From The History of the Phillips Curve: Consensus and Bifurcation by Robert J. Gordon (PDF, 41 pages, 2008):
Introduction
The history of the Phillips curve (PC) has evolved in two phases, before and after 1975, with a widespread consensus about the pre-1975 evolution, which is well understood...
The pre-1975 history is straightforward and is covered in Section I. The initial discovery of the negative inflation–unemployment relation by Phillips, popularized by Samuelson and Solow, was followed by a brief period in which policy-makers assumed that they could exploit the trade-off to reduce unemployment at a small cost of additional inflation. Then the natural rate revolution of Friedman, Phelps and Lucas overturned the policy-exploitable trade-off in favour of long-run monetary neutrality. Those who had implemented the econometric version of the trade-off PC in the 1960s reeled in disbelief when Sargent demonstrated the logical failure of their test of neutrality, and finally were condemned to the ‘wreckage’ of Keynesian economics by Lucas and Sargent following the twist of the inflation–unemployment correlation from negative in the 1960s to positive in the 1970s.
The history of the Phillips curve (PC) has evolved in two phases, before and after 1975, with a widespread consensus about the pre-1975 evolution, which is well understood...
The pre-1975 history is straightforward and is covered in Section I. The initial discovery of the negative inflation–unemployment relation by Phillips, popularized by Samuelson and Solow, was followed by a brief period in which policy-makers assumed that they could exploit the trade-off to reduce unemployment at a small cost of additional inflation. Then the natural rate revolution of Friedman, Phelps and Lucas overturned the policy-exploitable trade-off in favour of long-run monetary neutrality. Those who had implemented the econometric version of the trade-off PC in the 1960s reeled in disbelief when Sargent demonstrated the logical failure of their test of neutrality, and finally were condemned to the ‘wreckage’ of Keynesian economics by Lucas and Sargent following the twist of the inflation–unemployment correlation from negative in the 1960s to positive in the 1970s.

From After Keynesian Macroeconomics by Robert E. Lucas and Thomas J. Sargent (PDF, 34 pages, 1978):
1. Introduction
We dwell on these halcyon days of Keynesian economics because, without conscious effort, they are difficult to recall today...
That these predictions were wildly incorrect, and that the doctrine on which they were based is fundamentally flawed, are now simple matters of fact, involving no novelties in economic theory. The task which faces contemporary students of the business cycle is that of sorting through the wreckage, determining which features of that remarkable intellectual event called the Keynesian Revolution can be salvaged and put to good use, and which others must be discarded.
We dwell on these halcyon days of Keynesian economics because, without conscious effort, they are difficult to recall today...
That these predictions were wildly incorrect, and that the doctrine on which they were based is fundamentally flawed, are now simple matters of fact, involving no novelties in economic theory. The task which faces contemporary students of the business cycle is that of sorting through the wreckage, determining which features of that remarkable intellectual event called the Keynesian Revolution can be salvaged and put to good use, and which others must be discarded.

For the record...
The "negative" correlation of the Phillips Curve is the tradeoff: a little more inflation and a little less unemployment, or the reverse. The "positive" correlation is when both inflation and unemployment increase, or both decrease.
The negative correlation displays the tradeoff for given economic conditions. The positive correlation shows what happens when economic conditions improve or get worse. When people deny the existence of the tradeoff, it is because they are considering the long term, during which the condition of the economy varies. Even so, the tradeoff still applies to the short term.
But the proper fix for our economy is not to fiddle with the short-term tradeoff. The proper fix is to figure out how to improve conditions over the long term.