On implicit contradictions

Question

Does the premises

The roses are red.
The roses are blue.

involve an implicit contradiction? I mean, if roses are blue, doesn't this imply that "roses are NOT red"?

Assume that the premises are not open to interpretation and really mean what they say. That is, when I say that "roses are red" I mean that they are completely red, not partially red.

Edited to reply to Ludwig: It is convenient to distinguish between what is an explicit and implicit contradiction.

When you say that there is no contradiction between premises 1 and 2, you mean that there is no explicit contradiction between them. I totally agree with that and with your p^q formulation. But, and I see that you agree with that, the "roses are blue" implies that "roses are NOT red", (q => ¬p). According to what I have read, in order to formalize an implicit contradiction it is necessary to add an additional, necessarily true premise.

In the example of my question it would be: (p ^ q) ^ (q => ¬p). But this last formula can be reduced to a formal contradiction, p ^ ¬p, by modus ponens.

How can an implicit contradiction be formalized in classical formal logic? — Emmanuel José García, Oct 23 '22 at 15:59
@MauroALLEGRANZA: You first say: "Not in formal logic". Then it seems to suggest that it can, but by adding the axiom "Red is NOT blue". How is this consistent? — Emmanuel José García, Oct 23 '22 at 16:12
Colours are not part of formal logic; we need a "theory of colours" with suitable axioms. — Mauro ALLEGRANZA, Oct 23 '22 at 16:19
@MauroALLEGRANZA: Please, see that I have edited my question to clarify my point since I also agree that there is NOT explicit contradiction but an implicit one, which must be formalized adding a 3th premise. — Emmanuel José García, Oct 23 '22 at 17:27
@MauroALLEGRANZA: According to you, no implicit contradiction can be formalized or your objection only applies to colors? — Emmanuel José García, Oct 23 '22 at 17:38
In entirely red light, everything that reflects light looks red. Ditto blue. You are underspecifying the circumstances to establish a contradiction. When we think hard, we find 'true' isn't **in** the 'facts', it is about the whole situation of evaluating, & never stands separately from that. https://philosophy.stackexchange.com/questions/81655/why-is-a-measured-true-value-true/81664#81664 — CriglCragl, Oct 23 '22 at 18:25
Consider https://en.wikipedia.org/wiki/Magenta & more generally https://en.wikipedia.org/wiki/Spectral_color#Extra-spectral_colors & https://en.wikipedia.org/wiki/Impossible_color Colour perception is a substantial topic in philosophy, used to probe our assumptions & intuitions about how perception works. See eg https://philosophy.stackexchange.com/questions/73253/could-color-be-a-fundamental-thing-about-the-universe/73259#73259 or https://plato.stanford.edu/entries/color/ — CriglCragl, Oct 23 '22 at 18:34
You gave many issues going on at the same time. Are you aware what some call formal logic is actually a different type of logic named Mathematical logic. There are other logic systems. In Philosophy contradiction has a very specific & exact definition. The way average folk use the word contradiction is wrong. You may be one of those. Contradiction means you are comparing TWO propositions where it is impossible for BOTH propositions to be TRUE AND impossible for BOTH propositions to be false simultaneously. Your rose example is horrible because there can be white Rose's correct? — Logikal, Oct 23 '22 at 22:02
You also leave out specific details which would immediately qualify a proposition to be true or false. For instance there are quantifiers such as All or Sone you could have made direct. Did you purposely leave that out because you think the meaning is obvious? Well in rational though do not do that. Be direct and explicit as possible so no one will misinterpret your words or sentences. If you mean all Rose's are red you must include the quantifier and not assume the audience will get it. The Rose's example leaves the possibility BOTH propositions are false. No they are not contradictions. — Logikal, Oct 23 '22 at 22:06
@Logikal: I believe that reality is not contradictory since reality does not make assertions. It is people who contradict themselves by saying things about reality. Suppose I telephone a florist and ask the seller what color of roses are available. The seller tells me that he has red roses. After a few minutes I go back and ask the seller about the color of the roses available and he tells me that he has blue roses. Even if in reality the roses available were white or non-existent, I would find what the seller tells me (implicitly) contradictory. I guess in your case you wouldn't flinch. — Emmanuel José García, Oct 23 '22 at 23:27
@Logikal: are atheists unable to find contradictions between Christianity and Islam? — Emmanuel José García, Oct 23 '22 at 23:46
You cannot make up YOUR OWN definition of a widely know word. You are using some kind of street slang & then say the street slang trumps the academic definition. You cannot leave out the quantifier!!!! You need to ask if all the Rose's are red or are other colors available. There is no contradiction as you state it. I still have no idea what is implicitly contradictory means. What city is that lingo used? Contradictory means one thing to people in academia. You have some ghetto hood version. Aristotle gave us a definition that still applies more than 2,000 years ago. — Logikal, Oct 24 '22 at 01:13
Well, more than 1 share the same definition: https://www.quora.com/Can-two-contradictory-statements-be-false — Emmanuel José García, Oct 24 '22 at 03:03
The trick is "implicit"... formalization means "explicit": thus, if we have to make explicit the contradiction, the only way is to assume the principle that the [primary colors](https://en.wikipedia.org/wiki/Primary_color) are clearly identified and mutually exclusive. See [this post](https://philosophy.stackexchange.com/questions/94314/the-logic-of-analytic-inferences) for a similar discussion. — Mauro ALLEGRANZA, Oct 24 '22 at 08:01
You do not cleanly know the correct terminology. If two propositions are both false the relationship is NOT contradictory. If you were to look up Aristotle & the Square of Opposition you will find where Philosophy gets the single definition of what contradictory means. To YOU if I say all s are p and no s are p, YOU would think they are contradictory because they both cannot be true. The CORRECT relationship is they are CONTRARY to each other. Contrary means two propositions cannot both be true BUT BOTH CAN BE FALSE. You can simply look this up instead of making up your own definition. — Logikal, Oct 24 '22 at 14:07
Even though what you've said has been helpful to me, I think you've missed my point (or don't want to). The premises **"roses are red [all over]"** and **"roses are NOT red [all over]"** cannot both be true and neither can both be false. If roses are white, then "roses are NOT red" is true. There is no explicit contradiction in premises 1 and 2, I understand that. But by adding the third premise **"if roses are blue, then roses are NOT red"**, the premises are reduced to a formal contradiction as I indicated in my original question. I think the real problem here has been pointed out by Mauro. — Emmanuel José García, Oct 30 '22 at 14:09

