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It makes absolutely zero sense to me.
It would make sense if "I am in America" is the antecedent and the consequent is the former.
Even though it wouldn't be sound, it would make logical sense.
I hope someone could explain it in a way someone would to a beginner in logic.
Thanks
This is an example of the confusion inherent in switching between a natural language like English, and a formal language of logic.
The formulation
X only if Y
is rare in spoken English, but perfectly grammatical, and it typically has a logical meaning equivalent to
If X then Y
Both statements are saying you can't ever have X without Y. However, at first glance it looks closer to
If Y then X
which is entirely different. This represents how English has many different ways of saying the same thing (with incidental connotations and subtleties of meaning that are completely stripped out when you translate to a formal language).
Consider the sentence:
If I am in America then I am in New York.
One could make the antecedent, "I am in America", true by being in Chicago. But then the consequent, "I am in New York", would be false. So this conditional would be false unless we are given other information, such as travel plans, in addition to knowing that I am in America.
However, consider this sentence:
If I am in New York then I am in America.
Now whenever the antecedent, "I am in New York", is true, then so is the consequent, "I am in America". I don't need any additional information for that conditional to be true.
It would be similar for the following sentence:
I am in New York only if I am in America.
Here we are given that "I am in New York" and conclude that "I am in America". Except for English style this means the same as the previous sentence.
The authors of forall x provide a similar example using Paris and France in section "5.4 Condititional". They also provide this symbolization rule:
A sentence can be symbolized as A → B if it can be paraphrased in English as ‘If A, then B’ or ‘A only if B’.
P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2018 bis. http://forallx.openlogicproject.org/
"A only if B" and "if A, then B" mean the same.
The truth-condition for "if A, then B" excludes the case when A is True and B is False.
"A only if B" means that we cannot have A without B.
The two are equivalent.
The contrapositive of both statements is :
If I am not in America, then I cannot be in New York.
A conditional statement is logically equivalent to its contrapositive. It means both your statements are equivalent since they have the same contrapositive.
I see two interpretations of the sentence here. They mean logically different things. In both cases "only" is interpreted as "must be true and cannot be false".
I am in New York (only if I am in America).
If I am in New York, it can only be true that I am in America.
New York => America.
This is the interpretation everyone else is responding to. It is logically true.
I can be in (New York only) if I am in America.
If I am in America, then it can only be true that I am in New York.
America => New York.
This one is not logically true, you could be in Iowa.
These claims have distinctly different connotations. From a pure formal-logic perspective, the "X only if Y" is equivalent to "Y or not X" which is the same as "X implies Y", which is the same as "if X then Y". However, natural language carries more information than its simple-minded reduction to predicate logic.
The second formulation "If I am in NY then I am in USA" sounds like a simple statement of a containment relationship: it implies that "I" am an unbound variable and informs the listener that NY is within the USA.
The first formulation connotes something about the speaker's mental state: he entertains the possibility (perhaps even likelihood) of being outside the USA in a place confusingly-similar to NY.
One way of analyzing the statements is to look at a truth table. Let's make the following definitions:
A := "I am in New York"
B := "I am in America".
X := "I am in New York only if I am in America"
Y := "If I am in New York, then I am in America"
If both A and B are true, then X is true. We can write that as X(TT) = T. We have X(TF) = F (If you are in New York but not in America, then the statement "I am in New York only if I am in America" must be false). X(FT) = T and X(FF) = T; X makes a statement about what has to be true when you're in New York, so if you're not in New York, then X isn't telling you anything so it can't be proven wrong.
If you analyze Y, you'll find that all the values are the same:
X(TT) = Y(TT) = T
X(TF) = Y(TF) = F
X(FT) = Y(FT) = T
X(FF) = Y(FF) = T
Since no matter the truth values of A and B, X has the same truth value as Y, X and Y are equivalent; if you have two statements such that it's not possible for one to be true and the other false, then the two statements are saying essentially the same thing.
One thing to keep in mind is that in Formal Logic, statements of the form "If S1 then S2" are considered true any time S1 is false; that is, "If S1 then S2" is interpreted as meaning "Whenever S1 is true, S2 is also true". Because of this, "If S1 then S2" is equivalent to "Either S1 is false, or S2 is true" (if S1 is false, then the statement is automatically true, because it doesn't say anything about the situation of S1 being true). And "S1 only if S2 " is also equivalent to "Either S1 is false, or S2 is true".
To understand this more intuitively, I think it's helpful to use formatting help and rephrase this a little, while keeping the logic the same.
Start with this:
“I am in New York ONLY IF I am in America”
That means there is no option to be in New York without being in America. The reason why there is no other way is that "only"--it is there to indicate there are no other ways to be in New York and some other country. That's the work it does in this sentence.
Now consider the second sentence you gave:
"If I am in New York, then I am in America"
Let's rephrase that without changing the logic at all:
"If I am in New York, I MUST BE in America"
What both of these are saying is that being in New York necessarily entails being in America. You can't be in New York and be any other country. In other words, you have no other option. Which is just what I showed with the first sentence.
