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What is reductio ad absurdum? It will take a while to get to an answer to that question, and it ends up being beyond the scope of this post to get the answer. But the way to an answer is interesting.

Let’s get some background. Classical logic has a number of features. Relevant to this discussion are the Law of Non-Contradiction (LNC) and the Law of Excluded Middle (LEM). Speaking extremely loosely1, and speaking only about statements, sentences, propositions, or whatever can possibly be true or false (and so not talking about incomplete sentences, ill-formed statements, nonsense words or phrases, etc.), LNC says that nothing is both true and false, and LEM says that everything is either true or false. An interpretation assigns truth or falsity (at least one, by LEM, and no more than one, by LNC!) to the smallest “parts” in the language, called “propositions”. By rules for combining propositions, compounds are also assigned truth-values. Thus, if “p” and “q” stand for propositions, an interpretation assigns truth-values to “p” and to “q”, and a rule will tell us, for instance, that “p and q” is true if and only if “p” is assigned truth by the interpretation and “q” is assigned truth by that interpretation. The utility of this way of looking at the truth-values of all that can be said in a formal language is that truth-value assignment is defined only for the atomic parts of the language – anything more complex than that can have its truth-value derived from the truth-values of its constituent parts by analysis of the rules of the connective words or phrases in the complex statement. Thus, “if p then q”, “p and q”, “p or q”, “not-p” are true or false by virtue of the truth or falsity of “p” and “q” as well as whatever the rules for “if…then”, “and”, “or”, and “not-” are. Because the connectives take the truth-values of the smaller parts as inputs and map to exactly one of true or false as output, you sometimes see this logic called “truth-functional” – the truth-value of a compound is a function of the truth-values of its parts.

We need a little more background on the classical picture. “Valid inference”, again speaking extremely loosely, means something like this: B can be validly inferred from A if and only if all interpretations that make A true also make B true. A is the premise, B the conclusion (again, a rather loose way of speaking). If there are multiple premises, then let Σ be the set of those premises. Then B can be validly inferred from Σ if and only if every interpretation that makes each element of Σ true also makes B true.

There is one simple, but tedious, way to check whether an inference is valid – list every possible assignment of truth-values to the atoms of every premise in the set of premises, and check if all those assignments that make each premise true also make the conclusion true. As this is sometimes laborious, other methods exist, including natural deduction. Any classical method worth using exhibits the properties of Soundness and Completeness; it’s beyond the scope of this post to explain them, but essentially if a proof method is Sound, it will never mistakenly “prove” a conclusion that doesn’t have to be true, given a certain set of premises; if a proof method is Complete, anything that would be true under all interpretations that make a given set of premises true can be derived from the premises as a conclusion using the rules of the proof method.

Two deduction rules of classical logic are relevant here: ex falso quodlibet and reductio ad absurdum. Let’s add one bit of notation: ⊥ will stand for a proposition that’s true under no valuations. Ex falso quodlibet says that one can infer anything from ⊥; since no interpretation makes ⊥ true, in every interpretation that makes ⊥ true, A is true, for all A whatsoever. This is so because “every interpretation that makes ⊥ true” is a grand total of 0 interpretations; in all 0 of those, A is true! When would we have ⊥ as a premise from which to infer anything, though? Well, “⊥” stands in for whatever simply cannot be true, and by LNC, we know that something like “A and not-A” (assuming “and” and “not-” are truth-functions behaving in the expected way) comes out true if and only if A is both true and false. This will never happen. Thus, “A and not-A” and “⊥” can be substituted for each other.

Why would this be useful? Well, imagine we want to infer B from a certain set Σ of premises. By LEM, with p some proposition, p is true or false. If I can prove that p’s truth implies the truth of B, and that p’s falsity, together with the truth of each premise in the set of premises, implies a contradiction, then I can prove that p’s falsity and the truth of each premises in the set of premises also implies B! Once I get ⊥ inferred, I can infer B, while (in my hypothetical) B can be inferred from p’s truth as well, and since one of “p is true” or “p is false” must be true, I get to B one way or the other. Note that I had to use another rule – proof by cases – to make this work, but that detail isn’t important (here).

Reductio ad absurdum says that if one can infer ⊥ from not-A, then one can infer A. If all interpretations that make not-A true make ⊥ true, then none make not-A true. Since not-A must be true or false, it must be false. Given a standard interpretation of “not-“, A therefore must be true.

This is just the classical picture. Classical logic is not the only logic. There is a philosophical position called “dialetheism” that denies (in some sense) the LNC (in some form). In other words, the same thing may be true and false at the same time, although not everything is both true and false (the latter position is called “trivialism”).

