The shadow Foreign Secretary, who is on the party's centre right, promised that a Tory government led by him would mend Britain's "broken society". He also mapped out an agenda of low taxation, family values, small government, strong defence and Euroscepticism.
Dr Fox said tax cuts were "one of the elements of a prosperity agenda". He said the party needed to "make the case for lower taxes in the first half of a parliament, before we come to specific measures in the second."
The 43-year-old former GP said the Tories had spent too much time talking to themselves. "My aim in the next few weeks is to get the whole debate focused on what we will do in the country and not what we will do for the party," he said. Symbolically, he launched his campaign away from the Westminster village at the Hillside Clubhouse in Camden, north London, which helps people with mental health problems develop their skills.
Dr Fox, a former party co-chairman, claims the support of many Tory MPs. Michael Howard, the outgoing leader, is believed to favour David Cameron, the 38-year-old shadow Education Secretary, but is also an admirer of Dr Fox.
"In many parts of Britain today, social cohesion seems to be evaporating," said Dr Fox. "The alarming decline in secure family life, of good order in our schools and on our streets is increasingly destabilising our society. There is no clear vision for dealing with this."
"Britain is in danger of having a 'lost generation' of young people - men in particular - failed by family and education, and now venting their frustrations on society as a whole."
David Davis, the front-runner and shadow Home Secretary, highlighted his tax-cutting agenda during a visit to Scotland. He said: "Only a low-tax, light-regulation economy can provide the resources for good health care and education, roads and police, because those are the economies that generate sustainable wealth."
Mr Davis attacked Gordon Brown's "seven deadly sins" - big spending, big tax rises, big government, blocking public-service reform, loading regulation on business, growing welfarism and outdated thinking.
Ken Clarke, the former chancellor, played down the prospects of income-tax cuts in a speech focusing on the economy.
Mr Clarke said lower taxes would be good for the economy, but added: "We must be honest with people and tell them tax cuts can only come only when they are affordable. And probably the first area for tax relief, when we have done the necessary work to make it affordable, will need to be pensions and savings rather than cuts in direct personal taxation." Calling for more private-sector involvement in public services, he said: "We must first be prepared to ask the question of every activity, 'should the government be doing this at all?' If it is a responsibility that must fall upon the public sector, what is the best way to deliver it?"Reuse content