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Does the premises

  1. The roses are red.
  2. 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.

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  • Not in formal logic. Commented Oct 23, 2022 at 15:45
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    You need the additonal "axiom": red is not blue. Commented Oct 23, 2022 at 16:03
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    Colours are not part of formal logic; we need a "theory of colours" with suitable axioms. Commented Oct 23, 2022 at 16:19
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    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 are clearly identified and mutually exclusive. See this post for a similar discussion. Commented Oct 24, 2022 at 8:01
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    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
    Commented Oct 24, 2022 at 14:07

4 Answers 4

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

enter image description here

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.

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  • 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. Commented Oct 23, 2022 at 17:58
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    @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
    Commented Oct 23, 2022 at 18:08
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    @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
    Commented Oct 23, 2022 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
    Commented Oct 23, 2022 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
    Commented Oct 23, 2022 at 19:47
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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

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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.

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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) Commented Oct 23, 2022 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) Commented Oct 23, 2022 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
    Commented Oct 23, 2022 at 19:55
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You seem to be assuming that:

  1. All roses are either red or blue, not both.
  2. All roses are red.
  3. 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):

enter image description here enter image description here

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