If you want to tell the truth, you shouldn’t contradict yourself—that’s just common sense. A suspect who was home on the night of the crime can’t have been elsewhere, and whatever the weapon, we can rule out the hypothesis that it was both a candlestick and not a candlestick. But there are philosophers who claim we shouldn’t overgeneralize based on murder mysteries: some contradictions are true. Could a badly written law make the dastardly deed both legal and illegal? Do mathematical paradoxes create weird things that both do and don’t exist? If we embrace contradictions, will we still be able to tell the difference between truth and falsehood? Josh and Ray embrace contradiction with Graham Priest from the City University of New York, author of Doubt Truth to Be a Liar.
Josh Landy
How could something be both true and false at the same time?
Ray Briggs
If we accepted contradictions, would we end up believing just anything?
Josh Landy
Or would we simply be more open to the mystery of the world?
Comments (19)
Tim Smith
Wednesday, April 19, 2023 -- 5:30 AM
Emily Dickinson's poem, "ThisEmily Dickinson's poem, "This World is not Conclusion," alludes to the existence of a species beyond our comprehension.
This World is not Conclusion.
A Species stands beyond -
Invisible, as Music -
But positive, as Sound -
It beckons, and it baffles -
Philosophy, dont know -
And through a Riddle, at the last -
Sagacity, must go -
To guess it, puzzles scholars -
To gain it, Men have borne
Contempt of Generations
And Crucifixion, shown -
Faith slips - and laughs, and rallies -
Blushes, if any see -
Plucks at a twig of Evidence -
And asks a Vane, the way -
Much Gesture, from the Pulpit -
Strong Hallelujahs roll -
Narcotics cannot still the Tooth
That nibbles at the soul -
Perhaps the species Dickinson alludes to here are true contradictions.
True contradictions exist in logic, and translate to math, and even reality.
“This statement is not True,” is the classic example, and there are others. A sentence has long been a human attribute, though now machines mimic, and soon will attribute their own meaning to statements. But all purveyors of thought need to account for the paradoxical corners of Searle’s Chinese room.
Quantum entanglement is consistent with rules set by quantum mechanics, and is truly odd to human intuition. If people were not open to alternate rules then we wouldn’t ever have discovered quantum theory, which is the most complete that science currently offers. The mysteries of nature are too great to take the power of paraconsistent logic from our bag of tools that would allow us to contemplate our place in nature or more precisely our natural view.
Math is another matter. Russell’s paradox is a true contradiction, where set theory runs into the set of all sets that do not contain themselves. Zermelo-Fraenkel set theory, and others like it, exclude Russell’s problem by axiom. One could say this is cheating, but mathematics is grounded doing so, and it provides a framework for constructing new sets and exploring a much larger landscape of mathematical ideas that are the preponderance of our applied mathematics. That these maths give us our current standard of living, and confidence to set about our daily projects is not enough to exclude the idea that mathematical paradoxes do exist. There is room for these ideas, and possible benefits as yet unpicked.
Then there is Emily Dickinson. The World is not Conclusion. If true contradictions exist, and they do, there are no logical conclusions that can express the Species beyond us. . Theseus’ ship and Neurath’s boat change beneath our feet. Allowing for the possibility of paraconsistency, allows this change more readily. Dickinson was not given the paper to write her poems in the heat of her creativity. She resounds despite her oppression. True contradictions lie beneath the deepest passions of our humanity, and need nurture.
Graham Priest's ideas about true contradictions challenge conventional wisdom and invite us to consider new perspectives on truth and reality. By embracing contradictions, we can open our minds to new possibilities and foster a deeper understanding of the world around us.
Daniel
Friday, April 21, 2023 -- 2:34 PM
Is it a contradiction to sayIs it a contradiction to say something without meaning it? Take your final sentence above. You tell us that if you hug a contradiction, you'll understand the world. Is what precedes it supposed to support this claim? The examples you've provided are not accompanied by explanations which could accomplish this, but rather indicate a fairly clear disregard for those who might be interested in the subject. This is done by deploying summary judgments upon each one, a Dickinson poem, confusions in linguistic practices, a phenomenon in physics involving dual effects of singular causes, and paradoxes in set theory, only to claim at the end that "we", --whether you're speaking to a small group of readers or every sentient being in the universe is unclear, could never hope to understand the world unless these brain teasers are given a big hug. How is such a claim justified? Because your statements here are declarative and not interrogative, you must know the answer to this. Are you suggesting that an elder statesman who is appealed to as an authority on international affairs couldn't understand the world unless she/he understands the meaning of Russel's conclusion: "If it is it isn't, and if it isn't it is"? Certainly you would not deny this statesman the privilege of sharing in your wisdom on the matter.
