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I am having a hard time understanding this section in Wikipedia's article on Logical biconditionals:

Colloquial usage

One unambiguous way of stating a biconditional in plain English is of the form "b if a and a if b". Another is "a if and only if b". Slightly more formally, one could say "b implies a and a implies b". The plain English "if'" may sometimes be used as a biconditional. One must weigh context heavily.

For example, "I'll buy you a new wallet if you need one" may be meant as a biconditional, since the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional). However, "it is cloudy if it is raining" is not meant as a biconditional, since it can be cloudy while not raining.

My question is how can the plain English "if'" sometimes be used as a biconditional? I'm OK with the word "biconditional." I don't understand how the reader is to know the "speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional)" especially how this amounts to "(as in a conditional)".

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    I think this question belongs on english.stackexchange.com Jul 27 '12 at 12:49
  • Then again, that's more about expression. Jul 27 '12 at 13:16
  • I agree about migrating it to stack exchange. There isn't actually a philosophical question being asked. The question is entirely about how the English language lumps conditional and biconditional into the same word, relying on context.
    – Cort Ammon
    Feb 1 '15 at 17:21
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First thing, the title of your question seems to imply that your difficulty lies in translating the formal (bi-)conditional into plain English, however the body seems to imply that the problem is in translating plain English into formal logic.

Translating formal logic into plain English shouldn't be difficult and many sources will give one a list of accepted readings. Translating in the other direction (plain English to formal language) is a different story.

The biggest problem with translating any natural language as it is used in day to day life into any formal language is that natural languages tend to be ambiguous without context (this is in part why machine translation of natural language is so difficult).

Formal languages try to strip away as much context as possible in order to eliminate the ambiguity of natural language. Another example of this is how the English words, ‘and’, ‘but’, and ‘yet’ all typically get translates into the conjunction, as the attitudes or beliefs of the speaker carried with these words is unnecessary cruft as far as formal logic is concerned.

Another difficulty is that while formal logics usually only employ a single conditional construction with a clear and distinct meaning, English has several conditional constructions (counter factual conditional for instance) which, again, require context and carry with then implicit information.

So the long and short of it is that if you are looking for an easy/mechanical method for translating plain English into formal language then you're out of luck. Such translations require that you fully understand what you are translating and being able to fill in the gaps of what is not be said.

Quine's “Methods of Logic” [1] has a chapter devoted to the subject of translating plain English to formal logic which may help hone one's skill in this respect.

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  • Thank you for the Quine reference, it has completely cleared up my confusion. Aug 2 '12 at 18:01
  • On the bottom of page 54, can this be paraphrased to mean "only-if" means "then"? Aug 4 '12 at 18:38
  • Phrases such as ‘if A then B’, ‘$B, if A’, ‘A only if B’, etc. are all readings for the conditional sentence ‘A –> B’. Pay special attention to the order in which the variables, ‘$\phi$’ and ‘$\psi$, occur. Aug 6 '12 at 3:06
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My question is how can the plain English "if'" sometimes be used as a biconditional? I'm OK with the word "biconditional." I don't understand how the reader is to know the "speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional)" especially how this amounts to "(as in a conditional)".

This is really more of an English question than a logic question, so you may wish to run it by the English Language and Usage Stackexchange, but since you asked it here:

The reader knows this by context. Let's pretend, for the moment, that the speaker does not intend the "if" to represent a biconditional in the sentence "I'll buy you a new wallet if you need one." This would mean: "I will buy you a wallet if you need one, and I may buy you a wallet even if you don't need one." It seems terribly unlikely that this would be what the speaker intended; if it were the speaker's intent, there are many better ways to phrase the sentence. So, the more parsimonious reading would be to read it as a biconditional, meaning "I will buy you a wallet if and only if you need one; if you do not need a wallet, I will not buy one for you."

EDIT: Since the OP wants more detail, I will try to answer the additional question added.

Please explain how "if and only if" is used in daily Plain English and how this differs from its mathematical/logical meaning?

