Modus ponens

###
- If A, then B
- A
- Therefore B

Before being put into logical form the above statement could have been something like below.

- If Kelly does not finish his homework, he will not go to class
- Kelly did not finish his homework
- Therefore, Kelly will not go to class

The first two statements are the premises while the third is the conclusion derived from them.

Says that if one thing is true, then another will be. It then states that the first is true. The conclusion is that the second thing is true.

Modus tollens

###
- If A, then B
- Not B
- Therefore not A.

When modus tollens is used with actual content, it looks like below.

- If the Saints win the Super Bowl, there will be a party in New Orleans that night
- There was no party in New Orleans that night
- Therefore, the Saints did not win the Super Bowl

Another form of argument is known as modus tollens (commonly abbreviated MT). In this form, you start with the same first premise as with modus ponens. However, the second part of the premise is denied, leading to the conclusion that the first part of the premise should be denied as well. It is shown below in logical form.

Hypothetical syllogism

###
- If A, then B
- If B, then C
- Therefore if A, then C

When put into words it looks like below.

- If it rains today, I will wear my rain jacket
- If I wear my rain jacket, I will keep dry
- Therefore if it rains today, I will keep dry

Disjunctive syllogism

Disjunctive syllogism (sometimes abbreviated DS) has one of the same characteristics as modus tollens in that it contains a premise, then in a second premise it denies a statement, leading to the conclusion. In Disjunctive Syllogism, the first premise establishes two options. The second takes one away, so the conclusion states that the remaining one must be true.[3] It is shown below in logical form.

- Either A or B
- Not A
- Therefore B

When used A and B are replaced with real life examples it looks like below.

- Either you will see Joe in class today or he will oversleep
- You did not see Joe in class today
- Therefore Joe overslept

Disjunctive syllogism takes two options and narrows it down to one.

Constructive dilemma

###
- Either A or B
- If A then C
- If B then D
- Therefore either C or D

When content is inserted in place of the letters, it looks like below.

- Bill will either take the stairs or the elevator to his room
- If he takes the stairs, he will be tired when he gets to his room
- If he takes the elevator, he will miss the start of the football game on TV
- Therefore Bill will either be tired when he gets to his room or he will miss the start of the football game

There is a slightly different version of dilemma that uses negation rather than affirming something known as destructive dilemma. When put in argumentative form it looks like below.

- If A then C
- If B then D
- Not C or not D
- Therefore not A or not B