Is the core of morality logical?

"Logical" doesn't mean "devoid of compassion." It just means that conclusions are formed based on valid arguments.

There are a number of valid arguments for a number of different moral codes. Whether you subscribe to them or not depends on your understanding of the arguments and whether or not you agree to their premises.

Just because your moral code doesn't make use of a supernatural being or isn't based solely on your emotions, that doesn't mean it's not a moral code. You have just structured your code around a different set of principles. You still have an idea of what is "good" or "right" and what is "evil" or "wrong." The fact that it doesn't necessarily match that of most of the people known to you is inconsequential; it is still a moral code.

Ideally, every moral code should be based on logic, in that the conclusions follow from valid arguments. It is true that many do not. However, many do. And just because they make use of a supreme being or emotion, that doesn't make them invalid. Validity comes from the formulation of the argument.

The source of morality has been debated for centuries. From a purely secular standpoint, it can be difficult to satisfactorily explain morality. However, a simplistic explanation is that morality enables the cooperation of human societies, which in turn enable the continued existence of the species. In short, morality has proved useful to mankind, so it is something that has continued to be with us over the course of our evolution.

Of course morality arises from what I may call 'social logic'...... that which ensures the best desirable results for the society as a whole even if it implies certain sacrifices of individual interests and freedom with the overriding logic that the individual freedom, security and prosperity can not be sustained without the beneficial interests of the society as a whole being protected and catered to.

Morality is not inherently logical. It's probably created by exaggerating the value of certain actions.

From a religious Perspective, 'morality' is judging other people/stuff as 'good' or 'bad/evil'!

As a Xtian (or any other religion), we are warned against judging others;

"Judge not lest you be judged!"

Such is the sin of 'pride'!

'Pride' is the only sin (from which all others spring), yet the hypocrites flaunt their practices, joyfully, proudly, in the face of god!

You are told that;

"If you judge, judge with righteous judgment!"

And goes on to say that;

"None are righteous, no not one!"

Such 'morality' is certainly 'logically rationalized and defended'!

Whatever works.

Sin is sin.

according to biological psychology, b.f.skinner, a dissonance between any two instinctual actions, is the development of consciousness. on a daily basis, the concept of self as based in gestalt psychology, says that we understand the world around us, determined on how we define our true self. while carl jung's attribution theory says that we form our ego-boundary by making choices about the world around us. this aberration between items is the type of consciousness which skinner recognized, and the selection of the values which are meaningful to us, is the development of our ego and our idea of self. sometime the self-image does not match the ego-image, then people are unhappy. since most people like to be happy, let me tell you the story of BOB, the atheist. BOB, the atheist, is a common person who has common values. he is capable of making his own choices. he made his own values based on his own simple choices. for instance: good is better than evil. humbleness is better than arrogance. courage is better than cowardliness. unity is better than disunity. peace is better than unrest. consciousness is better than unconsciousness. respect is better than obnoxiousness. good is better than bad or evil. growth is better than stagnation. light is better than dark. humanism is better than animality. peace and calm is better than anger. reason is better than irrationality. caring compassion is better than apathy. love is better than hate. health is better than sickness. happiness is better than unhappiness. success is better than failure. competence is better than incompetence. intelligence is better than stupidity. wisdom is better than ignorance. appreciative compassionate benevolent action is better than lazy apathy. so, i think BOB, the atheist, has a great sense of value

You sound like a good consequentialist

Fear; otherwise, morality would be superfluous.