The phrase "if and only if" is used rarely in daily English; it tends to be used only for emphasis, or to avoid an otherwise unavoidable ambiguity.

When it is used, it has the same meaning as the mathematical/logical usage.

To summarize: "if and only if" is always a biconditional, both in mathematical/logical usage and plain English usage; "if" can be a biconditional in plain English usage, depending on context, but not in mathematical/logical usage.

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After reading the article in question, I can see why you're confused. The example's just flat-out wrong.

Converting any language into the syntax of formal logic is a messy business. Colloquial statements often carry implied statements along for the ride ("You really want to go out? It's raining outside!"), off-the-cuff idioms that resist quantification (I don't not like her), and assumed context (That's what she said!). The problems with the phrase you picked out there is that

a) it relies on you knowing unstated context about the speaker by any reading of the example, and b) the only implied statement the speaker makes as given is "I'll buy you a wallet only if you need one."

It's a simple conditional, not a biconditional, and you're absolutely right to be confused. I think the point it was trying to convey was, as someone else posted, that we use "if and only if" to make a point about consequences (You'll go to the party if, and ONLY if, you finish your homework, young lady!). It could be the case that someone drops the "only if" from their utterance, but I'd normally expect a vocal emphasis in that case (You'll go to the party IF you finish your homework, young lady!)

The Wikipedia example lacked this sense of emphasis entirely, and failed to supply enough relevant context. I wish I had a better answer for you than just "wikipedia's wrong", but sometimes that's just the case.

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I am not sure I understand the question:

You seem to be asking "How do I say this in plain English" and then give two examples of how to do so.

I can suggest others (I'll buy you a new wallet, but only if you need one.) but I can't suggest something answers your question until I know exactly what critera you place on a valid answer...

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  • "...buying the wallet whether or not the wallet is needed" cannot be the "exclusive-if," but then the wiki example has the parenthetical statement: "(as in a conditional)" this is where I get confused. What is the parenthetical "conditional" referring to? Jul 19 '12 at 16:36
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Perhaps you could use the word "unless", but you would need to rephrase your statement a little.

As in your example:

->I will buy you a wallet if (and only if) you need one

->I won't buy you a wallet unless you need one

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  • "...buying the wallet whether or not the wallet is needed" cannot be the "exclusive-if," but then the wiki example has the parenthetical statement: "(as in a conditional)" this is where I get confused. What is the parenthetical "conditional" referring to? Jul 19 '12 at 16:37
  • @Former_Math_Addict I understand this as how people make use of "if" in coloquial english... as it says there, when someone says "I'm only doing X if Y", in some circunstances this is understood as what is called a biconditinal in logic formalization. The coloquial value of "if" would comprise both "if" and "if and only if" of logic; that's why logicians need symbols to make a differentiation - "natural" language can be very ambiguous and relies mostly on context for the definition of meaning.
    – Tames
    Jul 19 '12 at 16:45
  • Y is the condition, which does not depend on you buying a wallet. Jul 27 '12 at 12:54
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"If X (you need a wallet) then Y (i'll buy you one)."

X is a condition under which you'd do Y.

Y does not depend on X. (Maybe you'd buy one for his birthday or something.)

"I'll *only* buy you a wallet if you need one."

In this case the other options are excluded, and buying depends on needing. Needing abstractly depends on buying for if a new one is bought then it would not be needed anymore.

Clearer still would be: "I'll buy you a wallet only if you need one."

Alas, logic dictates one can be rude about that by not buying one, so you'll have to use "if and only if" (aka "iff"):

I'll buy you a new wallet if and only if you need one.

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BASIC UNDERSTANDING

Bi-Conditional

  • Bi-Conditional, "p if and only if q"; p <-> q; for both (pq) ∧ (qp). The propositions are equivalent.