Dialetheism is “tolerant” of contradictions and thus it cannot regard “⊥” and “A and not-A” as equivalent – if “⊥” just stands in for something that is true under no interpretations, but a contradiction can be true in some (dialethetic) interpretations, then “⊥” won’t represent a contradiction…or the rules above have to be rejected.

The thought that started my current interest in reductio ad absurdum was my imagining an opponent of dialetheism saying, “If dialetheism were true, one would have to give up reductio ad absurdum. But to give up reductio is to give up an essential part of our inferential machinery. Many mathematical results are achieved by reductio. Since reductio works, and dialetheism says it doesn’t, dialetheism must be wrong.”

One can of course directly argue for dialetheism, but there is a possibly strategically better way to argue against this point – intuitionistic logic rejects reductio ad absurdum but has a much better reputation, not being as radical as dialetheism and having the inferential machinery necessary to prove important results in mathematics2. Intuitionistic logic rejects LEM; thus, reductio doesn’t work because showing that the negation of something leads to a contradiction shows only that the negation isn’t true; with LEM rejected, showing that the negation of something is not true, and thus that that thing isn’t false, doesn’t mean it has to be true, as there’s a third option – neither true nor false. Note that this doesn’t commit the intuitionist to thinking that anything actually is suspended between truth and falsity – it merely means that a negative proof showing that one option must be rejected is not yet a positive proof that the other option must be accepted (my “speaking loosely” caveat is necessary here, as I need to beg a great deal of charity in the interpretation of this – it does not, to put it mildly, give an entire clear and accurate picture of intuitionism).

Another thing came to mind, though – does dialetheism require rejecting reductio ad absurdum…at least of a sort? Dialetheism definitely rejects ex falso quodlibet, because, since some interpretations (may) make contradictions true, an arbitrary sentence is not vacuously true in all (i.e., all 0) interpretations that make a contradiction true. If “⊥” still stands for something that’s never true, it has to be “unhooked” from its equivalence to a contradiction, but even this may not be good enough – as ex falso quodlibet allows anything at all to be derived from “⊥”, the inferred statement may have nothing to do with the inferences that led to ⊥. The thought that premises and conclusion need to be related to each other takes us too far afield – into relevant (or relevance) logic, something I have to skip both because it’s not relevant (!) and because I’m largely ignorant (!!) of its details.

Come back to what I said in the previous paragraph – the meaning of “⊥” has to be unhooked from contradiction in a dialethetic logic. Reductio doesn’t fail in a dialethetic logic because dialethetic logics reject LEM – Graham Priest’s Logic of Paradox (LP) validates LEM, in fact. In LP, everything is either true, false, or both. Imagine we want to use reductio in LP. If “not-A” leads to “⊥”, which is now not a contradiction (because this would be fine in LP, and being able to infer ⊥ from not-A would not show not-A to lead to anything problematic, so it would be perfectly consistent with the falsity of A, which means we can’t prove A this way) but something else true in no interpretations, then, as this makes not-A true in no interpretations, but A must be true or false (and if not-A is true and negation works as we expect, A is not false), then A must be true.

This looks like reductio ad absurdum, but it’s obviously not the same, as ⊥ has had its meaning changed. It seems to me as if the natural language description of reductio is as follows: “If the non-obtaining of a state of affairs would imply an unacceptable state of affairs, the former state of affairs must in fact obtain.” In classical logic, “unacceptable state of affairs” means a contradiction. But I don’t think it needs to in all contexts.

In short, I’m not really sure what reductio ad absurdum is.

1. The Law of Non-Contradiction: New Philosophical Essays is relevant here, especially Grim’s chapter. What “the” LNC is is at least contentious. It doesn’t end up mattering here; I’m not talking about LNC (and LEM) themselves, but about some particular restrictions on logical interpretations they (seem to) enforce. If it’s not the LNC enforcing the restriction, so be it; what I care about in this specific context is the extent of the restriction. As an aside, fantastic book.

2. And I just learned while doing this research to minimize (oh, I hope) the number of stupid things I say that Euclid’s proof of the infinitude of primes was not by reductio but was “constructive” – in other words, it met the additional requirements imposed on proof in intuitionistic contexts vis-a-vis classical logic.

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Illustration of a theme, from Essential Topology by Martin D. Crossley:

Note that, in these examples, there are subsets which are both open and closed, such as the empty set and {0, 1}, and there are subsets which are neither open nor closed, such as {0} in the first example. So, just as with R, the terms “open” and “closed” are not opposite to each other, and we can only know if a subset is closed or not by looking at its complement.

