which the conclusion was obtained. Till anyone here concerned can experience the contrary of the generality just affirmed, the general principle just affirmed will be upheld to be of unfalsified certainty.
Furthermore, even if the reasoning is comprehensible to us, were the reasoning’s result to be impossible to comprehend we would also in turn be incapable of rationally discerning whether or not the reasoned outcome is correct relative to our wants or needs. For example, if we are in pursuit of accurate results, and if the reasoning used for this purpose consists of, “Sam is a better runner than Adam and zebras are native to Africa, therefore gibberish plus zebra is greater than running,” we will not be capable of rationally discerning whether or not the conclusion is correct due to our incapacity of comprehending it. Till anyone here concerned can experience the contrary of this second generality just affirmed, this second general principle just affirmed will be upheld to be of unfalsified certainty.
Given the validity of these just mentioned unfalsified certainties, the following conclusion becomes entailed: Relative to all those here concerned, because reasoning’s capacity to produce correct outcomes must be rationally discernable in order for it to be effective, and because we are incapable of so discerning whenever the given instantiation of reasoning or its outcomes is incomprehensible to us, it is a resulting unfalsified certainty that we must find reasoning and its outcomes intelligible in order for the reasoning to be effective for us.
In summation, if reasoning is to have any efficacy it must minimally hold the following two properties: 1) its instantiation must be rationally discernable as capable of obtaining correct outcomes and 2) both it and its outcomes must be comprehensible. In absence of these two properties, the addressed reasoning can then only be ineffective for us.
3.4. Why Correct Reasoning Cannot Allow Contradictions
For the purposes of this argument: Let reasoning be termed contradictory reasoning when its premises or evidence, its inferences or justifications, its outcomes, or any combination of these are contradictory. Let reasoning devoid of contradictions be termed non-contradictory reasoning. Let correct reasoning be understood to be an umbrella term for reasoning which is free form errors, i.e. for reasoning which is not fallacious. And let incorrect reasoning be understood to be an umbrella term for reasoning that contains errors and is thereby erroneous, i.e. for reasoning which is fallacious.
It is here first offered as unfalsified certainty that the relation between correct reasoning and contradictory reasoning can only take the following three possibilities: 1) correct reasoning will always be contradictory; 2) correct reasoning will sometimes be contradictory and sometimes will not be contradictory; and 3) correct reasoning will never be contradictory.
Were scenario (1) to depict what is ontically certain, correct reasoning could only obtain when it is contradictory—hence, when its premises or evidence, its inferences or justifications, its outcomes, or any combination of these are contradictions. As presented in §3.2, these contradictions shall be incomprehensible to those here concerned. As presented in §3.3, the incomprehensibility of premises or evidence, of inferences or justifications, of outcomes, or any combination of these shall result in the inefficacy of the given reasoning. Hence, were scenario (1) to be ontically certain, it would likewise be ontically certain that all instances of correct reasoning will be inefficacious relative to all those concerned.
Because all instantiations of correct reasoning would be inefficacious for everyone here concerned, the following is concluded as unfalsified certainty: If scenario (1) were to depict what is ontically certain, no one here concerned could make use of correct reasoning to determine which instantiations of reasoning are capable of producing correct outcomes—including for the purpose of obtaining this very conclusion, for this conclusion was obtained via the strict use of non-contradictory reasoning, which would in scenario (1) necessarily be incorrect reasoning. Per the arguments provided in §3.3, scenario (1) would thereby render all reasoning inefficacious.
Allowing so much as one instance of correct contradictory reasoning shall result in the world depicted by scenario (2). Again, in scenario (2) it will be ontically certain that correct reasoning sometimes takes the form of contradictory reasoning. Wherever contradictory reasoning is the correct form of reasoning to apply, non-contradictory reasoning will be the incorrect form of reasoning to apply (if, for example, it is deemed correct to apply both contradictory and non-contradictory reasoning at the same time and in the same respect, this would be one example of contradictory reasoning—for the application of each of these two forms of reasoning is discerned to exclude the other’s application, although they are here further discerned in need of being commonly applied at the same time). Likewise, whenever non-contradictory reasoning is the correct form of reasoning to apply, contradictory reasoning will be the incorrect form of reasoning to apply. Furthermore, if the form of reasoning applied for the purposes of making a determination is incorrect, then any instantiation of it for the purposes of making this same determination will likewise be incorrect; differently expressed, if it is known that use of type A reasoning in general for the purpose of X is erroneous, this entails that any instantiation of type A reasoning for the purpose of X will itself be erroneous.
Given the just stated characteristics of scenario (2), rationally discerning when contradictory reasoning is the correct form of reasoning to apply and when it is not will be impossible for anyone here concerned: Use of contradictory reasoning to make this determination will always result in the inefficacy of the reasoning used—for the incomprehensibility of the reasoning used or of its outcomes renders the reasoning ineffective in facilitating our understanding of when contradictory reasoning is the correct form of reasoning to apply. In turn, use of any non-contradictory reasoning to make this determination will not establish whether non-contradictory reasoning is the correct form of reasoning to apply for this particular determination—for, in scenario (2), it remains a viable possibility that contradictory reasoning is the only correct form of reasoning to use for the purposes of this determination. And, if non-contradictory reasoning happens to be the incorrect form of reasoning to apply, then the specific instantiation of non-contradictory reasoning used shall itself be incorrect.
Hence, for all those here concerned, scenario (2) results in the incapacity to rationally discern if any instantiation of reasoning makes use of the correct type of reasoning and, in consequence, whether or not it is correct reasoning. Because this hypothetical world makes it impossible for anyone here concerned to rationally ascertain which instantiations of reasoning are erroneous and which are not, no one here concerned could in this world then rationally discern which instantiations of reasoning obtain correct outcomes. Per the arguments provided in §3.3, this would in turn render all reasoning inefficacious.
While no epistemic certainty can be currently found for scenario (1) or (2) not being ontically certain, the following can nevertheless be concluded:
It is an unfalsified certainty that upholding the inefficacy of all reasoning is not something psychologically practicable by anyone here concerned—for, in searching for a justifiable alternative to this just mentioned unfalsified certainty, one would be in need of deeming the reasoning via which the alternative becomes justified to be in some manner efficacious, thereby evidencing the just stated unfalsified certainty. Likewise of unfalsified certainty is that the worldview espoused by scenario (3) facilitates the efficacy of at least some forms of reasoning for those here concerned. And, only scenarios (1), (2), and (3) are discernable by anyone here concerned as possibilities regarding correct reasoning’s relation to contradictory reasoning.
Then, the following is presented as this chapter’s concluding unfalsified certainty: For us to rationally maintain that at least some forms of reasoning are efficacious, we will need to intentionally maintain that correct reasoning can never allow contradictions in its premises or evidence, in its inferences or justifications, and in its outcomes. More succinctly expressed, we will need to intentionally uphold that scenario (3) accurately depicts that which is ontically certain in order for us to maintain the efficacy of reasoning in general.
This chapter’s concluding unfalsified certainty—if it will withstand scrutiny—thereby demonstrates one form of the law of non-contradiction. This demonstration of the LNC’s epistemic certainty will thereby be sufficient for the derivation of subsequent unfalsified certainties which require that the law of non-contradiction be applied.
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