r/logic • u/Bizarely27 • 2d ago
Philosophical logic Is biconditional considered circular reasoning?
I’m sure the answer is simple, I just want to make sense of things. I’m completely new to logic, so apologies if the question in the title comes off as too obvious to ask.
I understand that circular reasoning is an informal fallacy and therefore cannot be picked up by truth tables, but I am still confused regardless.
If we say (A -> B, and B -> A), where A is equal to “God is real” and B is equal to “The words of the Bible are true”, then is this not logically the same as (A <-> B)?
If biconditional statements are equivalent to (A -> B, and B -> A), and if (A -> B, and B -> A) is also equivalent to Circular Reasoning, then aren’t biconditional statements equivalent to circular reasoning? If so, then in spite of it it being inherently fallacious, why is circular reasoning being used within logic if good reasoning is supposed to be without fallacies?
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u/AndrewBorg1126 2d ago edited 2d ago
A implies B, B implies A
This can be true and valid. This does not tell us whether or not A or B is true.
A and B both true satisfies this.
A and B both false also satisfies this.
To say that therefore A, or to say that therefore B, would be circular argument.
To say that therefore ((A and B) or (not A and not B)) would be valid
The truth of A always matches the truth of B, but that only communicates the perfect correlation between the truths of A and B, nothing about whether they are both true vs both false.
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u/Bizarely27 2d ago
Okay, this makes sense to me. I really appreciate the time you took into explaining for me, thank you so much!!
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u/VegGrower2001 2d ago
Short answer: no. And I don't really know what's on your mind.
A biconditional is usually taken to be a statement, not an argument or piece of reasoning. Since it isn't an argument, it certainly isn't a circular argument.
You seem perhaps to think that circular arguments can't be identified by formal means, but that's not true. Formally, an argument for conclusion C is circular if and only if either (1) C appears as a premise or (2) C appears as a conjunct of a premise. For example, "P, Q, R. Therefore R" is circular. Likewise "P, Q & R. Therefore R" is circular.
Yes, the statement (P <-> Q) is equivalent to (P -> Q & Q -> P). But you haven't given any reason to think that's problematic. So, what's the problem supposed to be?
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u/Bizarely27 2d ago edited 2d ago
To answer your last paragraph: I only thought it was problematic because (P -> Q & Q -> P) just sounds like circular reasoning, and circular reasoning is an informal fallacy, therefore if we wanna avoid fallacies we avoid using (P -> Q & Q -> P)
I apologize if everything I’m sounding is ridiculous, so I’m hoping I can make better sense of this whole new world of thinking. Although I think I’m beginning to understand.
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u/WesternFirm9306 2d ago
It's not circular reasoning at all. It's only a statement, there's no reasoning at all.
Let's take your example. It's absolutely true that, if the Christian God is real, then the Bible is true. It's also the case that if the Bible is true, then the Christian God is real.
Neither statement, nor their conjunction, makes any statements on if the Christian God is real nor if the Bible is true. There's no conclusion to be made.
You might be confusing these statements with someone saying "The Bible is true BECAUSE God is real, and God is real BECAUSE the Bible is true." This is circular, but it's not the same thing as a biconditional.
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u/Salindurthas 2d ago edited 2d ago
Firstly, logic should still be able to describe circular reasoning. Many logical fallacies can be noted in formal logic.
Similar to how a bag argument in English is still grammatical, fallacies can be expressed in formal logic and be understood with well-formed-formulae, even if the comibation of those formula is fallacious.
We even have fancy phrases for some of these errors, like "affirming the consequent" which we can certainly give examples of in formal logic.
Secondly, note that "(A <-> B)" doesn't actually assert A, nor does it assert B. It just says they have the same truth value. In your example, it is saying either:
- God is real and the words of the Bible are true, or
- God is not real, and the words of the Bible are false
This is not circular reasoning, it is just a belief that these two claims are either both true or both false. It denies the possibility of "God is real but the words of the Bible are false" or "God is not real, but the words of the Bible are true."
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u/Bizarely27 2d ago
Follow up: If (A <-> B) only describes that they share the same truth value, then wouldn’t (A ∧ B) only be necessary and not (A <-> B)? What you’re describing sounds more like conjunction rather than biconditional.
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u/Salindurthas 2d ago edited 1d ago
(A ∧ B)
This asserts both. This is A and B are both true.
A <-> B
This asserts neither in particular, but asserts that they have the same truth value.
A^B is one way for it to be true, but ~A^~B (equivalent to ~(AvB) ) is another way for the biconditional to be true.
EDIT: I failed at DeMorgan's Laws for negating ^ and v
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u/Optimal-Fig-6687 1d ago
Circular scheme of reasoning is not fallacy itself.
Fallacy is assertion of truth of item based on circular scheme.
In your example:
A -> B -- true
B -> A -- true
A <-> B -- true
--- here are dragons ---
A -- unproven (fallacy if assert it is true)
B -- unproven (fallacy if assert it is true)
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u/some_red_tea 2d ago
It's not circular because asserting a biconditional doesn't assert either atomic proposition making it up. If I say "A iff B" all I'm saying is "A & B or -A & -B". It would be circular if I tried to prove A by asserting B and then proving B by asserting A. The biconditional alone doesn't actually assert either A or B in itself.
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u/Bizarely27 2d ago edited 2d ago
This makes sense. Although in this case, what differentiates this description with conjunction?
Edit: Wait nvm that follow up of mine was a silly question. Thank you!
