In a medium saucepan, combine the blackberries, sugar, and 1 cup of water.
Heat over medium heat, gently mashing the blackberries with a fork or potato masher until they release their juices and the mixture starts to simmer.
Add the sage leaves to the saucepan and simmer for an additional 5 minutes, allowing the sage to infuse its flavor into the syrup.
Remove from heat and strain the blackberry-sage syrup through a fine-mesh sieve into a bowl, pressing the solids to extract as much liquid as possible. Discard the solids.
In a large pitcher, combine the strained syrup with the freshly squeezed lemon juice and remaining 3 cups of water. Stir well to combine.
Taste and adjust sweetness or tartness by adding more sugar or lemon juice as desired. Chill in the refrigerator for at least 30 minutes before serving.
Serve over ice cubes in glasses, and garnish each glass with a lemon slice and a sprig of fresh sage.