  • For q has attribute "if and only if", therefore for p must has attribute "if and only if"

    • "p (if and only if q)" or "q (if and only if p)"
  • On plain English "if and only if" must be replaced with "ONLY IF". It's for the antecedent ("if and only if" q)

  • For the consequent (p) must be replaced with another term that has the same assertion as "ONLY IF" but without adopt "ONLY IF" to avoid confusion. For this, we need understand the nature of assertion that has the sameness with "ONLY IF". This can be asserted by understanding about "Consciousness".

"Consciousness"

Consciousness has abilities to:

  • Aware of something

    • It's perceive differences
  • Feeling something

    • It has relation with emotions

Further, it asserts:

  • Aware (Empirical)

    It has relation with necessity for aware of something.

    • ONLY IF something (specific alternate) then ...

    • (necessity) there MUST be something (else)

    Only if We perceive, then, there must be

  • Emotional

    Emotion has degree of strength. Emotion has threshold. Sensing something that lead our emotional to a higher level, until it (emotional level) reach the threshold and produce necessity, "I INSIST". See emotional demands.

    • ONLY IF our emotional level is sifted to or beyond threshold then ...

    • there is strong force to us to do something. At this state, there is "necessity" that "We MUST" do something.

    Only if I needed, then, I must

Therefore,

  • On plain English, for the consequent p has the same conditional as q (asserted with "ONLY IF"), by adopting "IS A MUST".

IMPLEMENTATION

Logically:

Stating a bi-conditional in plain English can be provided by:

  • Emotionally: An "act emotionally" is equal to "MUST ACT". And a conditional is equal to "ONLY IF"
  • (ONLY IF) P then Q (IS A MUST)

    • Q (IS A MUST) (ONLY IF) P
  • (ONLY IF) Q then P (IS A MUST)

    • P (IS A MUST) (ONLY IF) Q
  • Aware (Empirical): ONLY IF there is something then there MUST be something else
  • (ONLY IF) P (is perceived) then Q (IS A MUST) (be perceived)

    • Q (IS A MUST) (be perceived) (ONLY IF) P (is perceived)
  • (ONLY IF) Q (is perceived) then P (IS A MUST) (be perceived)

    • P (IS A MUST) (be perceived) (ONLY IF) Q (is perceived)

Practically:

Stating a bi-conditional in plain English can be provided by:

Emotionally

To avoid ambiguous, "I'll buy you a new wallet if you need one" must be replaced with:

P = "need a wallet"

Q = "buy a wallet"

  1. (ONLY IF) (P) then (Q) (IS A MUST) =

    (ONLY IF) (you need a wallet) then (I buy a wallet) (IS A MUST) = "ONLY IF you need it then I MUST buy a new wallet for you"

    • (Q) (IS A MUST) (ONLY IF) (P) =

    • (I buy a wallet) (IS A MUST), (ONLY IF) (you need a wallet) = ""I MUST buy you a new wallet ONLY IF you need one""

  2. (ONLY IF) (Q) then (P) (IS A MUST) =

    (ONLY IF) (I buy a wallet) then (you need a wallet) (IS A MUST) = "*ONLY IF I buy a new wallet then (it indicates) a new wallet MUST BE needed by you *"

    • (P) (IS A MUST) (ONLY IF) (Q) =

    • (You need a wallet) (IS A MUST), (ONLY IF) (i buy a new wallet) = ""You need a wallet MUST BE concluded, ONLY IF (because) I buy you a new wallet""

Aware (Empirical)

If there are perceiving functions, then there are perceiving existence.

