PostgreSQL - select only when specific multiple apperance in column

How about this:

   SELECT DISTINCT person
     FROM tableName
    WHERE ingredient IN('carrot', 'salt')
 GROUP BY person, recipe
   HAVING SUM(CASE WHEN ingredient = 'carrot' THEN 1 ELSE -1 END) <= 0 
          AND
          COUNT(DISTINCT ingredient) > 1;

I admit I don't have much experience working with PostgreSql, but the query seems to give the results you require in this SQL Fiddle (credit to @JohnWoo for providing the one to begin from).

I have updated the answer; before it returned users with only salts in some of their recipes as legit ones. The second HAVING clause filters out such cases.

UPDATE: The previous query returned all the owners which have at least one recipe which follows the rule ("for each added carrot add salt also"). But you (seem to) actually need the ones with all the recipes following the rule. So the query looks like...

SELECT DISTINCT person 
  FROM tableName
 WHERE person NOT IN (   
     SELECT person
       FROM tableName
      WHERE ingredient IN('carrot', 'salt')
   GROUP BY person, recipe
     HAVING SUM(CASE WHEN ingredient = 'carrot' THEN 1 ELSE -1 END) > 0
 );

SQL Fiddle to play with.

Try:

SELECT person from
(SELECT person, recipe, COUNT(DISTINCT ingredient) ingredients
 FROM tableName
 WHERE ingredient IN ('salt', 'carrot')
 GROUP BY person, recipe
 HAVING MAX(CASE WHEN ingredient = 'carrot' THEN 1 END) = 1) p
group by person
HAVING MIN(ingredients) = 2

This seems a variation of the relational division problem.

A double-nested NOT EXISTS solution:

SELECT DISTINCT person
FROM tableName AS t
WHERE NOT EXISTS
      ( SELECT *
        FROM tableName AS chosen
        WHERE chosen.ingredient = 'carrot'
          AND chosen.person = t.person
          AND NOT EXISTS
              ( SELECT *
                FROM tableName AS required
                WHERE required.ingredient = 'salt'
                  AND required.recipe = chosen.recipe
                  AND required.person = chosen.person
              ) 
      ) ;

And a JOIN:

SELECT DISTINCT 
    t.person
FROM 
        tableName AS t
    LEFT JOIN
            tableName AS chosen
        LEFT JOIN
            tableName AS required
          ON  required.ingredient = 'salt'
          AND required.recipe = chosen.recipe
          AND required.person = chosen.person
      ON  chosen.ingredient = 'carrot'
      AND chosen.person = t.person
      AND required.ingredient IS NULL
WHERE
    chosen.ingredient IS NULL ;