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I am not asking for a defense of or pro/con of the existence of an omnipotent (or multiple omni-x) being, or for the existence of square-circles or any other similar thing. These arguments are well documented within this site, for example Is the definition of God consistent?

My question concerns the terminology associated with a logically inconsistent definition or an argument flowing from it, and whether or not assigning a truth value to the conclusion of such argument is a named logical fallacy.v

The basic laws of logic indicate:

  • a valid argument is one such that if all propositions are true, then the conclusion is true.
  • if any proposition to a valid argument is false, then we cannot determine whether the conclusion is true or false. It may remain true even though a proposition is false.
  • with an invalid argument, it doesn't matter whether the propositions are true or false - we can never determine the truth of the conclusion.

Questions

Given the above, what happens if I define something in a way which is logically inconsistent and use that definition in an initial premise/proposition for an argument?

An illogical/incoherent thing is not able to be addressed by logic, other than perhaps to assign it to the set of objects which are incoherent. So how does an incoherent definition flow in an argument?

  1. If a definition used in a premise/axiom/proposition of a logical argument is illogical/incoherent/paradoxical, then do we say that the proposition itself is incoherent or paradoxical as a result?

  2. Continuing on to the logical argument that flows out of such a proposition,**

    • Do we say that such an argument is also incoherent or paradoxical because one of its propositions is?
    • Or is it more correct to say that such an argument is simply invalid? Which is to say we cannot establish the validity of such an argument (it is outside the realm of logic to determine it's validity)
    • Or something else?
  3. Is there a name for the fallacy of attempting to determine the logical truth value for the conclusion of such an argument?


Note: One may also be able to discuss this in mathematical terms, with the concept of infinity, division by zero and similar concepts which can be used to show impossible things (i.e. 2 + 2 = 5, etc.) by using improper or illogical definitions at the start of the proof

Note 2: I don't think this requires going to a formal system of symbolic logic - if it does, please help me understand why

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An argument that contains an inconsistent definition is guilty of "equivocation" because it fails to use the same term with the same meaning throughout. This is a type of "informal fallacy" because disputants could in principle disagree about whether what is happening is a material equivocation or meaningless (where the change in definition between statements/premises/conclusion does not matter).

Whether this has an impact of the validity of a deductive argument, the soundness of a deductive argument, or the strength of other arguments will depend on what happens when/if you resolve the equivocation.

  • Does this mean that proofs by contradiction equivocate when they reason about things like rational number with square 2? – Conifold Feb 9 '17 at 2:01
  • I'm not seeing the connection between proof by contradiction and equivocation. Maybe you could spell it our more? – virmaior Feb 9 '17 at 2:37
  • If inconsistent definition is guilty of equivocation then "rational number with square 2" is guilty of equivocation. This seems to mean that Euclid's proof of irrationality of the square root of 2 equivocates when it defines such a number and then derives a contradiction by reasoning about it. If the equivocation is in using rationality in some parts of the proof and square 2 in others (although, frankly, they mix in this case) then any conjunctive definition can be said to equivocate. – Conifold Feb 9 '17 at 2:52
  • I don't think that's an accurate description of what's happening there. Or may be to add something, an equivocation is when you accomplish your conclusion by changing the definition of the term. The proof of the existence of irraitonal numbers does not depend on that. – virmaior Feb 9 '17 at 3:30
  • We may disagree on that, but why does it matter if it is an accurate description of what Euclid does, or whether the proof depends on it? The point is that it can be (and often is) described in this manner, along with many other contradiction proofs. Where then is the definition of the term changed? It seems the same (inconsistent) conjunction is used throughout. – Conifold Feb 9 '17 at 3:59
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There is a related fallacy of "defining into existence" when implicitly defined entities are illicitly declared existent, versions of the ontological argument are often accused of defining God into existence. Kant clearly expressed the issue in his thesis that "existence is not a predicate". Even for ideal objects in mathematics it must be proved from axioms that objects fulfilling the defining conditions exists, the object is then said to be "well-defined". For example, Euclid defines equilateral triangle as a triangle with equal sides, but he gives a straightedge and compass construction of it before using it in demonstrations (in modern texts the two steps are often combined into a single "theorem-definition").

But in itself giving contradictory definitions with non-existent referents, and reasoning about them, is not a fallacy, although it does pose an old philosophical puzzle. Quine in On What There Is gave it a nickname that stuck:"Nonbeing must in some sense be, otherwise what is it that there is not? This tangled doctrine might be nicknamed Plato's beard". Plato mused over the nature of fleeting "sensibles", and famously assigned to them less than being, the becoming. This was a major point of difference between him and Aristotle, who saw becoming as a form of being, and argued against its dismissal by Parmenides and Plato. But objects nonexistent due to inconsistency, like round squares, pose the same logical problem: if round square is not what is it that there is not?

One solution is due to Meinong: objects in logic may not exist but only "subsist", this is Meinong's version of becoming, but it also covers all sorts of fictions and absurdities. If you take this route you have to give up existential generalization, P(a) does not imply existence of x with property P, and allow contradictory sentences, P(a) and ¬P(a) may both hold if a is non-existent. If you give an argument with subsistent objects in the premises you may conclude all sorts of things about them, but it will not get you very much since none of them have to exist. To move from subsistence to existence would be exactly to commit the "defining into existence" fallacy.

