Forms of Argument  

Statement variable: a letter used to represent a simple statement, most commonly from the middle of the alphabet (ex. p, q, r)

Some common patterns of valid inference (in arguments which fit these patterns, when the premises are true, the conclusions must be true, too):  

Case 1: Modus Ponens
1. If p, then q.
2. p.
3. Therefore, q.

Case 2: Modus Tollens
1. If p, then q.
2. Not q.
3. Therefore, not p.

Case 3: Disjunctive Syllogism
1. p or q.
2. Not p.
3. Therefore, q.

Case 4: Hypothetical Syllogism
1. If p, then q.
2. If q, then r.
3. Therefore, if p, then r.

Case 6: Simplification
1. p and q.
2. Therefore, p.

Case 7: Addition
1. p.
2. Therefore, p or q.

Case 8: Conjunction
1. p.
2. q.
3. Therefore, p and q.

Case 9*
1. All logicians are people.
2. All people have blood.
3. Therefore, all logicians have blood.

*We can symbolize Case 9 as follows:
1. All F's are G.
2. All G's are H.
3. Therefore, All F's are H.
Note that "F", "G", and "H" do not stand in here for statements (unlike "p", "q" and "r" which do stand in for statements).  They stand, rather, for common nouns, e.g., dogs, philosophers, vegans and so on.)

Case 10
Reductio ad absurdum: a method of argumentation by which a statement S is argued against by showing that absurd consequences follow from S, that is, a consequence known to be false; in the example statement “if p, then q”, if it can be shown that q is false, then the statement p is false.  

V. Interlude: Things Aren't Always Easy