Muddle the mint in the bottom of a cocktail shaker with the Simple Syrup. Add ice to the shaker, then add the gin, rum, triple sec and vodka and shake vigorously. Strain into 4 rocks glasses filled with ice. Top each with some lemon-lime soda and garnish with a lime wedge.
Combine the sugar and 1 cup water in a small saucepan and bring to a boil. Cook, stirring, until the sugar dissolves. Remove from the heat and let cool completely. Keep covered in the refrigerator up to 1 month.