2

Here are four of what I would call logical paradoxes:

  1. The Liar
  2. Curry's sentence
  3. Biscuit conditionals
  4. Bertrand Russell's set which contains all sets which do not contain themselves

We can perhaps define a logical paradox as a statement which doesn't seem to make any logical sense and perhaps makes us doubt that language is logical, that humans are logical, or even that logic itself is consistent.

Logical paradoxes are to be distinguished from paradoxes about the physical world, for example Zeno's paradoxes (from Greek philosopher Zeno of Elea c. 490–430 BC).

Are there paradoxes outside the four I just listed which are regarded as logical paradoxes, and why are they?

Thank you for any scholarly references.

Speakpigeon
  • 7,364
  • 1
  • 11
  • 26
  • 1
    See Paradoxes and Contemporary Logic. – Mauro ALLEGRANZA Mar 25 '24 at 07:38
  • 3
    Oxford reference: semantic paradoxes:"Following Ramsey and the Italian mathematician G. Peano (1858–1932) it has been customary to distinguish logical paradoxes that depend upon a notion of reference or truth (semantic notions), such as those of the Liar family, Berry, Richards, etc., from the purely logical paradoxes in which no such notions are involved, such as Russell's paradox, or those of Cantor and Burali-Forti." IEP, Logical Paradoxes has more. – Conifold Mar 25 '24 at 07:44
  • 2
    Wikipedia also has an extensive list, with a category for logical paradoxes. It is somewhat arbitrary as to what exactly constitutes logical paradoxes. I wouldn't say that biscuit conditionals are a paradox at all. If you ask a separate question about those I will endeavour to explain them. – Bumble Mar 25 '24 at 11:58
  • How is a biscuit conditional a paradox? – David Gudeman Mar 25 '24 at 13:46
  • 1
    @Bumble "I wouldn't say that biscuit conditionals are a paradox at all." I agree but Austin noted that they made no logical sense. 2. "If you ask a separate question about those I will endeavour to explain them" Done! I expect you will address the question of their logic . . . – Speakpigeon Mar 25 '24 at 17:04
  • @David Gudeman "How is a biscuit conditional a paradox?" I dunno. Austin thought they made no logical sense. – Speakpigeon Mar 25 '24 at 17:05
  • If the Liar and Curry are logical paradoxes, then the Yablo is also a logical paradox. The standard Yablo paradox can be constructed through a process of "unwinding" from the Liar, and there are variants of the Yablo that comes from the Curry paradox. But none of these paradoxes are strictly "logical" if by logic you mean a formal system. They all involve a truth predicate that is not strictly needed in a formal system. If you say they are logical paradoxes, then maybe you'd agree the knower and knowability paradoxes are logical paradoxes as well (see next comment...) – ayylien Mar 25 '24 at 17:13
  • since we'd have similar motivations to include them in our formal system (because they are present in natural language and we intend to model natural language) and because they follow similar derivations rules as the truth predicate (i.e. P(φ) ⊢ φ and if ⊢φ then ⊢P(φ)) – ayylien Mar 25 '24 at 17:16
  • @ayylien "if by logic you mean a formal system" No, by "logic" I mean "logic". - 2. "none of these paradoxes are strictly "logical"" I explained exactly what I meant by "logical paradox". - 3. "They all involve a truth predicate that is not strictly needed in a formal system." Nothing is strictly needed in any formal system unless we use it to mean something, so it depends on what you want to mean. - 4. "the knower and knowability paradoxes " Do they make us doubt that language is logical, that humans are logical, or that logic itself is consistent? – Speakpigeon Mar 25 '24 at 17:42
  • @Speakpigeon I doubt we would make any headway if we go down the route of defining what you mean by logic, but maybe we don't need to. Your reply to 3 is fair, and in that sense (what we want something to mean in a formal system), the knower and knowability paradoxes are similar enough to the liar and curry. I'll say this, and you can judge if they count as a "logical" paradox or not: The liar and curry arises when our language is semantically closed (it can refer to its own truth). Knower and knowability paradoxes arises when our language can refer to what is known and knowable... – ayylien Mar 25 '24 at 18:22
  • When you have a truth predicate (a predicate that applies to sentences, and tells you if the sentences are true or false), you'd want to say "If φ is provable, then 'φ is true' is provable" which is if ⊢ φ, then ⊢T(φ). Similarly, for the knowner paradox, "If φ is provable, then 'φ is known' is provable" (also called the principle of necessitation of knowledge), which is if ⊢ φ, then ⊢K(φ). T or K are predicates that we would include in our formal system if we want to model reasoning about truth and what's known. They follow similar derivation rules that give rise to their respective paradoxes. – ayylien Mar 25 '24 at 18:28
  • Let us continue this discussion in chat. – ayylien Mar 25 '24 at 18:44

3 Answers3

1

This is a really nice article which might inform your thoughts on that: Logic and Ontology

Overall, we can thus distinguish four notions of logic:

(L1) the study of artificial formal languages

(L2) the study of formally valid inferences and logical consequence

(L3) the study of logical truths

(L4) the study of the general features, or form, of judgements

Basically, it seems like prior to your question is needing to make a deep distinction between strictly “logical” paradoxes vs. non-logical ones.

