and To Assist the Reader (and yourself) In Following Its Logic and Structure Rick Garlikov (this list is not complete; it is just what I can think of now; if you have more categories or words and phrases that would be appropriate here, please email them to me and I will add them.) When what you have presented leads up to, or supports, or makes a
case for what you are about to say:
When you have given your conclusion first and want to then give your
evidence, support, justification for it:
To do these sorts of things in one sentence, you can use words such
as:
When you are going to "contradict" what has been said before [or
contradict what you are about to say]:
To link together similar things (whether ideas or reasons):
To say that something is true in "both directions"
To say it is true in only one direction:
To explain something further:
To change topics:
Using physical structure, rather than words  often good for introducing
remarks related to a particular point or place, but remarks which are distracting
or somehow peripheral even if they are important:
1. In normal language, "converse" is not always used in the sense it is in doing mathematical or philosophical "logic" as a formal study. We may say, for example, in normal language that the groom not only gave the bride a ring at the wedding, but conversely she gave him one also. Or we may say that Sue loves Bill, and conversely, he loves her. The Hatfields do not trust the McCoys, and conversely. In this sense, converses are reciprocating actions or characteristics. In formal logic, converses have a specific meaning, along with some implications of that meaning. The formal or logical converse applies to statements such as "if a (is true), then b (is true)", where the converse is "if b (is true), then a (is true)". We can leave out the parenthetical part and just say, the more usual, the converse of "if a, then b" is "if b, then a". Moreover, since "if a, then b" is generally the same as "all cases of a are also cases of b", or, for short, "all a's are b's", the converse is "all b's are a's." And, another way of saying "if a, then b" is "a is sufficient for b," since if whenever a is true we know b is true, it is sufficient to know a is true in order to know b is also true. Now when "a is sufficient for b's being true" it follows that "b is necessary for a's being true." Consider: "knowing someone was murdered is sufficient for knowing they are dead." This implies that "knowing someone is dead is necessary for knowing they were murdered." Finally in this regard, "if a then b" also implies that "a is true only if b is true". Consider: since "if you were murdered, you are dead." Then that implies you were murdered only if you are dead. [The "if/only if" relationship and the "necessary/sufficient" relationships are not always intuitive or easy to see how to word. I usually have to go through the "murdered implies dead" analogy to keep it straight. The converse of "if you were murdered, you are dead" is not necessarily true, because you can be dead without having been murdered. An example of converse statements which are both true can be found in testing to see whether a baking cake is finished baking or not. You can test the doneness of a cake by sticking a toothpick in its center and pulling it out. If the toothpick comes out clean, the cake is done; if the cake is done, the toothpick will come out clean. There are other relationships (contradictions, contrapositives, contraries, etc.) in logic; there is no need to go into them here.] 
or
or
or

or
or
or

(which is the same as b, if and only if a) or
or