Tim Smith
Thursday, May 18, 2023 -- 1:46 PM
This was a compelling episodeThis was a compelling episode, and I appreciated the introduction to Graham Priest, whose arguments I found convincing. Josh's reference to Borges, Ray's discussion of infinitesimals, and Graham's response reminded me of Emily Dickinson's poetry and the concept of entanglement in quantum physics. While I'm not entirely persuaded, I'm open to the idea of para-consistency not to be confused with inconsistency.
I recently watched a podcast featuring Priest and Joel David Hamkins, who, despite my admiration for him, seemed to falter in that particular discussion. However, it's interesting to note that Hamkins just posted about paraconsistency today, suggesting he may have been more receptive to the conversation than he appeared during the podcast.
I've engaged with a lot of material on this topic and find myself partially inclined towards dialetheism. Or perhaps I embody a contradiction by subscribing to both Aristotelian logic and dialetheism. Speaking of Aristotle, it's important to note that his stance on contradictions wasn't crystal clear and most restatements of the principle of non-contradiction may be somewhat misguided. For more on this, here is a link to a Stanford Encyclopedia of Philosophy entry on Aristotle's principle of non-contradiction.
Ian Shoales
Tuesday, May 2, 2023 -- 6:15 PM
Yeah! Good stuff. ProvingYeah! Good stuff. Proving if we all learn how to take a Yes or No approach to a Yes and No approach, we'll be safe from entanglement for the near future. That's my two cents!
Daniel
Thursday, May 4, 2023 -- 5:17 PM
The last sentence aboveThe last sentence above strikes me as a contradiction which is so regularly embraced that it requires special effort to see it as such. This is the fact that only the token exchange-value of two pennies can be owned by their possessor. The material of which they are composed, the copper in which they were struck, belongs to the federal government and obtains commodity-value which is different from that of the standard currency unit. If the exchange value of their currency-designation and the commodity-value of their material content is ever the same, it could only be due to an accident. Is this correct? Written as a proposition, one might say that
1) Person x has two cents, (modus ponens, or assertion).
2) Person x does not have the quantity of copper of which two pennies are made, (modus tollens, or negation).
3) Person x both has two cents and does not have two pennies, (modus ponendo tollens, or negation by assertion).
Premise (3) is only a contradiction if two cents is the same thing as two pennies, which is commonly assumed to be the case. On this example, then, it would appear that participant Smith's formula could be reversed by saying that, instead of understanding the world by embracing contradictions not yet embraced, one might have more luck in disentangling the contradictions which are already assumed. Does the tension between commodity- and exchange-value in this case indicate a much larger contradiction in market economies generally?
Tim Smith
Sunday, May 7, 2023 -- 7:27 AM
Hey Ian,Hey Ian,
I'm not sure you are replying to my post above, but if you are I appreciate it. I mentioned entanglement there, but to be clear, entanglement is just the poster child of a logic that is foreign. No change in view is going to change reality there, but it might prevent our having a look.
In reading thru that post - it kinda sorta has a Shoalesian bent. I don't think that is without cause. Thanks for all you do - keeping it real.
Gotta go,
Tim
Daniel
Sunday, May 7, 2023 -- 2:22 PM
What's the basis of yourWhat's the basis of your claim that Russel's paradox is a true paradox? Because it can be demonstrated that this is not the case, your answer to this question would be very informative.
Tim Smith
Sunday, May 7, 2023 -- 6:16 PM
https://plato.stanford.eduhttps://plato.stanford.edu/entries/russell-paradox/
So... what your saying is "^^^This^^^ link is not true."?
This show is going to be fun ;-)
Daniel
Monday, May 8, 2023 -- 4:05 PM
What I said is that Russel'sWhat I said is that Russel's paradox is not a true paradox, and that this fact can be demonstrated to be the case. This is the reason why I asked you for your reasons for understanding the contrary to be true. Is one to assume that because you refer to someone else's reasons that you don't have any of your own? Such a supposition seems to me exceedingly unwise, convinced as I am that they must exist. So if you will forgive the importunity of having to repeat a question asked with collegiate sincerity, --what's the basis of your claim in this regard?
