A Hypothetical Proposition Is A Conditional Statement Which Takes The Form 1q1s23

A hypothetical proposition is a conditional statement which takes the form: if P then Q. Examples would include: If he studied, then he received a good grade. If we had not eaten, then we would be hungry. If she wore her coat, then she will not be cold. In all three statements, the first part (If...) is labeled the antecedent and the second part (then...) is labeled the consequent. In such situations, there are two valid inferences which can be drawn and two invalid inferences which can be drawn - but only when we assume that the relationship expressed in the hypothetical proposition istrue. If the relationship is not true, then no valid inferences can be drawn. A hypothetical statement can be defined by the following truth table:

P T T F F

Q if P then Q T T F F T T F T

Assuming the truth of a hypothetical proposition, it is possible to draw two valid and two invalid inferences: The first valid inference is called affirming the antecedent, which involves making the valid argument that because the antecedent is true, then the consequent is also true. Thus: because it is true that she wore her coat, then it is also true that she will not be cold. The Latin term for this, modus ponens, is often used. The second valid inference is called denying the consequent, which involves making the valid argument that because the consequent is false, then the antecedent is also false. Thus: she is cold, therefore she did not wear her coat. The Latin term for this,modus tollens, is often used. The first invalid inference is called affirming the consequent, which involves making the invalid argument that because the consequent is true, then the antecedent must also be true. Thus: she is not cold, therefore she must have worn her coat. This is sometimes referred to as a fallacy of the consequent. The second invalid inference is called denying the antecedent, which involves making the invalid argument because the antecedent is false, then therefore the consequent must also be false. Thus: she did not wear her coat, therefore she must be cold. This is sometimes referred to as a fallacy of the antecedent and has the following form: If P, therefore Q. Not P. Therefore, Not Q. A practical example of this would be: If Roger is a Democrat, then he is liberal. Roger is not a Democrat, therefore he must not be liberal. Because this is a formal fallacy, anything written with this structure will be wrong, no matter what you use to replace P and Q with. Understanding how and why the above two invalid inferences occur can be aided by understanding the difference between necessary and sufficient conditions. You can also read the rules of inference to learn more. Inference Rules

Complex deductive arguments can be judged valid or invalid based on whether or not the steps in that argument follow the nine basic rules of inference. These rules of inference are all relatively simple, although when presented in formal they can look overly complex. Conjunction: 1. P 2. Q 3. Therefore, P and Q. 1. It is raining in New York. 2. It is raining in Boston 3. Therefore, it is raining in both New York and Boston Simplification: 1. P and Q. 2. Therefore, P. 1. It is raining in both New York and Boston. 2. Therefore, it is raining in New York. Addition: 1. P 2. Therefore, P or Q. 1. It is raining 2. Therefore, either either it is raining or the sun is shining. Absorption: 1. If P, then Q. 2. Therfore, If P then P and Q. 1. If it is raining, then I will get wet. 2. Therefore, if it is raining, then it is raining and I will get wet. Modus Ponens: 1. If P then Q. 2. P. 3. Therefore, Q. 1. If it is raining, then I will get wet. 2. It is raining. 3. Therefore, I will get wet. Modus Tollens: 1. If P then Q. 2. Not Q. (~Q). 3. Therefore, not P (~P). 1. If it had rained this morning, I would have gotten wet. 2. I did not get wet. 3. Therefore, it did not rain this morning. Hypothetical Syllogism:

1. If P then Q. 2. If Q then R. 3. Therefore, if P then R. 1. If it rains, then I will get wet. 2. If I get wet, then my shirt will be ruined. 3. If it rains, then my shirt will be ruined. Disjunctive Syllogism: 1. Either P or Q. 2. Not P (~P). 3. Therefore, Q. 1. Either it rained or I took a cab to the movies. 2. It did not rain. 3. Therefore, I took a cab to the movies. Constructive Dilemma: 1. (If P then Q) and (If R then S). 2. P or R. 3. Therefore, Q or S. 1. If it rains, then I will get wet and if it is sunny, then I will be dry. 2. Either it will rain or it will be sunny. 3. Therefore, either I will get wet or I will be dry. The above rules of inference, when combined with the rules of replacement, mean that propositional calculus is "complete." Propositional calculus is simply another name for formal logic. To say that it is "complete" means that, in this system, the axioms used are sufficient to demonstrate any true proposition or to justify any valid argument. The question of necessary and sufficient conditions for something relates to how we evaluate arguments. A necessary condition for an event is something which is absolutely required to exist or happen if the event is to occur. An example of this would be the existence of oxygen for human life - without oxygen, we could not live. Therefore, the existence of oxygen is a necessary condition for human life. A sufficient condition for an event, on the other hand, does not have to exist for the event to occur, but if it exists, then the event will occur. To continue with our above example, the absence of oxygen is a sufficient condition for human death. Human death can happen without the absence of oxygen and does not require the absence of oxygen. However, if oxygen disappears, then people die. Thus, an absence of oxygen is a sufficient condition for human death. A more formal way for saying that one thing, p, is a sufficient condition for some other thing, q, would be to say "if p then q," which is a standard hypothetical proposition. Confusing necessary and sufficient conditions is one way to understand how some of the rules of inference with hypothetical propositions can be violated. The fallacy of affirming the consequent, for example, makes the assumption that a sufficient condition is also a necessary condition.