We can perhaps define a logical paradox as a statement which doesn't seem to make any logical sense and perhaps makes us doubt that language is logical, that humans are logical, or even that logic itself is consistent.

I am pretty sure you are staunchly residing in the L4 view of logic (above), which the articles links to Kant.

It’s an interesting question, since trying to resolve it through some kind of formalization may be deeply missing the point - your question is about logic as the science of judgment, which in a way is far above the science of formal systems.

Julius Hamilton
  • 1,559
  • 4
  • 29
  • "L1 L2 L3 L4" Same thing . . . There is only one logic, namely human logic. L1, L2, L3, L4 are each only considering logic under one angle. A remake of the parable The blind men and the elephant but with logic in the role of the elephant. – Speakpigeon Mar 25 '24 at 17:26
  • 1
    Do you think a different intelligent organism could have a different logic, or only the same as the human one? – Julius Hamilton Mar 25 '24 at 17:53
  • "a different intelligent organism could have a different logic" I don't know and nobody human does, but it seems plausible to me that cows and pigs and just as logical as we are, and given our shared biology, they probably have exactly the same logic. Some Alien species, on a different planet, who knows, although the same causes produce the same effects. – Speakpigeon Mar 26 '24 at 10:57
1

Logical paradoxes are to be distinguished from paradoxes about the physical world, for example Zeno's paradoxes (from Greek philosopher Zeno of Elea c. 490–430 BC).

Zeno's paradoxes are not necessarily about the physical world, as they may instead be construed as being about "how does one pass from information about finite initial segments of an infinite sequence to information regarding the whole 'completed' sequence?", and there's a rather precise sense in which this is not possibly achievable by 'constructive'/'finitary' reasoning - the kind the Greeks and most everyone else was used to until around the 19th century - alone, so it may well be one example to

Are there paradoxes outside the four I just listed which are regarded as logical paradoxes, and why are they?

besides, regarding

We can perhaps define a logical paradox as a statement which doesn't seem to make any logical sense and perhaps makes us doubt that language is logical, that humans are logical, or even that logic itself is consistent.

it's well known by now that there are many different logics, some of which are not consistent by design

edit: OP answers that

"Zeno's paradoxes are not necessarily about the physical world" You may interpret them as you please but they are ostensibly about the physical world. - 2 "it's well known by now that there are many different logics" No, it is not "well-known" since it is not true. There is only one logic that we know of, human logic. Anything else cannot therefore by regarded as logic. Mathematicians call some of their theories "logic", but they are not.

being completely oblivious to the fact that extensions of classical logic (modal ones, arithmetic/set theories, higher-order-type theories, and whatnot), restrictions of it (positive, constructive/intuitionistic, paraconsistent), plus not-directly comparable ones (non-monotonic) have been around and will continue to be around for quite a while, and claims 'human logic' is in fact The One and Only True Logic: fucktard

ac15
  • 828
  • 10
  • 1
    "Zeno's paradoxes are not necessarily about the physical world" You may interpret them as you please but they are ostensibly about the physical world. - 2 "it's well known by now that there are many different logics" No, it is not "well-known" since it is not true. There is only one logic that we know of, human logic. Anything else cannot therefore by regarded as logic. Mathematicians call some of their theories "logic", but they are not. – Speakpigeon Mar 25 '24 at 17:17
  • No need to engage with him on the last point; he's well known for being vocal about his opinions but refuses to elaborate on anything (even though he wouldn't admit that, perhaps because he's doing his best). – ayylien Mar 25 '24 at 18:19
  • 1
    @ayylien really had no idea, very new here, thanks – ac15 Mar 25 '24 at 18:33
0

Logical paradoxes are ones that come from an application of the rules of a logical system. They are typically found in informal logic, because formal logical systems are designed to prevent the possibility of paradox. For instance, "I am a liar" is a logical paradox, because our naive notions of logical rules allow us to evaluate it as both true and false simultaneously. However, you wouldn't be able to state a sentence like that in most formal logical systems (because of deliberate exclusions).

Notably, Bertrand Russell attempted to create a formal logical system robust enough to encompass all mathematics as a subset, but Godel proved, using a logical paradox, that no such system could be completely consistent.

Chris Sunami
  • 29,852
  • 2
  • 49
  • 101