3 Formal Logic in Philosophy
Bahram Assadian
This chapter discusses some philosophical issues concerning the nature of formal logic. Particular attention will be given to the concept of logical form, the goal of formal logic in capturing logical form, and the explanation of validity in terms of logical form. We shall see how this understanding of the notion of validity allows us to identify what we call formal fallacies, which are mistakes in an argument due to its logical form. We shall also discuss some philosophical problems about the nature of logical forms. For the sake of simplicity, our focus will be on propositional logic. But many of the results to be discussed do not depend on this choice, and are applicable to more advanced logical systems.
Logic, Validity, and Logical Forms
Different sciences have different subject matters: physics tries to discover the properties of matter, history aims to discover what happened in the past, biology studies the development and evolution of living organisms, mathematics is, or at least seems to be, about numbers, sets, geometrical spaces, and the like. But what is it that logic investigates? What, indeed, is logic?
This is an essentially philosophical question, but its answer requires reflection on the status and behavior of logical rules and inferences. Textbooks typically present logic as the science of the relation of consequence that holds between the premises and the conclusion of a valid argument, where an argument is valid if it is not possible for its premises to be true and the conclusion false. If logic is the science of the relation of consequence that holds between the premises and the conclusion of a valid argument, we can say that logicians will be concerned with whether a conclusion of an argument is or is not a consequence of its premises.
Let us examine the notion of validity with more care. For example, consider the following argument:
- If Alex is a sea bream, then Alex is not a rose.
- Alex is a rose.
- [latex]/ \therefore[/latex] Alex is not a sea bream.
It can be shown that it is not possible for (1) and (2) to be true yet (3) false. Hence, the whole argument is valid. For convenience, let us represent each sentence of the argument into the standard propositional logic, which aims to analyze the structure and meaning of various propositions. To do this, we must first introduce the language of our logic.
The alphabet of propositional logic contains letters standing for sentences: A, B, C, and so on. For example, we can translate “Alex is a rose” by just using B. Similarly, we can use S to translate “I would love to smell it.” The alphabet of propositional logic contains other symbols known as logical connectives. One is a symbol for “not” or negation [latex](\neg )[/latex]. When we say that Alex is not a rose, we, in effect, say that it is not the case that Alex is a rose. If we translate “Alex is a rose” by B, we translate “Alex is not a rose” as “[latex]\neg B[/latex].” Another