Paul Ross · Answer 1 · 2022-10-23T18:37:20.977

If I might go a step further, how would you account for this, if roses being red and being blue together imply a contradiction?

E: Being more specific, the idea is that whatever account one might give about why the above is either not red, not blue or not roses, it does not seem to be that what has been presented is a logical impossibility. We might give empirical arguments for why roses cannot be like this, or why colour attributations have certain relations to general sensory boundaries, but by and large it’s recognised that any such theory has to be grounded in a rigorous, evidenced scientific discovery, rather than pre-baked into logic.

From the article:

This means that there is usually a generalization, or principle, that the person is committed to which implies a statement that is inconsistent with (a) particular facts (like the tuna example) or (b) statements implied by other principles that the person also accepts (like the atheist example.)

The problem, as well indicated by this roses example, is that the domain of “particular facts” is difficult to cash out. Unless we have some pre-arranged protocol to determine the boundaries of interpretation, then there’s little sense in presenting a programme of evaluation.

Classical First Order Logic is couched within the terms of mathematically precise definitions and axioms, and as such it makes some sense to talk about logical models and proof theories. Logical truths, we sometimes say, are those that are true in every interpretation.

I think if you want to make progress here, we might usefully key in with questions around Realism; statements about some particular domain of interest have to be True in order for there to be any kind of implied contradiction. But where there is disagreement about realism in a given domain, I don’t think you can expect to be able to determine any kind of contradiction algorithm.