Tim Smith
Tuesday, May 9, 2023 -- 2:07 PM
Daniel,Daniel,
I'm not going to recreate the wheel here. We've been through this before. If you won't read the source material - I can only help so far.
Russell's Paradox is called a paradox because it leads to a contradiction within set theory, once thought to be a consistent foundation for mathematics. The Paradox arises when we consider the concept of sets that contain themselves as members and sets that do not have themselves as members.
The Paradox is named after the philosopher and logician Bertrand Russell, who discovered it in 1901. It can be stated as follows:
Consider the set R of all sets that do not contain themselves as members. Now, ask the question: does R have itself as a member?
In either case, we encounter a contradiction, which is why Russell's Paradox is called a paradox. It exposed a fundamental flaw in the naive set theory and led to more rigorous set theories, such as the Zermelo-Fraenkel set theory, which avoids the Paradox by imposing restrictions on the types of sets that can be formed.
Yes, some philosophers and mathematicians have debated the nature of Russell's Paradox and whether it should be considered a true paradox. Some argue that the Paradox results from a misunderstanding or misuse of set theory rather than an inherent problem with the theory itself.
One such view is that of the philosopher and mathematician Willard Van Orman Quine, who suggested that the Paradox arises from a confusion between the notions of "membership" and "subclass." He argued that sets should be considered subclasses of a larger class. The Paradox could be resolved by distinguishing between a set being a member of another set and a set being a subclass of another set.
Others have questioned the legitimacy of self-reference in formulating the Paradox, arguing that it's an invalid construction that leads to the paradoxical result. They propose alternative approaches to set theory that avoid self-reference or carefully regulate it to prevent paradoxes from arising.
While these alternative viewpoints exist, the majority of mathematicians and philosophers accept Russell's Paradox as a genuine paradox that exposed flaws in naive set theory. The best reason to hold this view is that the work on the Paradox has led to the development of more robust set theories, such as the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), which forms the foundation for most of contemporary mathematics.
Daniel
Wednesday, May 10, 2023 -- 6:24 PM
Now I'm more confused than INow I'm more confused than I was before. I didn't ask you what Russel's paradox was. In fact, I've run across it before and read a little about it. My inquiry concerned rather the basis of your claim that it's a genuine one, i.e. that it's viciously circular. What you've done above instead is to repeat something like it along with a rather pedestrian review of some associated aspects. I conclude therefore that you are unable to provide any basis for your claim on your own and that it derives accordingly from what someone else has said or written for which you are not responsible.
To help you escape then from what seems a severe epistemic pathogenesis, allow me if you will to pose a different question. Consider two different collections of objects. One consists of all the plastic spoons in the world that has ever existed; the other of everything in the world which has ever existed which are not plastic spoons. How do these two sets differ from one another? In one we know what's in it, and in the other we know what's not. Now apply the distinction to the paradox. What's the axiomatic assumption we are asked to accept in order that the paradox be intelligible? What is the primary distinction in this case without which no contradiction can be asserted?
Approached from another direction, take the concept of a self-membered set. In the plastic spoon example, is the set defined by their exclusion one of this variety? Most would say so, since a set is also not a plastic spoon, but would you say so as well?
Tim Smith
Wednesday, May 10, 2023 -- 9:34 PM
Daniel,Daniel,
This is the sort of approach and needed clarification that ZF takes, along with others. However, not all self-referential sets can be replaced by well-defined sets without losing meaning and/or function. The significance/existence of Russell's Paradox and the role of self-reference in such situations remain.
Take care,
Tim
Daniel
Friday, May 12, 2023 -- 3:21 PM
Self-reference has widerSelf-reference has wider extension. Self-membership constitutes a special kind of self-reference, namely that which applies to collections. Two questions were asked above:
1) On what is the distinction between self-membership and heterogenous membership based; and
2) can self-membership derive from its heterogeneity with another?
The answer provided in your reply is to distinguish between self-membership and clear membership, which fails (by my lights) to answer either. A multiple choice format is therefore provided below, in reversed order:
1) Is the set of all sets which do not have plastic spoons as members a member of itself?
a) Yes, because it is not a plastic spoon.
b) No, because a set's membership can not be stipulated by what it's not, (since the number of sets is formally infinite), and
c) Neither yes nor no, since both must assume that this is a different kind of collection from the set of sets, which could be an unwarranted assumption.