To most people, it would seem obvious that “open” and “closed” are contraries – both cannot be true, although certainly both could be false, or perhaps just inapplicable to a certain situation. In any case, if a door is open, it’s not also closed. And, to most people, the contrary meanings of the words should carry over into any context – “open” and “closed” are so unambiguously opposite to each other that any context into which these words could be imported cannot help but to import their contrary nature. So, either topology should not allow a space to be both open and closed, or it should not use these words at all.

Yet the quoted text refutes that commonsense notion.

And now to come to the point of this post – “subjective” and “objective” have the same relation as “open” and “closed” – not always contrary, in other words.

Now, dictionary definitions are not arguments, but they are useful starting points for discussing the meaning of words; whoever wrote the definition thinks that other people actually do, or ought to, regard that word as having that meaning. That said:

1. existing in the mind; belonging to the thinking subject rather than to the object of thought (opposed to objective ).
2. pertaining to or characteristic of an individual; personal; individual: a subjective evaluation.
3. placing excessive emphasis on one’s own moods, attitudes, opinions, etc.; unduly egocentric.
4. Philosophy. relating to or of the nature of an object as it is known in the mind as distinct from a thing in itself.
5. relating to properties or specific conditions of the mind as distinguished from general or universal experience.

subjective. (n.d.). Dictionary.com Unabridged. Retrieved February 14, 2014, from Dictionary.com website: http://dictionary.reference.com/browse/subjective

4. being the object or goal of one’s efforts or actions.
5. not influenced by personal feelings, interpretations, or prejudice; based on facts; unbiased: an objective opinion.
6. intent upon or dealing with things external to the mind rather than with thoughts or feelings, as a person or a book.
7. being the object of perception or thought; belonging to the object of thought rather than to the thinking subject (opposed to subjective ).
8. of or pertaining to something that can be known, or to something that is an object or a part of an object; existing independent of thought or an observer as part of reality.

objective. (n.d.). Dictionary.com Unabridged. Retrieved February 14, 2014, from Dictionary.com website: http://dictionary.reference.com/browse/objective

Note that definition 1 of “subjective” and definition 5 of “objective” certainly aren’t contrary to each other: the example used for the latter, an opinion, has to be the opinion of someone, of some thinking subject. So, ipso facto, that “objective opinion” is also subjective. I think the supporter of the idea that the two terms are contrary would not be satisfied with this. I’m not either.

How about definition 1 of “subjective” and definition 7 of “objective”? After all, each definition says it is “opposed” to the other. Well, the supposed opposition is not as strong as it appears. Take an example. I am thinking right now about a book I’ve bought and that’s being shipped to me (I hope!) within the next few days. The book itself has a certain weight, a certain appearance, a certain texture to the paper, etc. My thoughts about it include the anticipation of receiving it, the hope that I can understand it without too much trouble, and, of course, various thoughts about the objective qualities mentioned above. “weighs 1.2 pounds” is a fact about the object of this thought (although I suppose it applied to me a very long time ago, before birth…considered as an object, that is, as it makes little sense to say that I, as subject, weigh anything…but that’s another discussion). “is wondering when to read the thing that weighs 1.2 pounds” is a fact about the subject of this thought. These facts are not in any sense contrary to each other; indeed, they have to work together to complete the experience. I think about an object, making it an object of my thought; the subjective directing of my attention toward the object, and the object’s being its objective self, combine to create the experience. The two aspects are opposed to each other, but in no way that generates contradiction.

Imagine two football players, one a wide receiver, the other a cornerback. The receiver runs a route; the corner, in man coverage, shadows the receiver’s movement. The point of the receiver’s movement, which might include fakes or adjustments that are not strictly provided for in the prescribed route, is to get open. The point of the cornerback’s movement is not to allow the receiver to get open. So the two aspects are opposed, but they create one event, which we might name “the coverage of WR by CB during play N of drive M”, and I think the name has gone long enough – you can see the point.

The point of this example isn’t to be a perfect image of the subjective/objective mix making up the experience; it’s merely to show that being opposed is not sufficient to generate contrariety or contradiction. Indeed, imagine another person asks me what I’m thinking about, and I tell him. Then, “thinking about that book” is a subjective fact insofar as it’s considered from my point of view, but an objective fact when considered from the point of view of my inquisitor. I can extend the insight, too. “I am anxious about my first year of college” is subjective from my point of view, but objective from the point of view of a therapist. One might call it a psychological fact. And if a hundred thousand such subjective-but-also-objective facts are considered together, they might make up a sociological fact. But the names aren’t important; what is important is that the duality of aspect of the thought(s) makes them both subjective and objective by necessity.