I said in my question that the premises are not open to interpretation. That is, when I say that "roses are red" I mean that they are completely red, not partially red. — Emmanuel José García, Oct 23 '22 at 17:58
@EmmanuelJoséGarcía, This requires a further premise that being red implies the impossibility of being another colour. Not only is that not a logical principle, there are good reasons to think it is not even true. Why are the above roses not red? — Paul Ross, Oct 23 '22 at 18:08
@EmmanuelJoséGarcía, I’ve broadened the scope out a bit, but I do think the example is instructive. I think you’re chasing a leprechaun here. — Paul Ross, Oct 23 '22 at 18:38
I like the point this is making. Just like "There is one person playing the guitar" and "There are two persons playing the guitar" technically is both true as long as there are *at least* two persons playing the guitar, even if it could be 5, 10, or 1000. You just cannot insist on illocutionary force being different when it comes to logical analysis of propositions. — Philip Klöcking, Oct 23 '22 at 18:51
My word, I'm out of practice at this game. I should have carefully said "red all over" or "blue all over". But I notice that the original question says "when I say that "roses are red" I mean that they are completely red, not partially red". I may have thought that covered it. — Ludwig V, Oct 23 '22 at 19:47
Illocutionary force probably covers this. But I was under the impression that illocutionary force couldn't be formalized. Of course, it can be made explicit. But then, what about the illocutionary force of the additional propositions? Do they not have any? — Ludwig V, Oct 23 '22 at 19:48

score 1 · Answer 2 · answered Nov 09 '22 at 21:56

As noted by others, without further assumptions, this is not a contradiction in classical logic.

However, we hold that the two propositions stated by you imply a contradiction. This failing of classical logic to analyse everyday reasoning/natural language is not an isolated case. For example, it is intuitively logical to derive 'there is an animal in the room' from 'there is an elephant in the room' without any further premises. A framework that satisfactorily analyse these aspects of reasoning is inquisitive logic (https://www.dropbox.com/s/3dfw5x04fhiiobx/Inquisitive%20Logic%20preprint.pdf?dl=0 for an introduction). It does not analyse truth values and truth preservation, but information and s ⊨ φ is read as “s supports φ". It can also identify implicit contradictions and most probably distinguishing them by explicit ones. Notably, it can also analyse questions, a type of sentence neglected in logic since they are not truth-apt. Here follows the website of the broader framework called inquisitive semantics, i.e. 'a semantic framework that is based on a formal notion of meaning which, unlike the traditional truth-conditional notion, captures both informative and inquisitive content' https://sites.google.com/site/inquisitivesemantics/Home

score 0 · Answer 3 · answered Oct 23 '22 at 16:18

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In formal logic, the two propositions you cite would have to be treated as distinct, i.e. would have to have different variables in any formula. "Roses are red and roses are blue" would have to be treated as "p & q". "Roses are not red" would have to be "not-p", which does, of course, contradict "Roses are red", but doesn't assert "Roses are blue". So there's no contradiction.

However, "Roses are red" does exclude "Roses are blue" because "Roses are red and blue all over" is meaningless.

This is called "the colour-exclusion problem". The fact that it could not be formalized is one of the main reasons why Wittgenstein abandoned the logical atomism of the Tractatus Logico-Philosophicus.

answered Oct 23 '22 at 16:18

Ludwig V

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It is convenient to distinguish between what is an explicit and implicit contradiction. When you say that there is no contradiction between premises 1 and 2, you mean that there is no explicit contradiction between them...(1/2) – Emmanuel José García Oct 23 '22 at 17:18
I totally agree with that and with your p^q formulation. But, and I see that you agree with that, the "roses are blue" implies that "roses are NOT red", (q => ¬p). According to what I have read, in order to formalize an implicit contradiction it is necessary to add an additional necessarily true premise. In the example of my question it would be: (p ^ q) ^ (q => ¬p). But this last formula can be reduced to a formal contradiction, p ^ ¬p, by modus ponens. (2/2) – Emmanuel José García Oct 23 '22 at 17:18
You are right that this makes the implied contradiction explicit. But don't you need to find a way to restrict the domain of the variables? It isn't obvious to me that the additional "necessarily true" premiss would necessarily be true for any possible substitution. Maybe I've misunderstood something here. – Ludwig V Oct 23 '22 at 19:55

score 0 · Answer 4 · answered Dec 10 '22 at 02:23

You seem to be assuming that:

All roses are either red or blue, not both.
All roses are red.
All roses are blue.

We can then infer by contradiction that there exists no roses.

PROOF

Using a form of natural deduction, we have (where '|' = OR-operator):

On implicit contradictions

4 Answers4