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FOUR Complex Propositions

The syllogism is one type of argument that can be proved to be deduc- tively valid or invalid in a given case by specifying its form. There are also many other common forms of argumentation that display a deductively valid form that is easily recognizable. These forms of argument are built up of complex propositions of certain kinds. Some propositions are com- plex, in the sense that they are formed from simple propositions by certain logical operations. One of these operations we have already encountered above is negation. ‘Snow is not white’ is formed from the simple propo- sition ‘Snow is white’ by negation. If a proposition is true, its negation is false. And if a proposition is false, its negation is true. So the opera- tion of negating a proposition gives the resulting (negated) proposition the opposite truth value. Another logical operation is that of conjunction, which takes two propositions and asserts that both are true. For example, the conjunctive proposition ‘The book is on the table and the pencil is on the desk’ takes the two component (simple) propositions, and forms their conjunction, using the word ‘and’. A conjunctive proposition can also include a negated proposition, as in the sentence ‘The book is on the table and the pen is not on the desk’. Another kind of logically complex proposition is formed by the disjunction of propositions. For example,

the disjunctive proposition ‘The book is on the table or the pencil is on the desk’ takes two component (simple) propositions and forms their dis- junction, using the word ‘or’. Disjunctive propositions can also include negated propositions, as in the sentence ‘The book is on the table or the pen is not on the desk’. A conjunctive proposition is only true if both (or all) the propositions contained in it are true. But a disjunctive proposi- tion is true if either of the contained propositions is true. So generally speaking, conjunctive propositions are harder to prove than disjunctive sentences, because they claim that more is true.

Another kind of complex proposition that is very important in under- standing logical reasoning is the conditional type of proposition. A con- ditional proposition (statement) is a proposition of the form ‘If A, then B’ where A and B are component propositions in the conditional proposi- tion. The component that directly follows the ‘if’ (the A, in this case) is called the antecedent (hypothesis). The other proposition (the B, in this case) is called the consequent. An example is the statement:

If Bob is in Miami, then Bob is in Florida.

In this example, ‘Bob is in Miami’ is the antecedent, and ‘Bob is in Florida’ is the consequent. Conditional propositions, like conjunctive and disjunc- tive propositions, can include negated propositions. For example, the con- ditional proposition ‘If Bob is in Miami, Bob is not in Texas’ contains a negated proposition as its consequent.

In fact, conjunctive, disjunctive, negated, and conditional proposi- tions can all be combined, as in the following proposition: If Bob is in Miami or Bob is in Tampa, then Bob is not in Texas and Bob is not in Vermont. In stating this complex conditional, we have written everything out explicitly, but normally the sentence would be expressed in a more compact form using the pronoun ‘he’: If Bob is in Miami or Tampa, then he is not in Texas and not in Vermont. In this case, the whole sentence has the form of a conditional proposition, but the antecedent part of the conditional is a disjunctive proposition, and the consequent part of the conditional is a conjunction.

The most important thing about conditionals is that the antecedent proposition is not (necessarily) being claimed (asserted) to be true. Generally, it is only an assumption or hypothesis. What is being claimed is that if the antecedent proposition is true, then the consequent propo- sition is true as well. Such a claim is hypothetical in nature. For exam- ple, if I say, ‘If Bob is a gangster, then he should be prosecuted’, I am not (necessarily) saying that Bob is a gangster. I am only claiming that if he is a

5. Some Other Common Forms of Deductive Argument 61

gangster, then he should be prosecuted. The whole conditional proposi- tion can be true, even if the antecedent proposition is false. Even if, in fact, it’s not true that Bob is a gangster, it could still be true that if he is a gangster, then he should be prosecuted.

What is important to note is that there is a difference between assump- tions and assertions. If I assume that a proposition is true, such an assump- tion is only a hypothesis, and if anyone questions it, I don’t have to prove it. But if I assert that a proposition is true, I have made a claim. So if any- one questions this claim or asks me to prove it, then either I have to give reasons (evidence) that support it, or I have to give up my claim that it is true. So assumptions are ‘free’, whereas assertions have a ‘cost’ attached. You either have to prove (justify) them or give them up (as assertions). In this respect, assumptions are more like questions than assertions.

EXERCISE 2.4

Identify the simple propositions in the following complex propositions and show how the complex propositions are made up from the simple ones, identifying each logical operation used.

(a) If antibiotic use is not restrained, tuberculosis and polio will become impossible to control.

(b) If eating bran cereal every day reduces the chances of colon can- cer, then not eating bran cereal every day is a way of decreasing longevity.

(c) Either Bob is a gangster or Bob is working for the FBI and is engaged in undercover work.

(d) Madagascar is south of the equator.

(e) Courage is a form of addiction to combative action, and it needs to be restrained, or it may threaten those who fear its dangers.

(f) If this object is metal, then it will be attracted by a magnetic coil, if one is placed in its vicinity.

(g) Some coins are made of silver and some are made of gold.

(h) Either Ted is not coming to the party, or Linda is staying home and waiting for a call from Bob.

Outline

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