The presidential campaign initiated by the Republican convention last week promises to be the meanest in half a century. For the two main candidates exhibit weakneses to so glaring as to compel attack. Which leaves an opening for the third-party candidate, John Anderson.

Carter's albatross is his performance on the major issues. Inflation has doubled during his term, and is now contained only by a recession bound to drive unemployment over 8 percent. The country remains heavily dependent upon foreign oil. The Soviet Union and its allies have made gains in Asia, Latin America and Africa at the expense of America. This country's closest allies are more and more going into business on their own.

A widespread theory is that the president will redeem the past the some spectacular development -- an October miracle. In fact, events now run the other way. The release of one hostage from Iran underlines the plight of 52 others. The opening of the Olympics in Moscow combines with the legal ups and downs of draft registration to highlight anew the weak American response to the Soviet invasion of Afghanistan. Billy Carter's troubles serve to remind the world of the low quality found in many of those around the president.

Ronald Reagan and the Republicans can hardly be expected not to batten on the Carter record. But it is not as though they have much that is positive to offer. On the contrary, a fair reading provides no evidence that the Republicans will do better than the Democrats in addressing the country's pressing problems.

Inflation especially. If elected, Reagan would inherit a deficit running at over $50 billion annually. He has promised a tax cut of $36 billion and increases in defense spending of at least $10 billion. That leaves a budget deficit of close to $100 billion.

Maybe Reagan could balance the budget by severe spending cuts. But probably not. A far more realistic expectation is a series of heavy deficits sure to stimulate more inflation.

Energy conservation, by reducing demand and fostering a glut in the international market, could yield the most immediate relief from rising prices. But the Republican approach to the energy problem features incentives to produce more oil, and a turn away from conservation symbolized by the call to end the 55 mph speed limit. Because of physical limits, it seems very doubtful that more oil can be drawn from American soil by higher prices. In the process of trying, however, conservation will be de-emphasized, consumption will rise and the international oil cartel, OPEC, will be in better position to hike prices -- thus forcing a new burst of inflation.

The same conflict between promise and prospect characterizes the international security field. Reagan addreses strong language to the Russians. But he has backed away from the grain embargo and draft registration. This country's friends and allies -- the Germans, the French and the Chinese, in particular -- already show signs that they place less confidence in Reagan than they do in Carter. They continue to do business on their own. So what the Republicans offer in foreign policy is provocation of the Russians -- not deterrence.

Anderson, of course, is not free from blemishes. He is in constant danger, as witness his recent trip to Europe and the Middle East, of identifying himself with narrow causes dear only to small minorities.

But Anderson -- and Anderson alone -- stands for a truly constructive approach to the central knot of national problems. His proposal for a 50-cent-a-gallon tax on gasoline emphasizes the best short-term approach to the energy problem -- conservation through higher prices to the consumer. By using the revenues to offset payroll taxes, he -- and he alone -- moves to constrain without promoting unemployment. tThe combination of decreased dependence on foreign oil and steady economic growth provides the only good base for dealing with foreign problems. It is typical that Anderson does not feel constrained to follow Carter in appeasing this country's foes, or Reagan in refusing to accept the logic of the grain embargo.

To be sure, the prospect that Anderson can win the election is remote. He has no capacity to publicize himself by such circuses as the national conventions. His organization remains poor.

Still, he enjoys advantages not previously available to third-party candidates. He will have available substantial sums -- over $5 million at least -- for use in television toward the end of the campaign. He will be able to target his appeal to the states where he has a chance -- Massachusetts, Connecticut, New York, Illnois, Michigan, Oregon and Washington.

So the congressman from Illinois can make a difference. He can build a constituency for the policies required to have this country from four more fallow years -- four more years of slow growth, high inflation and decline abroad.