Inference

Once the premises have been agreed upon, the argument proceeds using a step-by-step process called inference. In inference, you start with one or more propositions that have been accepted, and then you use these propositions to arrive at a new proposition. If the inference is valid, the new proposition should also be accepted. You may later use the new proposition for inference. This means that initially you can only infer things from the premises of the argument. However, as the argument proceeds, the number of statements available for inference increases. Various types of valid and invalid inferences will be discussed later. Inference steps are often identified by using phrases such as "therefore" or "implies that."

Conclusion

If all goes well, you will arrive at a proposition that is the conclusion of the argument (the result you were trying to prove). The conclusion is the result of the final step of inference. It is only a conclusion in the context of a particular argument; it could be a premise or an assumption in another argument. The conclusion is said to be affirmed on the basis of the premises, and the inference from them.

Implication

Clearly, you may build a valid argument from true premises, and arrive at a true conclusion. You may also build a valid argument from false premises, and arrive at a false conclusion. You may also start with false premises, proceed using valid inference, and reach a true conclusion. However, you cannot start with true premises, proceed using valid deductive inference, and then reach a false conclusion. These results may be displayed as a Truth Table for implication. In the table, "A" is the premise, "B" is the conclusion, and "=>" denotes implication.

**Truth Table for Implication**
Line	Premises	Inference	Conclusion
	A	A => B	B
1	False	True	False
2	False	True	True
3	True	False	False
4	True	True	True

If the premises are false and the inference valid, the conclusion can be true or false. (Lines 1 and 2.)
If the premises are true and the conclusion false, the inference must be invalid. (Line 3.)
If the premises are true and the inference valid, the conclusion must be true. (Line 4.)

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