Neither!

As a general rule, most natural language operators do not have a formal equivalent. In your case, it is even more unlikely, since "until" involves time, while logical implication is blind to time and change.

First Proposed Definition

In propositional terms, is X until Y equivalent to if Y then not X ?

Equivalence is too strong: "If it rains, then the ground is wet" does not imply "The ground is not wet, until it rains".

The left-to-right-direction makes less trouble: in many cases "X until Y" seems to assert that Y is (not yet) the case, which makes the implication true. A necessary condition is obviously not enough for a definition, but see below in Temporal Logic.

Second Proposed Definition

or is it just equivalent to if Y then [(] X or not X [)]?

As above, we can immediately exclude equivalence, since the proposed logical implication is a tautology. That means this is true for any X and Y, and you would be forced to make X until Y a tautology. (You won't be healthy, until you eat ice cream). This refutes both directions!

Why does it not work?

The "until" operator is not truth-functional. There are 16 truth-functional 2-ary operators, and none of them will represent "until" appropriately. One could go through each case and find a counterexample, which seems tedious. But there are several clues why it is not truth-functional: (1) "X until X" seems to be meaningless, not true or false. (2) Nesting "until" does not produce meaningful sentences ("My light is on until (my light is on until the sun rises)" ???). (3) If X can happen only in an instance, then "X until Y" makes no sense. All these cases suggest that the structures of X and Y matter, not only their truth values.

Temporal Logic

Temporal/Modal Logic offers a possibility to add the until operator.

Pragmatics, imperatives and the boss

Uttering the same sentence in different circumstances can mean different things. In your case, without knowing the circumstances, I can see several possible meanings (1) A quite literal one: "Work from home, but as soon as you recover, work in the office" (2) With different intonation it could also mean: "Please work from home even if you are sick" (3) Not much thought was put into the wording, simply meaning: "You are sick, work at home, we will talk when you recover".

Was the conversation in english? It also seems to be a bit strange requesting from a sick person to work...

_{Get well soon! Also: way to find an interesting philosophical question in everyday life :)}

Neither!

As a general rule, most natural language operators do not have a formal equivalent. In your case, it is even more unlikely, since "until" involves time, while logical implication is blind to time and change.

First Proposed Definition

In propositional terms, is X until Y equivalent to if Y then not X ?

Equivalence is too strong: "If it rains, then the ground is wet" does not imply "The ground is not wet, until it rains".

The left-to-right-direction makes less trouble: in many cases "X until Y" seems to assert that Y is not (yet) the case, which makes the implication true. A necessary condition is obviously not enough for a definition, but see below in Temporal Logic.

Second Proposed Definition

or is it just equivalent to if Y then [(] X or not X [)]?

As above, we can immediately exclude equivalence, since the proposed logical implication is a tautology. That means this is true for any X and Y, and you would be forced to make X until Y a tautology. (You won't be healthy, until you eat ice cream). This refutes both directions!

Why does it not work?

The "until" operator is not truth-functional. There are 16 truth-functional 2-ary operators, and none of them will represent "until" appropriately. One could go through each case and find a counterexample, which seems tedious. But there are several clues why it is not truth-functional: (1) "X until X" seems to be meaningless, not true or false. (2) Nesting "until" does not produce meaningful sentences ("My light is on until (my light is on until the sun rises)" ???). (3) If X can happen only in an instance, then "X until Y" makes no sense. All these cases suggest that the structures of X and Y matter, not only their truth values.

Temporal Logic

Temporal/Modal Logic offers a possibility to add the until operator.

Pragmatics, imperatives and the boss

Uttering the same sentence in different circumstances can mean different things. In your case, without knowing the circumstances, I can see several possible meanings (1) A quite literal one: "Work from home, but as soon as you recover, work in the office" (2) With different intonation it could also mean: "Please work from home even if you are sick" (3) Not much thought was put into the wording, simply meaning: "You are sick, work at home, we will talk when you recover".

Was the conversation in english? It also seems to be a bit strange requesting from a sick person to work...

_{Get well soon! Also: way to find an interesting philosophical question in everyday life :)}

Neither!

