Logic itself postulates nothing. It is simply reasoning which allows you to come to correct conclusions (assuming you keep it within the rules).

People do the postulating and then apply the rules as they are aware of them. Thus,

1.X = Y
2.Y = Z
3.X = Z

is just as valid as

1.X = Y
2.Y ≠ Z
3.X ≠ Z

The problems arise when the syllogisms must express conditionals:

1. All X = Y
2. Some Y = Z
3. Therefore some X = Z

But unspoken is "but not all X = Z" and when we attempt to ignore that we get into trouble. Check out https://owl.english.purdue.edu/owl/resource/659/03/ for a list of common logical fallacies