Zinoviev was born in the middle of a family of 11 to a carpenter father and peasant mother in the village of Pakhtino, in the Kostroma region of central Russia. Academic success took him to the Institute of Philosophy, Literature and History at Moscow University in 1939. However, his outspoken criticism of Stalin led to his expulsion from the institute and from the Komsomol, the Communist youth league, and finally to his arrest. He escaped being sent to the camps, and re-emerged in 1940 in the Red Army, first as a sergeant in a tank regiment, then as a fighter pilot. His distinguished war record - he saw action at Stalingrad, and took part in the capture of Berlin and the liberation of Prague - redeemed him in the eyes of the authorities, and in 1946 he was allowed to resume his studies. He gained a doctorate from Moscow State University in 1951, writing his thesis on Karl Marx's Das Kapital.
This marked the start of what promised to be a brilliant academic career. A post at Moscow University was quickly followed by membership of the Soviet Academy of Sciences and ultimately a chair in logic in the department of philosophy. Zinoviev was a prolific contributor to his field. He published more than 100 articles on logic, many of which appeared in the west. His Philosophical Problems of Many-Valued Logic (1960), a landmark book, won him invitations to international conferences, but although Zinoviev's name appeared occasionally on lists of delegates he never actually travelled outside the Soviet Union. Although initially he was not an open critic of Nikita Khrushchev or Brezhnev, his department proved a haven for more outspoken colleagues and he resisted pressure from party apparatchiks to provide information on their activities. However, as Brezhnev's grip on power increased and crackdowns on political and academic freedoms gathered strength, Zinoviev became more involved in the human rights movement during the 1970s. He resigned from the board of a leading philosophical journal and in turn was isolated from colleagues.