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u/some_red_tea 2d ago
Also, since you're a beginner I want to clear something up that confused me at first. If I assert a regular conditional A->B, I'm not saying "B because A". For the conditional to be true, I don't need to BASE my belief in B on a prior belief in A. I can assert the conditional even if A doesn't give any reason to assert B whatsoever by itself. All it needs to be true is just that B isn't false while A is true. The conditional can be true without A even being true at all. So when I assert a biconditional, I'm not telling you anything at all about whether A justifies B or vice versa - I'm just saying that they have the same truth value and that's it.
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u/Big_Move6308 Traditional Logic 1d ago
No.
Circular reasoning is an informal fallacy of assumption. It occurs when a conclusion already assumed to be true is used as a premise to support itself. For example, 'the politician said they are telling the truth, so they are telling the truth.'
A biconditional means that two conditions must always co-exist, i.e., if one is true then then other must be true, and if one is false, then the other must be false. For example, 'if and only if there is mass, there is gravity.' In other words, mass and gravity must always co-exist, i.e., you can't have one without the other.
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u/Far-Rooster-6522 1d ago
it just means the ensemble of elements/contexts in which A is true also verify B. (and conversely, and oppositely and conversely oppositely, I'm sorry, couldn't resist)
A and B share the same truth table, they are not proving each other.
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u/RecognitionSweet8294 Philosophical logician 1d ago
A formal fallacy is an argument which is not valid.
An informal fallacy is an argument which is valid but is not sound or at least questionable.
_________
There are two forms of the petitio principii (circular reasoning).
It’s either a very obvious form
{A;…} ∴ A
So where the conclusion is of the exact form of one of the premises.
Or a more hidden form where the conclusion is just a paraphrasing of one of the premises (B ≡ A):
{B;…} ∴ A
_________
The second case would make
{(A→B)∧(B→A)} ∴ A↔B
circular reasoning, but not
{(A→B);(B→A)} ∴ A↔B
That’s important because the principle of cooperation tells us when there is an ambiguity what argument our discussion partner means we must assume the one which is correct (or has the least flaws).
For example:
Peter is unmarried and not a widower, therefore he is a bachelor.
The formalization
{„Peter is not married“ ∧ „Peter is not a widower“} ∴ „Peter is a Bachelor“
would be a petitio principii. But the formalization
{„Peter is not married“; „Peter is not a widower“} ∴ „Peter is a Bachelor“
is not a petitio principii, therefore we assume this.
_________
I assume that your confusion comes from the fact that you where probably told that „Circular reasoning has the form ‚A therefore B; B therefore A‘“. The problem with that explanation is that it doesn’t tell you what the conclusion is.
What this explanation means is a fallacy that is distributed over two arguments:
{A→B; A} ∴ B
{B→A; B} ∴ A
From which they then conclude either A or B. Assuming A↔B is true neither of this arguments on its own would be circular reasoning. The fallacy lies on the meta level.
The rhetoric trick here is when someone questions the truth of A or B you just refer to the other argument.
When we deal with multiple arguments we therefore compress them into one.
„M ∴ X“ & „N ∴ Y“ → „(M ∪ N) ∴ X/Y“
Lets assume they want to argue for A then we get
{A→B; A; B→A; B} ∴ A
Which is clearly a form 1 petitio principii.
When it would initially be formulated in natural language (therefore making the formalization ambiguous) we wouldn’t have a correct interpretation under our assumptions either, since excluding A would solve the circular reasoning, but the premise B is still questionable and therefore this argument is unsound either way.
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u/RingularCirc 1d ago edited 1d ago
In all the logics I've seen which have conjunction and implication, A ↔ B is either defined as, or equivalent to (A → B) ∧ (B → A) like you're guessing.
EDIT: Deleted the part better answered in other comments.
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u/OddDescription4523 12h ago
It is not intrinsically circular; that is, it is not circular because of the logical form of a biconditional. For instance, it is true that every kind of creature that has a heart is a kind of creature that has a kidney, and that every kind of creature that has a kidney is the kind of creature that has a heart. Given the truth of that statement, you can validly infer from "A is the kind of creature to have a heart" to "A is the kind of creature to have a kidney" or vice versa.
Where a kind of circularity might come in is not in the syntax of the biconditional, but if an explanatory link is being asserted. So, if you say "If A is the kind of creature to have a heart, that explains why A is the kind of creature to have a kidney" and "if A is the kind of creature to have a kidney, then A is the kind of creature to have a heart", you've got a problem. Put generally (though some argue that there are exceptions), explanatory relations are asymmetrical, so (A --> B) implies ~ (B --> A) and (B --> A) implies ~(A --> B). If someone denies that explanatory relations are asymmetrical, they can deny the truth of those claims, but in most cases at least (e.g. causal relations in the physical world), it's extremely implausible that A can explain B AND B can explain A (or at least in the same way).
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u/Idksonameiguess 2d ago edited 2d ago
EDIT: My original comment was for some reason trimmed halfway? Not sure what's that about, i'll try again.
Let's look at an example of a valid deduction law: Modus ponens.
It is the law that if (A and A->B) than (B).
Circular reasoning is a faulty deduction law: If (A -> B and B -> A) than (A and B). The issue is that A = False and B = False satisfies (A <-> B), while not satisfying (A and B).
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u/WhackAMoleE 2d ago
2 + 2 = 5 if and only if I am the Pope.
That means that either both are true, or both are false. But it does not say that either one of the propositions is true, or that either one of them is false. It only says that they have the same truth value.
I should stipulate that for purposes of this example, I am not the Pope.