It makes more sense if you use "only if" in conditions e.g. I'll come with you only if you kiss me. A kiss is necessary for whoever I is to accompany whoever you is. In logic then I'm accompanying you means you kissed me (the rather romantic condition was fulfilled) i.e. IF I'm accompanying you THEN you kissed me. IF you didn't kiss me THEN I'm not in your company.
Draw a Venn diagram of all things that are in NY, all things that are in the US, draw in points representing people that are in neither, in one but not the other, and so on. Light will dawn upon you.
Nelson Lande's Classical Logic and Its Rabbit-Holes (2013) expatiates on the difference between "P only if Q" vs. "P if Q" the best! I forgo blockquotes, for the sake of readability.
Be prepared in this section to run up against your first conceptual speed bump. (If you aren't fully alert, don't read on; take a nap instead.) Consider the following sentence:
(1) Bobo is a widow only if Bobo is a woman.
Suppose that we dispense with propositional symbols just this once, for the sake of clarity. Interpret 'WIDOW to mean 'Bobo is a widow', and interpret 'WOMAN' to mean 'Bobo is a woman'. Translating (1) into Loglish--our halfway house (once again) between purely logical notation and English--yields the following:
(2) WIDOW only if WOMAN
Translating (1) and (2) into full-blown logical notation is a bit tricky. No doubt your first inclination may be to translate it as follows:
(3) WOMAN → WIDOW
But (3) can't be right; i.e., it can't possibly capture what (1) and (2) are claiming. Think of what (3) says: 'If Bobo is a woman then Bobo is a widow'. (1) and (2) are truths: if you know nothing about Bobo and I tell you that Bobo is a widow, you immediately know that Bobo is a woman. Why? Because being a woman is a necessary condition of being a widow: you can't be a widow unless you're a woman. (Once again: a man in a comparable position is a widower, not a widow.) (3), however, is a falsehood: if, once again, you know nothing about Bobo and I tell you that Bobo is a woman, you know nothing about her marital status. The claim that if she's a woman then she's a widow is simply false. So if (1) and (2) are truths and (3) is a falsehood, then (3) must be a mistranslation of (1) and (2). So then how do you translate (1) and (2)?
It turns out that, your intuitions to the contrary, 'Bobo is a widow only if Bobo is a woman' has the very same meaning as 'If Bobo is a widow then Bobo is a woman'--in which case the correct translation of (1) and (2) is the converse (the very opposite) of (3):
(4) WIDOW → WOMAN
In the following paragraph, I shall write very loosely--so that you'll have an easier time grasping what's at issue. In the paragraph following the following paragraph, I'll express the same thought without the looseness.
The loose version first. All of the following statements have the exact same meaning; i.e., it should strike you that (6) has the same meaning as (5), that (7) has the same meaning as (6), etc.
(5) P only if Q.
(6) You have 'P' only if you have 'Q'.
(7) You have to have 'Q' in order to have 'P'.
(8) You can't have 'P' without 'Q'.
(9) It's not the case that you have 'P' without 'Q'; i.e.,--(P∧ -Q).
(10) If you have 'P' then you have 'Q'.
(11) If 'P' is true then 'Q' is true.
(12) If P then Q.
(13) P → Q.
So (5) has the same meaning as (12); i.e., 'P only if Q' has the same meaning as 'If P then Q'. Therefore because we translate 'If P then Q' as 'P→ Q', we translate 'P only if Q' as 'P→ Q'.
The non-loose version next. All of the following statements, (14) | through (19), have the exact same meaning; i.e., it should strike you that (15) has the same meaning as (14), that (16) has the same meaning as (15), etc.
(14) P only if Q.
(15) The truth of 'Q' is necessary for the truth of 'P'.
(16) It's not the case that 'P' is true and that 'Q' is not true; i.e., -(P∧ -Q).
(17) If 'P' is true then 'Q' is true.
(18) If P then Q.
(19) P → Q.
Once again, we see that 'P only if Q' has the same meaning as 'If P then Q'. And (once again) because we translate 'If P then Q' as 'P-→ Q', it follows that we translate 'P only if Q' as 'P-→ Q'.
If you're still unconvinced then it's time to beat a dead horse.... Notice first that the sentence 'If P then Q' amounts to the following two claims:
(20) If 'P' is true then 'Q' is true; i.e., the truth of 'P' guarantees (i.e., is sufficient for) the truth of 'Q'.
(21) If 'Q' is not true then 'P' is not true; i.e., the truth of 'Q' is necessary for the truth of 'P'.
Notice next that the sentence 'P only if Q' amounts to the following two claims:
(22) If 'Q' is not true then 'P' is not true; i.e., the truth of 'Q' is necessary for the truth of 'P'.
(23) If 'P' is true then 'Q' is true; i.e., the truth of 'P' guarantees (i.e., is sufficient for) the truth of 'Q'.
