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I am having trouble understanding the notion of logically possible or impossible when it comes to concepts given that all concepts (including mathematics) require a form of language.

For example, we consider a square triangle to be contradictory since a square has four sides and a triangle has three sides. But this hinges upon the notion of a side, triangle and square. The notion of a side or a triangle or a square may easily change in the future in a way that makes this logically possible.

What, then, really is the practical difference between a concept like a square triangle and a human coming alive after he dies. If, as understood, the definition of a human includes a person dying and staying in that state forever, then the notion of revival after death does become logically impossible. One can of course change this definition the same one can change the definition of a square or triangle.

What really is the inherent difference if both are language games for concepts we don’t know to be true. We don’t know what it would mean for a side or “square” or “triangle” to be defined in such a way where a “square triangle” is not a contradiction. Similarly, we don’t know what it would mean for a human to die and come alive.

  • Different concepts of "possibility" here: physical, biological, etc according to the laws involved. Thus, a logical impossibility is a concept violating logical laws. – Mauro ALLEGRANZA Sep 15 '23 at 15:22
  • If we agree that triangle implies not-square, a squared triangle violates NC. – Mauro ALLEGRANZA Sep 15 '23 at 15:23
  • Contradiction applies not to sentences but to propositions they express, it is the same distinction as between words and concepts. That the concept attached to "side" or "square" can change in the future is beside the point, it only means that the same sentence will express a different proposition, not that the original proposition will cease to be contradictory. The difference between mathematical vs colloquial concepts and propositions is not in that, but in the latter being vague already in the present, so that deciding possibility may be frustrated by uncertain boundaries. – Conifold Sep 15 '23 at 17:45
  • So then why isn’t a human who doesn’t revive after he dies not considered logically impossible? Usually, death comes part and parcel with what it means to be human. Given our current understanding of a human, it is impossible. The same way that given our conceptual understanding of a square, it also being a triangle is impossible. So ultimately, it does just depend on the kinds of concepts people refer to with words, yes? –  Sep 15 '23 at 19:00
  • Permanent death does not come "part and parcel" with what it means to be human, folklore and world religions are full of stories about human resurrections. But that is exactly what vague and ambiguous mean, some include mortality into the concept expressed by "human", others do not, and yet others do not have any established view because it usually does not matter anyway. This sort of divergence and fudging are systematically eradicated from mathematical concepts. – Conifold Sep 15 '23 at 20:33
  • You’d be playing the challenging game of decreeing to others their statements don’t mean such and such proposition through your own statements, while it’s far easier for the other side to say they meant what they said. – J Kusin Sep 15 '23 at 21:47
  • @Conifold Yes but the propositions we truly care about the truth for aren’t in mathematics. It’s easy to remove ambiguity from a system when the whole point of the system is to remove it. Nevertheless, my point was that people coming up with stories in regards to humans living after death seem to make its possibility more justified than say a square triangle existing. Practically though, I don’t see a difference. Similar concepts such as immaterial beings causing physical things fit in well here. –  Sep 15 '23 at 22:30
  • Apparently, you do see the difference. Stability of bridges, say, depends on accurate engineering calculations that everybody is on the same page about. So we do care practically about math too. And even somewhat vague concepts can have sufficiently outlined boundaries to make some propositions about them as contradictory as existence of square triangles. Science would not be possible without that. So the answer to your title question is yes, it can. It is true that vagueness is itself vague, and no bright boundary between sharp and vague concepts exists, but it does not mean no differences. – Conifold Sep 15 '23 at 23:58
  • "Whatever Logic is good enough to tell me is worth writing down," said the Tortoise. "So enter it in your notebook, please." – Charles Hudgins Sep 16 '23 at 01:58

4 Answers4

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The obvious answer to your question is that the logical possibility or impossibility of an unambiguous statement can be decided on the current definition of its constituent terms. Given that we currently define a square as a shape with four side and a triangle as one with three, we can definitively state that a square triangle is a logical impossibility. Clearly if you change the meaning of the words square and triangle then you are effectively replacing the original phrase 'square triangle' with a different homophone.

Marco Ocram
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When considering "a square triangle" one should take these words to have their current commonly-understood meanings, based on the premise that we're interested in analysing the underlying meaning, rather than analysing the words themselves. If some of their meanings change, the words as written may no longer be true, but this wouldn't affect the underlying truth that 3 straight lines aligned to be non-parallel and connected at their endpoints cannot also be a shape with 4 sides.

Similarly, if someone says life after death is impossible, this would relate to how biology and the laws of reality work, without much concern for how the definition of the word "death" might change in future. Although rather than "impossible", I'd probably go with "implausible", "unrealistic" or "unfalsifiable and contradicted by empirical evidence in as far as it's possible to disprove it".

If you do care about the words themselves, then it's fairly trivial to haphazardly redefine words such that any given statement would become true or false or impossible or necessary.

NotThatGuy
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Let's start with the concepts of true and false, before moving on to possible and impossible. If we take the sentence, "Sometimes it rains in London", we may say that in the ordinary meaning of these words, this sentence is true. It could come out false if we changed the meanings, e.g. if we understood 'London' to mean a crater on the moon, or 'rains' to mean is covered in lava. It could also become false in future if London were relocated or there was some huge climactic change.

In formal logic, we use the concept of interpretation to distinguish betweem terms that require an assigned meaning, and logical constants, which do not. Whether the sentence A is true depends on how you interpret A. Whether the sentence A ∧ ¬B is true depends on the interpretation of A and B, but not on the interpretation of ∧ or ¬ because those are logical constants and do not need interpretation. Their meaning is fixed by the choice of logic.

We are now able to say what logical impossibility is: a sentence is logically impossible if it is false under all interpretations. So, while A ∧ ¬B might be true, depending on the interpretation of A and B, A ∧ ¬A is always false in classical logic, no matter how we interpret A, so A ∧ ¬A is logically impossible.

This is a fairly narrow and formal account of logical impossibility, and some accounts use the term more broadly to include things that are 'conceptually impossible'. But there are other kinds of impossibility that should not be confused with it. It is distinct from physical impossibility, for example. Whether it is logically impossible for a square to be a triangle might depend on just how you conceive the relationship between logic and mathematics. Whether a person may survive death is not an issue of logic, but it might well be physically impossible.

Bumble
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Formal languages, used for mathematics and logic, are quite different from natural languages like English, Chinese or Swahili. Formal languages are internally consistent, rule-governed systems, and they address only a narrow, tightly defined set of statements. Although you can say much fewer things in a formal language, you can say those things with certainty.

When you talk about the impossibility of the square triangle, what you mean is that "triangle" is defined in one way, and "square" in an incompatible way, and the rules of the (mathematical) language in which we have defined them don't allow for any intersection or overlap.

The discussion about the reanimated dead person, on the other hand, is conducted in English, and while it may violate our understanding of the world, it doesn't break any rules enshrined in our language. Of course, you can translate statements about life and death into a formal logical context, but at that point you'd primarily be examining the relationship between those terms as you've captured them formally, rather than any independent facts-of-the-matter about the real world.

Chris Sunami
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  • A formal language can still be re-interpreted. I’m not sure why the distinction matters –  Sep 15 '23 at 19:04
  • No, if a formal language is re-interpreted, it becomes a different formal language. A formal language is has set, immutable definitions, that's part of what makes it a formal language. // Similarly, if you redefine an object in a formal language, it becomes a different object. // The reason it makes a difference is because natural language doesn't have those constraints and properties. – Chris Sunami Sep 15 '23 at 19:42