Mryntu

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 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).

"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.

See necessary and sufficient.

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.

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.

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.

LOL - The two statements are not equivalent.

You could be in New York - Lazio - Italy

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".