Sylvan was born Francis Richard Routley in Levin, New Zealand, and his early work is cited with this surname. He studied at Victoria University, and then Princeton University, before taking positions successively at several Australian institutions, including the University of Sydney. From 1971 until his death in Bali, Indonesia, he was a fellow at the Research School of Social Sciences at the Australian National University in Canberra.

Sylvan was married to the philosopher/environmentalist Val Routley (later, Val Plumwood), which whom he occasionally worked. After his divorce from Plumwood, he married Louise Sylvan (née Merlin) in 1983 and adopted the last name Sylvan (an English word meaning "of the forest") to reflect his commitment to environmentalism.

Sylvan was instrumental in the development and study of relevant logic. In 1972, Sylvan (in a paper co-authored with Plumwood) proposed a semantics for certain relevant logics that had been developed by American philosophers Nuel Belnap and Alan Ross Anderson. His logic work helped make ANU a center for the study of non-classical logic in general. His work had particular influence for Graham Priest, a well-known proponent of non-classical logic; Sylvan and Priest edited a well-regarded volume on the topic.

Sylvan's studies ranged over a variety of topics in logic and the philosophy of logic. He wrote important papers on free logic, general modal logic, and natural deduction systems. However, much of his most important work in logic was dedicated to relevant logic, for which he authored numerous papers (both technical and expository).

From early in his career (and for many years after), Sylvan defended a sophisticated Meinong-inspired ontology (which he called "noneism"), first presented in his 1966 paper, "Some Things Do Not Exist." After several more papers in the 1970s, the theory was given a book-length treatment in 1980, "Exploring Meinong's Jungle and Beyond." The view ... also defended in recent years by Priest ... utilizes a modal theory including "impossible worlds" to deal with supposed objects, like the "round square." Sylvan's formulation is logically consistent, and avoids certain paradoxes associated with Meinong's original ontology; although, like many Meinongian views, it faces criticism due to its presumed ontological implausibility.

Outside of logic and metaphysics, Sylvan was a proponent of deep ecology in the study of environmental ethics. Beginning in the 1970s, he published several notable articles and books on the topic, and he co-authored the 1994 book "The Greening of Ethics," with David Bennett. From his work in environmental ethics, Sylvan took an interest in anarchism, contributing an often-cited entry on the subject to A Companion to Contemporary Political Philosophy.