P = "functions"

Q = "existence"

  1. (ONLY IF) (P) (is perceived) then (Q) (IS A MUST) (be perceived) =

    (ONLY IF) functions (are perceived) then existence (IS A MUST) (be perceived) = "ONLY IF there are perceiving functions, then there MUST BE perceiving existence"

    • (Q) (IS A MUST) (be perceived) (ONLY IF) (P) (is perceived) =

    • existence (IS A MUST) (be perceived) (ONLY IF) functions (are perceived) = "there MUST be perceiving existence, ONLY IF we are perceiving functions"

  2. (ONLY IF) (Q) (is perceived) then (P) (IS A MUST) (be perceived) =

    (ONLY IF) existence (is perceived) then functions (IS A MUST) (be perceived) = "ONLY IF there is perceiving existence, then there MUST BE perceiving functions"

    • (P) (IS A MUST) (be perceived) (ONLY IF) (Q) (is perceived) =

    • functions (IS A MUST) (be perceived) (ONLY IF) existence (are perceived) = "there MUST be perceiving functions, ONLY IF we are perceiving existence"

If they were dead, then they were not functioning in this reality at all

P = "dead"

Q = "no function"

  1. (ONLY IF) (P) (is perceived) then (Q) (IS A MUST) (be perceived) =

    (ONLY IF) dead (is perceived) then not functioning in this reality at all (IS A MUST) (be perceived) = "ONLY IF there is dead, then there MUST BE no functions in this reality at all on it"

    • (Q) (IS A MUST) (be perceived) (ONLY IF) (P) (is perceived) =

    • there is no functions in this reality at all (IS A MUST) (be perceived) (ONLY IF) dead (is perceived) = "there is no functions in this reality at all MUST BE concluded to them, ONLY IF they were dead"

  2. (ONLY IF) (Q) (is perceived) then (P) (IS A MUST) (be perceived) =

    (ONLY IF) no functions in this reality at all (are perceived) then dead (IS A MUST) (be perceived) = "ONLY IF there is no functions in this reality at all, then it MUST BE concluded that they were dead"

    • (P) (IS A MUST) (be perceived) (ONLY IF) (Q) (is perceived) =

    • dead (IS A MUST) (be perceived) (ONLY IF) there is no functions in this reality at all (is perceived) = "dead MUST BE concluded, ONLY IF there is no functions in this reality at all can be perceived from them"

Stating Bi-Conditional in plain English

  1. (ONLY IF) (there is the need) (Stating a Bi-Conditional in plain English), then, (there MUST BE) (there is implementing a Bi-Conditional using "OnlyIF-Must")

    • (there is implementing a Bi-Conditional using "OnlyIF-Must") (MUST BE provided) (ONLY IF) (there is the need) (Stating a Bi-Conditional in plain English)
  2. (ONLY IF) (there is implementing a Bi-Conditional using "OnlyIF-Must"), then (there MUST BE) (there is the need, indication) (Stating a Bi-Conditional in plain English)

    • (there is the need) (Stating a Bi-Conditional in plain English) (MUST BE concluded) (ONLY IF) (there is implementing a Bi-Conditional using "OnlyIF-Must")

On Math

P = (x + 1) = 2

Q = (x = 1)

  1. (ONLY IF) (x = 1) then (x + 1 = 2) (IS A MUST) =

    • (x = 1) (IS A MUST) (ONLY IF) (x + 1 = 2)
  2. (ONLY IF) (x + 1 = 2) then (x = 1) (IS A MUST) =

    • (x + 1 = 2) (IS A MUST) (ONLY IF) (x = 1) =

CONCLUSIONS

The points are:

  • Stating Bi-Conditional using plain English must be provided by

    • Replacing conditional "IF" with "ONLY IF",

    • Further (this is crucial), consequence "THEN" must be replaced with "MUST".

  • Forming using "OnlyIF-Must", may be replaced with "one and only-insist", or "the only reasonable alternate-the only consequence", and many more similar to these.

    But id' rather choose "OnlyIF-MUST" because

    • Each of two terms "ONLY" & "MUST" is a single term, which is easier to be captured (perceived) quickly as an understanding

    • The two terms "ONLY" & "MUST" are most popular, mostly used on our everyday life, therefore those terms are quickly to be applied on conversations.

  • There is no differentiation in between math and plain English in this case, both assert the same essential understanding (same possible directions of assertion) but with different notations (since math using logical symbol, and plain English using language).

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A and B must be either both true or both false.

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