2) Is the set of sets a member of itself?
a) Yes, because it is a set.
b) No, because, if one takes (1.a) to be incorrect, this is a collection for which by definition only one member is possible, and as a tautology is uninformative.
c) Yes and no. Yes as a naturalized composite which generates an observable distinction between identity-condition and the membership-singularity, (again, under the assumption that (1.a) is false); and no if membership is determined by the definition alone.
These questions are not directed to the paradox itself, but to the distinction which it must assume. If a self-membered set can have more that one member, and in that connection there can be more than one set of that kind, then the assumption appears warranted. If on the other hand a self membered set is the same thing as the definition of its singular member, --itself, then no such set exists comparable to those with heterogenous membership, and therefore no paradox results. The solution by this reading hinges on the answer to the first question, which is also the one asked at the end of my 5/10/23 post above, and which determines for the examinee how the second will be approached. Is there any impropriety is soliciting an answer a second time?
Tim Smith
Saturday, May 13, 2023 -- 4:02 AM
Daniel,Daniel,
True contradictions and paradoxes occur all around us. We can shape our views to avoid them, and yet they exist. Accepting their existence, understanding their origin, and handling true contradictions is propitious, as is propriety. Russell's Paradox is not the primary concern here, but I will give this one more whack and leave you the opportunity for a last word.
The complexity and inherent issues associated with self-reference are central to Russell's Paradox. As you mention, Zermelo-Fraenkel and the Axiom of Choice (ZFC) address these issues by disallowing self-membered sets. This exclusion provides a generally accepted logical foundation for math, despite having its own paradoxes, such as the Banach-Tarski Paradox, which suggests that a ball can be separated and reassembled into two identical balls, a consequence of the Axiom of Choice. Contradictions arise regardless of the view you take. If that last statement seems paradoxical, it is not. Russell's and Banach-Tarski's Paradoxes are unrelated and true all the same.
You correctly identify self-membership as a specific type of self-reference. Let's reconsider your thought experiment involving sets and plastic spoons and add heterogeneity. Let's imagine we have sets of objects, each defined by the color of plastic spoons it contains. We then define a unique set, set S, which includes all sets that do not contain a plastic spoon of their own defining color. The Paradox arises when we question whether set S includes a plastic spoon corresponding to its defining characteristic (i.e., not containing a spoon of its own color). This question leads to a contradiction, demonstrating that the issue at the heart of Russell's Paradox is not necessarily the distinction between self-membership and heterogeneous membership but rather the self-referential nature of the set's definition.
In response to your two questions:
==> The distinction between self-membership and heterogeneous membership is based on whether a set is a member of itself or not.
==> The notion of self-membership does not inherently derive from its heterogeneity with another set.
Crucially your questions imply that understanding and resolving Russell's Paradox depends on the distinction between self-membership and heterogeneous membership. However, the colored spoon example demonstrates that the Paradox arises from the self-referential nature of a set's definition rather than the membership type.
Regarding your multiple-choice questions, the options provided seem to misunderstand certain aspects of set theory. They appear to treat sets as if they were physical objects like plastic spoons, which is inaccurate. Crucially, a set is defined by its criteria, not by its specific members, and the members of a set are the result of the requirements and not the other way around.
Similarly, in ZFC, the "set of all sets" does not exist, as it leads to contradictions akin to Russell's Paradox. Therefore, it cannot be a member of itself. The membership of a set is determined not solely by its definition even but also by how it fits within the framework of the theory.
While your inquiries illuminate the complexities of set theory, the broader topic here is true contradictions. Self-reference, present in fields as diverse as linguistics, mathematics, computer science, biology, art, and literature, can lead to paradoxes or complications. However, it is an inherent aspect of many real-world phenomena and can provide valuable insights or solutions in specific contexts. Gödel's Incompleteness Theorems, for example, constructively use self-reference. These theorems demonstrate that any sufficiently complex formal system (i.e., a system with enough resources to do arithmetic) will have true statements that cannot be proven within the system itself. These theorems, though not paradoxical, highlight the limitations and complexities inherent in our attempts to formalize mathematics. They underscore that self-reference isn't merely a source of paradoxes but can also be a profound tool for revealing fundamental truths about logical systems.
