The Difference between Syllogisms, Modus Ponens, and Modus Tollens

The concept of Syllogisms, Modus Ponens, and Modus Tollens are quite alike in that they are forms of deductive logic. Deductive logic is a process by which an argument flows from a general to a more specific or particular stand. It is an argument or claim that is made by a series of ideas that build on each other. Deductive logic uses knowledge that is already known in order to make connections. Let's begin with the concept of syllogisms.
Syllogisms are the linking of two statements in order to draw a conclusion. Because it uses deduction, a general idea will be the major premise or first statement. The following statement will be a particular idea. This will be known as the minor premise. Following both of these arguments is the conclusion. The conclusion will be drawn based on the information provided in the first two statements. For instance:

(Major Premise) All dogs eat dog food.

(Minor Premise) Barker is a dog.

From these two statements, what is apparently true to us is that dogs eat dog food and Barker is a dog. Based on these two arguments, we can come to the conclusion that:

(Conclusion) Barker eats dog food.
Next we must decide whether the argument made is true. Can we trust that these statements are true? "Truth refers to content, substance, and accuracy of the statement..." Based on what we know, it is safe to agree that content of the subject matter is accurate. All dogs eat dog food and Barker is a dog. Because both statements are deemed the truth, then the conclusion must be valid. Validity refers to procedure, to form, and to the way the statements are linked together. First, the first statement made is a generalized statement, "All dogs eat dog food." We do not know specifically what kind of dog, who the dog is, where the dog comes from, etc. It is a general statement that all dogs eat dog food, regardless of these other aspects because we are primarily speaking about dogs in general. Next, the second statement, or the minor premise, is a specific dog. The issue of dogs is no longer generalized, rather, we know a specific dog. Because both these statements were already deemed true, the conclusion drawn from these two arguments may be deemed valid. The conclusion is drawn from a general statement to a particular statement. Even when the first two statements are deemed untrue, if the conclusion is accurately based upon the two statements made, it may be deemed valid. Categorical syllogisms, specifically, follow a certain form that may be recognized in all true and/or valid syllogisms. Terms are used to determine this. In the major premise, we must find the major term. Referring back to the argument about dogs, the major term would be the dogs that eat dog food. This is because our focus is mainly about the dogs that eat dog food. The major term will be seen in the major premise, and then again in the conclusion. The middle term, then, would be all dogs. This term will appear in both premises, for it is the middle term that will link or connect the statements to one another. The connection made will become the conclusion. The minor term is found in the minor premise. In this case, Barker is the minor premise. The minor term will be seen in the minor premise and the conclusion. When these terms are found in the correct format, we are able to conclude that the statements made in relevance to the conclusion are valid. This can be expressed as a mathematical equation:

A = major term

B = middle term

C = minor term

(Major Premise) A=B

(Minor Premise) B=C

(Conclusion) therefore, A=C

The major term appears in the major premise and once again in the conclusion. This is because the conclusion is the connection between the major premise and the minor premise. In which case, when the conclusion is drawn, the major term will reappear in the conclusion along with the minor term. If we were to take the mathematical proof as statements being made to reach a conclusion, we understand that A is equal to B. This is the generalized statement. A particular statement would be that B is equal to C. In this case, we are able to conclude that A is equal to C.

Modus Ponens are rules of logic. They are unlike syllogisms because there is only one statement being made. In addition, the statement made is the cause and effect. A conclusion is made as a result of the effect. Because the statement is the cause and effect within itself, there will be a use of the terms if and then. For instance:

If I eat the sandwich, it will be gone.

I ate the sandwich.

... therefore, it is gone.

The cause is me eating the sandwich. The effect is that the sandwich will be gone. As a result of me eating the sandwich, the sandwich is gone. The conclusion is the result of the cause happening.

The difference between syllogisms and modus ponens are that syllogisms are drawing a new conclusion from the general statement to particular statement being made, while modus ponens already determine the outcome as a result of the a prior cause.

Modus Tollens are different from modus ponens because they are the reciprocal of it. The statement made is like modus ponens, the cause and the effect are stated. "If this happens, then that will happen." However, the result is negative though. Because that did not happen, this did not happen. Because the effect did not take place, then the cause did not take place in order to cause the effect. (Haha, get it?) For instance:

If I eat that rotten banana, then I will get a stomach ache.

I did not get a stomach ache.

... therefore, I did not eat the rotten banana.

The statement made is still a cause and effect. However, modus tollens are the negatice reciprocal. The cause is eating a banana and the effect is a stomach ache. And because I did not get a stomach ache, I did not eat the rotten banana.

These methods of deduction are used everyday in order to draw conclusions and make logical decisions.