What is the logical operator for but?

An alternative way of conveying the same information would be to say "I am fine and he has flu.".

Often, the word but is used in English to mean and, especially when there is some contrast or conflict between the statements being combined. To determine the logical form of a statement you must think about what the statement means, rather than just translating word by word into symbols.

This seems like an exercise in semantics. I cannot think of a logical operator which fits other than $\land$.

However, if we define the predicate $\operatorname{Fine}(x)$ which holds if and only if $x$ is fine, then we can assume "has the flu" is $\lnot\operatorname{Fine}(x)$.

In which case we can write the sentence:

$$\operatorname{Fine}(\textbf{me})\land\lnot\operatorname{Fine}(\textbf{him})$$

If you want to distinguish $\operatorname{Flu}(x)$ from simply $\lnot\operatorname{Fine}(x)$, then we are reduced to: $$\operatorname{Fine}(\textbf{me})\land\operatorname{Flu}(\textbf{him})$$

This is indeed an exercise in semantics as mentioned by @Asaf. My interpretation would be: $$fine(\text{me}) \land flue(\text{him})$$ where $fine, flue$ are predicates and me, him are constants.

Wiktionary lists various semantic meanings of 'but'. Here is an attempt to translate them into logic:

• But as preposition:
  Everyone but Father left early: $\forall{X} [X \ne \text{father} \implies left(X)]$
  I like everything but that: $\forall{X} [X \ne \text{that} \implies i\_like(X)]$

• But as adverb:
  Since that day, my mood has changed but a little: $changed(\text{mood}) \land small(\text{change})$

• But as conjuction:
  I have no choice but to leave: $\lnot choice (\text{me}) \land leave(\text{me})$
  I am not rich but (I am) poor: $\lnot rich (\text{me}) \land poor(\text{me})$
  (or if we asume $ poor \implies \lnot rich$, just $ poor(\text{me})$ )
  She is very old but still attractive: $old(\text{she}) \land attractive(\text{she})$.

It is instructive to translate these logical clauses into Prolog and to play with them. Prolog and Natural Language Semantics is a good paper about translating English into Prolog.