This is true even if I’m lying. After all, take the claim “It is 40 degrees Fahrenheit at time t and location x” and let “t” and “x” be sufficiently precise for the relevant purpose. One wouldn’t call this an objective fact if it was actually 38 degrees, but by being untrue, it doesn’t stop being objective. So if I’m lying, nothing changes about the objectivity of “a hundred thousand people in the US are currently anxious about their first year of college.”

But again, I think the analysis focuses on the wrong definitions, and thus misses the mark. It’s true that I’ve shown that “subjective” and “objective” are not always contrary, but then I can’t see that anyone but a straw man was arguing that those senses of the words were the ones that never worked together.

So enough just reading off definitions; what is a person saying who says “opinions are subjective and nothing subjective can be objective,” or some such thing? I suppose that begs the question that that’s a clear and accurate presentation of the view of the person who makes the distinction. Let’s take what I think will qualify as a good example of a subjective judgment, in the sense in which such people take the word: “I like Led Zeppelin.” As before, whether this is true or not, it’s an objective fact that a person does or does not like the band, so when someone is expressing a “subjective” preference in such a manner, there is still objective content to it; it’s true or not, and surely it makes no sense to say that the truth of the fact depends on the subjective judgment of each person! I can sense the reader objecting: “But ‘I like Led Zeppelin’ is true for you but not for me, so of course it’s subjective-and-not-objective!” Imprecision of language, I say! To make “I like Led Zeppelin” into a neat logical sentence, suitable for receiving a truth value in plain old extensional logic, I need to name “I”. The speaker gets a name, let’s just say “c” (for “constant”, see how creative that was?), and now “c likes Led Zeppelin” does not depend on the subject uttering it or thinking it at all.

I think perhaps I haven’t been fair again. So let’s try two more claims:

“Led Zeppelin songs are enjoyable”

and

“Led Zeppelin is better than Pink Floyd”

If both are prefixed with something like “I find that” then my treatment immediately above should deal with the two. But that’s too easy. I see no reason to think that I should prefix them so.

I don’t know exactly what the first means. If it means “Led Zeppelin songs are enjoyed in some possible world” or “at least one person enjoys Led Zeppelin songs,” there’s an objective fact (again, true or false!) being stated. I think a trickier rendering is “although I lack concrete information about the tastes of most people, I enjoy Led Zeppelin songs, so most people must also,” which certainly seems objectionable. Perhaps worse, take it to mean “everyone ought to enjoy Led Zeppelin songs.” I think this hits upon the sense in which “subjective” and “objective” exclude each other – a subjective preference is only illegitimately hypostasized into an objective fact about all other subjects’ preferences.

So, “this is enjoyable” is subjective. And, sure, it’s subjective in a way that isn’t objective, or at least not in every sense – sure, it’s an objective fact that someone enjoys something, but not that the enjoyment is an inherent part of the object rather than the subject. And yet…I wonder about something. I wonder if there is a sense in which the duality of aspects appears again. If the majority of people do enjoy Zeppelin, there may be something in the object that triggers the same subjective preference in many people. Now, we have no reason to doubt that those who don’t enjoy the songs don’t enjoy them, nor should we ascribe bad taste to those who don’t enjoy them; therefore, the preference has to be called “subjective” for two reasons: it deals with the subject aspect of subject-experiencing-object, and it is a matter of that varies from subject to subject – it involves taste, opinion, or whatever. Yet there’s an objective element, if the same objective facts about the object cause the experience to be similar even in its “subjective-and-not-also-objective” elements.

And then there’s a normative statement like at least one meaning of “you should like Led Zeppelin songs.”

I think I’ve proved too much, or, really, claimed that “subjective” and “objective” are more compatible than they really are, and certainly more compatible than I’ve proved. So let’s return to the central issue. Now is probably the right time to explain the “XOR” in the title. I think the claim “judgments are either subjective or objective” is taken to involve exclusive disjunction. I’ve tried to argue that it need not. The strongest evidence for an exclusive reading is a judgment about preferences of taste. However, I think a few things need to be kept in mind. First, often “a subjective judgment can’t also be objective” merely begs the question. If “subjective” is defined to exclude objectivity, then the claim is tautological but simply ignores what the words actually mean in many contexts. I’ve demonstrated that above. Second, a focus on the subjective aspect of experience should not be taken to mean that the objective aspect doesn’t exist. Finally, and this is the most important point here, it should be recognized that there may be a basis in the object even for a subjective preference. So a subjectivity that excludes objectivity, even when legitimate, should not be extended beyond its proper limits; a person’s preference for a thing should not be construed to ignore some basis in the object, and thus an objective aspect, to the preference.

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