There are a number of what are called valid implicational forms that are part of the rules of inference in symbolic logic.
[1]. Modus Ponens (MP): if we are given [p 
q], and we know that [p], then we can conclude [q]. Or, if [p q], [p], [ 
q]. In other words, if we assume the following is true "If a cat purrs, then the cat is happy" and we know "A cat is purring" then we can conclude that "The cat is happy".
[2]. Modus Tollens (MT) (or Modus Tollendo Tollens): if we are given [p q], and we know that [~q], then we can conclude [~p]. Or, if [p q], [~q], [~p]. In other words, if we assume the following is true "If a cat purrs, then the cat is happy" and we know "The cat is not happy" then we can conclude that "The cat is not purring".
A potential problem with MT can be seen in the following case. Assume that "If it rains, then the ground will be wet". If we know that "The ground is not wet" we can conclude, by MT, that "It is not raining".
But, what if "The ground is not wet" is true and "It is raining is true" in the case that, perhaps, the ground is covered by a waterproof tarp?
I imagine that the reasoning behind MT is that the logical statement is true, regardless of the specific circumstances of the logical argument simply because it is stated and assumed.
In other words, I could assume "If the vacuum speaks, then the world will implode" and, if the vacuum does speak then the world will implode even though neither of these possibilities could occur given our scientific understanding of both vacuums and our planet.
[3]. Disjunctive Syllogism (DS): if we are given [p V q] and we know that [~p] then we can show that [q]. Also, if we are given [p V q] and we know that [~q] then we can show that [p].
For example, if I say that "the universe is either infinite or finite" and I know that "the universe is not infinite" then I can deduce that "the universe is finite" and, likewise, if I know that "the universe is either infinite or finite" and I know that "the universe is not finite" then I can deduce that "the universe is infinite".
In using the disjunction, it is important to note the various ways in which it can be used. [A or B] can be true if either [A is true] or [B is true] and if both [A and B are true].
But, some uses of "or" mean to say that either [A is true] or [B is true] but not both [A and B can be true] simultaneously. This latter form is called exclusive or and it is also called exclusive disjunction.
Exclusion means that exactly one possibility must be true for the statement to be true but not both. The statement "the bills are either paid or not" is either true because "the bills are paid" is true or because "the bills are not paid is true" but it can never be true that both "the bills are paid" and "the bills are not paid".
The inclusive disjunction is an "or" statement that can be true even if both sides of the disjunction are true. In other words, if I were to say "Tommy will be a lawyer or a politician" it is certainly not a false statement if Tommy becomes both a lawyer and a politician.
[4]. Simplification (Simp): if we are given [p & q] we can deduce that [p] and we can deduce that [q]. In other words, if we know that "the punch is spiked with LSD and the CIA experimented with LSD" then we know that it it obviously true that "the punch is spiked with LSD" is true and "the CIA experimented with LSD" is true independently of the conjunction.
[5]. Conjunction (Conj): if we are given [p] and we are given [q] then we can deduce that [p & q]. If I were to assume that "Bush is a war criminal" was true and "Tony Blair was Bush's lapdog" was true then it also be obviously true that "Bush is a war criminal and Tony Blair was Bush's lapdog" is true.
[6]. Hypothetical Syllogism (HS): if we are given that [p q] and we are also given that [q r], we can deduce that [p r]. If I said that "If I eat a whole pizza then I will be full" and "If I am full then I will fall asleep" it can be deduced that "If I eat a whole pizza then I will fall asleep".
[7]. Addition (Add): if we are given [p] then we can deduce [p V q] because the statement is true because only [p] need be true for the statement to be true. If it were true that "Maggie is yellow" then it is also true that "Either Maggie is yellow or farts are deadly" because the truth of "farts are deadly" is not necessary for the truth of "Either Maggie is yellow or farts are deadly".
[8]. Constructive Dilemma (CD): if we are given [p q] and we are given that [x y] and we are given [p V x] then we can deduce that [q V y]. Put linguisitically, assume the following three statements are true:
(i) "If Nixon's lips moved then he was lying", (ii) "If Kissinger spoke then babies cried", and (iii) "Either Nixon's lips moved or Kissinger spoke". We can then deduce that "Nixon was lying or babies cried".
