# How is the claim “I am in New York only if I am in America” the same as "If I am in New York, then I am in America?

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

• Already discussed many times on this site; see e.g. what-is-the-difference-between-necessary-and-sufficient as well as what-are-the-truth-tables-for-necessary-and-sufficient – Mauro ALLEGRANZA Apr 5 at 12:13
• Are you perhaps interpreting the word "only" to be qualifying New York? A comma would help to clarify, as would an appropriate pause in the spoken sentence. In other words, do you understand this sentence to be "I am in New York, only if I am in America" or "I am in New York only, if I am in America." If you understood it to be the latter, then I agree that it is illogical. If you understood it to be the former, then hopefully the existing answers have helped you. – Richard II Apr 5 at 15:09
• Technically if you were in New York you might be in a foreign embassy and not in "America" – Mark Schultheiss Apr 5 at 19:11
• @MarkSchultheiss To take your technicality futher, are you still in new york if you are in an embassy? Is yes, then you are also in america (as you are saying the politics are irrelevant). If no, then you are also NOT in america – user34150 Apr 5 at 19:23
• I'm a native English speaker, and to my ear "A only if B" sounds like a stronger version of "A if B". Like, not only are we told "A if B", we are also told "A only if B". So it sounds to me like "A only if B" should mean the same thing as "A if and only if B". However, it is a convention in math that the statement "A only if B" means "if A then B". – littleO Apr 6 at 8:48

"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.

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/

• This way of converting the sentence to logic does not do justice to the sentence's implications in English. The "only" version of the sentence could easily be read as "if and only if" (that is, a two-way implication). – Brilliand Apr 5 at 20:20
• I'd read "I am in New York only if I am in America" as "I cannot be in New York if I am not in America", which is (more or less) the contrapositive of "If I am in New York, then I am in America", and therefore (more or less) logically equivalent to it. The pragmatic content might differ but as a native English speaker I wouldn't consider the "if and only if" reading natural. – Unrelated String Apr 5 at 23:26
• @UnrelatedString I find phrases using "only if", "is a necessary condition", or "is a sufficient condition" to be difficult to understand as a native English speaker. I would ask the speaker for clarification if I were the listener. I only use "if-then" constructions to make sure I am understood by others. In this context I am using forall x to make a default interpretation of the sentence since I can't ask a speaker for clarification. – Frank Hubeny Apr 6 at 8:01
• @Brilliand I don't think the "only if" construction would be viewed as "if and only if" by a native speaker, but it might suggest that to the listener or cause confusion. However, I agree that symbolizing English sentences risks losing some of the native speaker's intentions. The writers of forall x warn about this as well. – Frank Hubeny Apr 6 at 8:11
• @UnrelatedString Exactly. – Eric Duminil Apr 6 at 13:39

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).

• I understand the logic arguments. And I concur that TECHNICALLY the logic of the two would be the same. However, in common usage, the phrasing of the 2nd clause, using "only" would be interpreted by US English speakers as defining oneself as being in New York because they are in America. Which is obviously not true. Regardless, the "only" usage would be widely misunderstood, whereas the "if . . .then" construction would be interpreted correctly. I suspect there is an error somewhere in interpreting the logic of the two to be the same. But said error is beyond me. – Corvus B Apr 5 at 23:43
• @CorvusB, huh. Now that you point it out, I can see superficially similar constructions such as "I will eat only if I am given yoghurt" that would indeed in common usage mean "if I am given youghurt, I will eat". But I can't make my brain see the sentence in the OP with anything other than its intended meaning, and I'm not sure why. – Harry Johnston Apr 5 at 23:55
• @HarryJohnston. Yes - you've brought an excellent example. I am pretty sure the first interpretation most Americans would give the "only" example would be "If I am in America, I am in New York". It might get marked incorrect on a test, but that is more like how people would "hear" the "only" statement. – Corvus B Apr 6 at 0:20
• @CorvusB The sense of "wrongness" you feel is a just a symptom of how formal logic isn't really native to natural language. The suggestion you find in beginner texts that there's a firm, reliable translation between certain English formulations and formal logical equivalents is misleading. – Chris Sunami Apr 8 at 13:15
• @ChrisSunami. Indeed. English is not inherently logical. What you've just said validates my point, yes? – Corvus B Apr 9 at 17:12

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.

• My reading of the OP's first sentence could be paraphrased as "I am in New York, unless I am not in America". It's logically equivalent to your second version, I think, although reads like a statement about a particular person rather than a general statement about everyone. – Brilliand Apr 5 at 20:16
• @Brilliand, agreed, I first interpreted it as something like "When I go to America, I only go to New York." – usul Apr 6 at 7:02

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.

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 think this answer is correct. – Mark Andrews Apr 5 at 21:23

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.

“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.

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