A more mainstream version of dealing with Plato's beard, one favored by Quine himself, is due to Russell. It involves eliminating defined objects from premises by using descriptions, before any logical analysis of arguments. Russell's way of talking about say round squares is to use a variable x with predicates R(x) and S(x), rather than a proper name with dubious existential status. The rest depends on how exactly you want to use round squares in premises. If you want to make any existential claim about them, e.g. "some round squares are green" ∃x(R(x)∧S(x)∧G(x)), then any premise involving it will come out as false, and any argument based on it will be unsound, even if valid. But something like "all round squares are round" ∀x(R(x)∧S(x) → R(x)) is not just true but even a logical tautology. For that matter, even "all round squares are green" ∀x(R(x)∧S(x) → G(x)) is a tautology, if we are assuming that R and S contradict each other.

Plato's beard has an interesting application in mathematics. In proofs by contradiction negation of the intended conclusion is treated as an additional premise, and an auxiliary valid, but unsound argument is given using it. The contradiction in the conclusion of the auxiliary argument is then interpreted as entailing the intended conclusion. But at the onset of the auxiliary argument we do invoke inconsistent objects, implicitly or explicitly. For instance, Euclid's proof of the irrationality of square root of 2 ostensibly involves defining a rational number with square 2, and then reasoning about it. This non-existent number can be interpreted in Meinongian or in Russellian manner.

See more in SEP's Nonexistent Objects, and Negative Existential Beliefs.

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I wouldn't call this a logical fallacy, because the logical reasoning very well may be correct. A logical fallacy is when there is something wrong with the logical form of the argument, not its descriptive contents.

The operative thing here is the Principle of Explosion, which says that from a logical contradiction, every proposition can be derived.

  • Thank you for the Principle of Explosion, I'm with you there. On the point of logical reasoning, though, can one really proceed with a logically sound argument if one of the core definitions you are using is incoherent? Isn't any step which has to refer back to that definition suspect? If the definition is by default unexplainable, how can you show that a step later in the logic that attempts to use any of its characteristics is valid? – LightCC Dec 31 '15 at 0:48
  • It depends on how the definition is being used. Consider this: Let $1=2$. Then $1=1^2=2^2=4$. Thus we get $0=1-1=4-1=3$. Thus $3=0$. The conclusion here is false, but it's false because we started off with something that is wrong. This isn't really an inconsistent definition, but an inconsistent definition can be used in exactly the same kind of way, creating a deduction that is valid but not sound. It's also possible that the argument is logically invalid... it just depends on the argument. – Stella Biderman Dec 31 '15 at 6:27
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The use of logically inconsistent definitions is the fallacy of four terms. The fallacy is also called equivocation, as virmaior pointed out. Generally, the use of four or more terms causes a break in the line of reasoning, because the incompatible terms prevent the premisses from linking together through a common term.

0

Square circles aren't possible in Euclidean Geometry - so they are logically inconsistent there; but they are possible in other geometries:

Equip the plane with the the L1 norm; and draw the circle; and then stand back and look at it - it's a square circle.

Paradoxes come in many forms - zen koans, physical singularities, logical absurdities, reductios and so on.

Sometimes they say: this far, and no further; at other times they are pregnant with thought.

Here is another example: division by zero; not 0/5, which is well defined within the formal context of arithmetic; but 0/0, which is not - it's undefined, it could be any number - but this turns out to be unuseful and unusable.

But 0/0, when considered as dx/dy, is not - it's fruitful, being the formal concept of the calculus; but why do I say this - after all it's not how historically calculus was invented.

Mathematicians like closure: where all operations and moves are well defined; a point of exception, instead is a challenge; and often proves to be the site of a new idea.

Thus 0/0, can be considered a site of exception, which when pushed through generates a new site, of a different order - not the arithmetic, but the analytic; this may, on the face of it, seem strange, or bizarre - but consider another site of exception, by way of comparison: the square root of -1, whose proper answer is i, the imaginary - it generates geometry: the argand diagram; it's very name signals the differance of degree that pushing past this site has provoked.

Proper points of exceptions in mathematics (as opposed to mere or illusionary such points), may be considered as sites, where several concepts fuse, in an alchemical act of the mathematical imagination, and reveal a hitherto hidden dimension of depth in the being that is mathematics.

  • Mozibur, while this may be true, it completely ignores my question. I'm not asking for a reformulation of an inconsistent definition, I'm asking how one defines an argument like that. What is it called, etc. For example, if you define a "a square circle is both a square and circle while existing only in a plane under Euclidean Geometry" (or whatever you have to do to ensure it is incoherent) and then proceed with a proof based on such an incoherent definition. – LightCC Dec 31 '15 at 0:18
  • @lightcc: it doesn't answer your question because there is no answer to it, in the terms you've put it - the point of the example is to show how thinking like that happens; there is no definitive name - perhaps a leap of the imagination. – Mozibur Ullah Dec 31 '15 at 0:28
  • The people who discovered this sense of the 'square circle', weren't setting out to look for it - for example: ie they didn't say, here's a logical inconsistency - how can we make sense of it. – Mozibur Ullah Dec 31 '15 at 0:32
  • Having said all this, I appreciate the clarification of your question. – Mozibur Ullah Dec 31 '15 at 0:36
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It could be considered Proof by Contradiction, namely proof that the proposition is false because its truth would imply or infer a contradiction.

  • That doesn't look right, based on the Wikipedia page, it appears to be the other way around - you show that when the proposition is false, it creates a contradiction, which implies it must be true. My question is what if the definition of the proposition is incoherent/contradictory, in and of itself - like 2+2=5, or better yet, 2+2=purple. – LightCC Dec 31 '15 at 5:01
  • 1
    Proof by contradiction is not a fallacy. – user2953 Dec 31 '15 at 5:50

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