We may still differ on this subject, but there are many resources where these concepts are laid bare. One of the best is https://plato.stanford.edu/ and https://plato.stanford.edu/entries/russell-paradox/. If we push beyond that, we must, at least, come to terms there. I would appreciate a reference for further reading if we still disagree.
You get the last word, but we should stick to true contradictions and get out of the Russell hole if possible.
Best to you, as always,
Tim
Daniel
Sunday, May 14, 2023 -- 4:56 PM
See Parmenides, On Nature, VISee Parmenides, On Nature, VI.1-VII.2; and Aristotle, Metaphysics, book IV. Your fifth paragraph above, (not including the two numbered questions), contains an indication of what seems to me a fundamental ontological question raised by the paradox, involving comparison between accidental form of an aggregate and pre-determined form of a paronymate. To restate my position in summary form, Russel's paradox is not a true contradiction because it equates what a collection is with something that itself can be collected. As noted, if a set could be a member of itself merely by virtue of its not containing what another set contains, ala the plastic spoon example, then my objection won't work. We both seem to agree however that just because a set contains no plastic spoons, this does not imply that it's a member of itself merely because a set is also not a plastic spoon. In this respect your original position seems to have shifted. Do I have that right? At any rate, before we can get to the meat-n-potatoes of paragraph five, please restate your so-called "reconsideration" of the thought experiment involving autogenous membership defined by heterogenous membership, which you offer in the third paragraph above. Whatever you're trying to convey there is completely unintelligible.
Daniel
Monday, May 8, 2023 -- 3:43 PM
In reply to the main pointIn reply to the main point contained in the second sentence, reference to approach indicates traversable distance and therefore the notion of temporal passage. This occurs, according to the description, as two positions which differ in their logical commitments and do not in the near term entangle each other. One is bi-valent and the other sub-tertiary. At some point however the two come into conflict due to a logical incompatibility, so that one position must exclude the other. An example of this clash of two different approaches can be seen to occur in slave economies. For the master class, the relation to the slave is bi-valent. A command is obeyed or not obeyed. But within the slave class a third possibility emerges: No command is given, and optional ground of agency belonging to the command-recipient is affirmed. If a slave prepares a duck for an owner's dinner, the rawness of the duck is negated by its being cooked. But by eating it the owner negates the thing itself, or the whole duck. In adding the negation of a property of a thing to the negation of the thing itself, one gets an affirmation: culinary skill, nutritional understanding, or what have you. The negation of a negation equals an affirmation, or, two negatives equal a positive; --but only on the side of the slave. The owner/master can't hold on to the third term.
In this way logic has a social history. In Aristotle duration of a substance can not be approached in bi-valent terms. "Socrates stands" is true so long as he's standing, but will become false when he sits down.* A time-order or chronology must therefore supervene upon bi-valent criteria for determinative judgement, if a substance can be predicated by duration. And "Socrates stands at time t" and "sits at time t'" is not sufficient, since t, in Aristotle, equates to a particular "now", defined as a contiguous division between earlier and later which itself has no duration, so that it could not be a predicate of anything which lasts. Stoicism in this relation represents individual duration or "the Self" as precisely that part of the world which expresses the limit of the world, with the result that no causal influence between optional agency and independent events can be recognized. And because the Stoic doesn't have to worry about the duration of agency for particular substances, space was opened up for propositional logic to replace Aristotle's logic of the combination of reference-tokens.
The inference drawn in conclusion is that sub-tertiary logic, the logic of Hegel or "both-and", overrides the law of bi-valence, re-instantiated by Kierkegaard as the law of "either-or", through the progress of a continually improved awareness of the latter's dependence on that to which the former applies. Progress towards universal slavery-abolition thus furnishes the necessary historical condition for a rigorous sub-tertiary logic. Would a contrary position in defense of strict bi-valence here have to include the historical context of its formulation?
_________
* Metaphysics 1051b/Categories 4a.
Daniel
Saturday, May 6, 2023 -- 3:29 PM
Is there a built-inIs there a built-in contradiction between the mechanism of perception and the awareness of what is perceived? E.g. in seeing something, say a book lying open on the seat of a chair, is there any sense to saying that one is also seeing one's seeing of an open book on a chair? If so, which is the real thing and which the mere means of becoming aware of it? This could be tested by checking to see if the book is a real book by reaching over and closing it up, thereby confirming that it's not an artificial imitation. Such a test however confirms both claims, indicating that the difference between them consists in their ontological commitments, and not what each predicts would be the case.