Of course, (20) simply is (23), and (21) simply is (22). It should come as no surprise, therefore, that 'If P then Q' and 'P only if Q' have the same meaning. Accordingly, because the correct translation of 'If P then Q' is 'P→ Q', the correct translation of 'P only if Q' must also be 'P → Q'.
The obvious question: Why do our intuitions have to be dragged kicking and screaming before they will acknowledge that 'If P then Q' and 'P only if Q' have the exact same meaning?
The non-obvious answer: In the course of using conditionals in everyday conversation, we presume that the speaker (or writer) believes that there is a connection of some sort between the state of affairs to which the antecedent of the conditional refers and the state of the affairs to which the consequent refers. For example, the connection might be of the causal sort, of the definitional sort, or of the logical sort--and the context will normally make clear exactly which sort of conditional the speaker (or writer) is using.
Suppose that a dog-walker reprimands you: "If you pull Old Fido's tail one more time then he'll bite you." Your presumption is that the dog-walker believes that there is a causal connection between the former state of affairs--your pulling Old Fido's tail--and the latter state of affairs--Old Fido's biting you--such that the former will be the cause of the latter. But here's the crucial point: Your presumption, as well as the dog-walker's belief, are distinct from the meaning of the conditional sentence itself. The meaning of the conditional sentence itself is what it shares with all typical conditionals. Because a reference to causality doesn't characterize the other sorts of conditionals, a reference to causality can be no part of its own meaning.
Its meaning--what it shares (once again) with all typical conditionals--is precisely this: It isn't the case that its antecedent is true and its consequent isn't true. Once you abandon the belief that there is a connection of some sort between the state of affairs to which the antecedent of the conditional refers and the state of affairs to which the consequent refers, then you should have no difficulty seeing that the meaning of a conditional consists exclusively in its not being the case that its antecedent is true and its consequent is not true.
Think of a conjunction. Suppose that on the first day of the semester, your instructor had walked into your class and said, "This is a course in formal logic and I shall now be taking roll." You would have found that entirely unsurprising. Suppose instead that on the first day of the semester your instructor had walked into your class and said, "This is a course in formal logic, and Lenin suffered his first stroke in May 1922." You would have found this more than a bit odd. Suppose that a short while later in the same class your instructor had then gone on to say, "There will be a quiz every other week, and Alexandria, Egypt, is named after Alexander the Great." At this point you would have begun to feel a bit uneasy and you would have looked around at the other students. Suppose finally that somewhat later, your instructor had then gone on to say, "The final exam will count for one-third of your course grade, and Euclid is credited with the proof that the square root of 2 is an irrational number." My guess is that at that point you and your fellow classmates would have started tiptoeing toward the exit. The collective bubble over all of your heads would have read: "What does the one thing have to do with the other? What's the connection between the first half of each of this instructor's sentences and the second half?" Or (if on that day you had known the terminology) your collective bubble would have read: "In each of the preceding conjunctions the two conjuncts have no relation to one another. Why, then, is this instructor conjoining such conjuncts?"
Your unease, however, would have concerned psychology (your instructor's) and not logic as such. Your confusion concerned not the meaning of your instructor's statements but rather your instructor's reasons for uttering them. The point is that you understood each of the sentences and you could have determined their truth-values without too much difficulty. Consider the sentence 'This is a course in formal logic, and Lenin suffered his first stroke in May 1922'. Had you known that each conjunct is true, you would have known in a jiffy that the entire conjunction is true. Whether the sentence is true or false is one thing; whether it's appropriate or inappropriate (i.e., bizarre) to utter it is another thing altogether. In logic our concern is exclusively with truth and falsehood, rather than with appropriateness and inappropriateness.
It simply doesn't matter--at least where the truth-value of the sentence is concerned--whether there's any connection between the left conjunct and the right conjunct in the conjunction 'This is a course in formal logic, and Lenin suffered his first stroke in May 1922'. By the same token, it simply doesn't matter--again, at least where the
truth-value of the sentence is concerned--whether there's any connection between the antecedent and the consequent in the conditional 'If Fido wrote the Iliad then the Moon is made of pink fluff.
Once you divest yourself of the view that there has to be a connection, it should become somewhat easier to see that there's no difference either meaning-wise or truth-value-wise between the sentence 'If Fido wrote the Iliad then the Moon is made of pink fluff and the sentence 'Fido wrote the Iliad only if the Moon is made of pink fluff. Each of these sentences has the exact same meaning as the sentence 'It is not the case both that Fido wrote the Iliad and that the Moon is not made of pink fluff. Now, since this sentence is true--Fido did not write the Iliad--each of the two former sentences is true as well. And once you see that, it should become easy-ish to see that there's no difference truth-value-wise between the sentence 'If you work hard next semester then you'll pass' and the sentence 'You'll work hard next semester only if you'll pass'. They both mean that it's not the case that you'll work hard and yet that you won't pass.