As a general rule, most natural language operators do not have a formal equivalent. In your case, it is even more unlikely, since "until" involves time, while logical implication is blind to time and change.

First Proposed Definition

In propositional terms, is X until Y equivalent to if Y then not X ?

Equivalence is too strong: "If it rains, then the ground is wet" does not imply "The ground is not wet, until it rains".

The left-to-right-direction makes less trouble: in many cases "X until Y" seems to assert that Y is (not yet) the case, which makes the implication true. A necessary condition is obviously not enough for a definition, but see below in Temporal Logic.

Second Proposed Definition

or is it just equivalent to if Y then X or not X?

As above, we can immediately exclude equivalence, since the proposed logical implication is a tautology. That means this is true for any X and Y, and you would be forced to make X until Y a tautology. (You won't be healthy, until you eat ice cream). This refutes both directions!

Why does it not work?

The "until" operator is not truth-functional. There are 16 truth-functional 2-ary operators, and none of them will represent "until" appropriately. One could go through each case and find a counterexample, which seems tedious. But there are several clues why it is not truth-functional: (1) "X until X" seems to be meaningless, not true or false. (2) Nesting "until" does not produce meaningful sentences ("My light is on until (my light is on until the sun rises)" ???). (3) If X can happen only in an instance, then "X until Y" makes no sense. All these cases suggest that the structures of X and Y matter, not only their truth values.

Temporal Logic

Temporal/Modal Logic offers a possibility to add the until operator.

Pragmatics, imperatives and the boss

Uttering the same sentence in different circumstances can mean different things. In your case, without knowing the circumstances, I can see several possible meanings (1) A quite literal one: "Work from home, but as soon as you recover, work in the office" (2) With different intonation it could also mean: "Please work from home even if you are sick" (3) Not much thought was put into the wording, simply meaning: "You are sick, work at home, we will talk when you recover".

Was the conversation in english? It also seems to be a bit strange requesting from a sick person to work...

_{Get well soon! Also: way to find an interesting philosophical question in everyday life :)}

Neither!

As a general rule, most natural language operators do not have a formal equivalent. In your case, it is even more unlikely, since "until" involves time, while logical implication is blind to time and change.

First Proposed Definition

In propositional terms, is X until Y equivalent to if Y then not X ?

Equivalence is too strong: "If it rains, then the ground is wet" does not imply "The ground is not wet, until it rains".

The left-to-right-direction makes less trouble: in many cases "X until Y" seems to assert that Y is (not yet) the case, which makes the implication true. A necessary condition is obviously not enough for a definition, but see below in Temporal Logic.

Second Proposed Definition

or is it just equivalent to if Y then [(] X or not X [)]?

As above, we can immediately exclude equivalence, since the proposed logical implication is a tautology. That means this is true for any X and Y, and you would be forced to make X until Y a tautology. (You won't be healthy, until you eat ice cream). This refutes both directions!

Why does it not work?

The "until" operator is not truth-functional. There are 16 truth-functional 2-ary operators, and none of them will represent "until" appropriately. One could go through each case and find a counterexample, which seems tedious. But there are several clues why it is not truth-functional: (1) "X until X" seems to be meaningless, not true or false. (2) Nesting "until" does not produce meaningful sentences ("My light is on until (my light is on until the sun rises)" ???). (3) If X can happen only in an instance, then "X until Y" makes no sense. All these cases suggest that the structures of X and Y matter, not only their truth values.

Temporal Logic

Temporal/Modal Logic offers a possibility to add the until operator.

Pragmatics, imperatives and the boss

Uttering the same sentence in different circumstances can mean different things. In your case, without knowing the circumstances, I can see several possible meanings (1) A quite literal one: "Work from home, but as soon as you recover, work in the office" (2) With different intonation it could also mean: "Please work from home even if you are sick" (3) Not much thought was put into the wording, simply meaning: "You are sick, work at home, we will talk when you recover".

Was the conversation in english? It also seems to be a bit strange requesting from a sick person to work...

_{Get well soon! Also: way to find an interesting philosophical question in everyday life :)}