The physics of very small things runs into a similar problem in trying to see how they move. Along the lines of a well known thought experiment, one can imagine a pebble being dropped into two open boxes and landing in one of them. The boxes are then closed and transported in opposite directions to different locations. During the trip and after arrival at their destinations, it's true of each that there's a fifty percent chance it contains the pebble. Only after one is opened would the claim be false. Does that mean it was never true?
In this particular case the answer is obvious because it's presumed that the boxes can without difficulty be opened by their proper recipients, so that a statement about the probability of their contents is not a statement about the things inside. But what if the boxes were so small that they're almost never opened? An inference from the probability of their contents would have to be made.
Would this be an inference from something actually observed? If all one sees of a particle's motion, say an electron, is a series of probable locations, each of which results in a different set of probable locations from their alternates, isn't the progression of alternate probability-sets just as much a part of nature as are the positions known with zero percent improbability? In this latter case the alternate probability-cones (or "wave-function(-s)") disappear as a result, making certain knowledge of behavior a preclusion of research-object observability, consisting, as it is and by analogy, of very small boxes which are almost always closed.
If the unrefined character of this account be forgiven, does it make sense to ask whether or not in some cases seeing more of something causes one to know less of it? Is there any contradiction between understanding something by not knowing it and knowing something because one ceases to understand it?
Harold G. Neuman
Sunday, May 7, 2023 -- 8:48 AM
I think I understand whatI think I understand what true contradiction is aiming at, from examples given and questions asked. But the statement: this sentence is false, makes no sense. Many utterences/sentences are either true , false, inconclusive or irrelevant. But, if like the example, they lack information, there is no means of determining anything about them, save perhaps that they are gibberish? "This sentence is false" makes no sense: a four word phrase which reveals nothing; neither true nor contradictory. I hope, and trust, the show will do better.
Daniel
Tuesday, May 16, 2023 -- 6:19 PM
In a post by participantIn a post by participant Smith of 4/19/23 above, a version of the Liar-paradox is offered in which the claim can be read as made that it would still exist even if nobody is aware of it. Since the example has already been introduced, then, indulgence in a brief analysis can perhaps be forgiven.
Two approaches are possible: One is as a practical matter, to resolve it so that it won't happen. The other is theoretical, to analyze it in order to understand why it does. The most well-known version runs something like "this sentence is false". In semantic context, the statement is not meaningless, since its frustration of a primary semantic assumption indicates a truth-value inapplicability, which issues the positive judgement of a contradictory sentence. As a matter of truth value however a determination can not be made for what is asserted. To review, "sentence x is false" says that it's true, which claims that it's false. Because a middle term is not permitted under the rule of bi-valence which assumes it, no truth value can be assigned, implying that truthless sentences which aren't lies inherit meaning, or positive semantic content, from typical assumptions which are frustrated without being weakened.
A third term certainly however appears to be possible, provided a distinction is made between paradoxes and contradictions. "This sentence is false" is both, but "this sentence is true", while it does not contradict itself, nevertheless remains a paradox on account of the fact that one can apply the values of both truth and falsity to it with equally good reasons to choose its opposite. But if one distinguishes propositional from semantic logic, the law of non-contradiction* does not seem to be sacrificed if one rejects the law of bi-valence and that of the excluded middle which it assumes,** for propositions in which no truth-value can be determined, so that a third term can be added to true and false, namely, "non-determinable".
While a sentence describes an object, a proposition can be said to indicate the truth-values of a description, which can accommodate paradox without contradiction, by reference to a representation-operator rather than a corresponding content, and as such a third term between true and false might be permitted. In a contrasting example, the paradox "0 = 0" implies that anything added to "0" equals zero, which is equal to anything subtracted from it, in turn implying that it is not identical with itself, and is therefore a false proposition. Here no middle term is needed because what is permitted according to the assumptions of language-use is eliminated by the meta-semantics of a propositional language. While some contradictions, then, appear to persist as non-determined, their overall number which occurs semantically is reduceable. From where could a rebuttal to this position arise, --irreducibility of independence of semantic contents, or the dependency of propositions on lingual practices?
__________
* "Both x and not x" is false.
** "x is either true or false" (bi-valence); and "x is true or not true, but can't be both and must be either" (excluded middle).