Philosophy 148

Chapter 3 (part 2)

Sentences that express multiple propositions.

• We have noticed that some sentences express propositions.

• Some sentences express several propositions and connect them together.

• For example, note that in the sentence ‘I am wearing a blue hat and a green jacket’ two things that might or might not be the case are expressed:– I am wearing a blue hat– I am wearing a green jacket– These two propositions are asserted together, connected

by the word ‘and’.

Symbolizing propositions

• Just as people save time by using letters to stand for any given number in algebra, logicians use letters to stand for any given proposition.

• So the compound statement on the previous slide can look like this: p and q (where p stands for the proposition that I am wearing a blue hat and q stands for the proposition that I am wearing a green jacket).

Connectives

• There are several ways for sentences that express propositions to be connected:– ‘not’– ‘and’– ‘or’– ‘if…then…’

Argument forms• We can apply this procedure to whole arguments, not just

statements to reveal the structure that lives behind our language.

• Example: 1. If it is raining, then the ground is wet.2. It is raining.C. The ground is wet.• Since the underlined sentences express propositions, they

can be replaced with propositional variables like so:1. If p, then q2. pC. q

Argument forms

• When the content of an argument is abstracted away, and only propositional variables and connectives remain, we have an argument form, and can see the structure that lives behind our language.

• As it happens the structure of an argument alone determines whether the argument is valid. This is a significant insight.

Forms that are always valid• There are indefinitely many valid arguments that are

not of the forms below, but the forms below are always valid:

• Modus ponens (affirming the antecedent):1. If p then q2. pC. q• Modus tollens (denying the consequent):1. If p then q2. Not qC. Not p

More valid forms

• Hypothetical Syllogism (chain argument)1.If p then q2.If q then rC.If p then r• Disjunctive Syllogism (process of elimination)1.p or r2.Not pC.r

Forms that are always invalid

• Affirming the consequent1.If p then q2.qC.p• Denying the antecedent1.If p then q2.Not pC. Not q

Organizing long arguments

1. Separate the claims into a numbered list2. Identify the conclusion, put it on the diagram

first.3. Draw an arrow from any claim that is

intended to be support for the conclusion from that claim, to the conclusion.

4. underline 2 claims that are meant to work together

Organizing long arguments

• Be as faithful to the original text as you can, but remember that sometimes authors repeat themselves in different ways, go on tangents, and bring in irrelevant information.

Example (claims that work together):

• (1) Bill is a student at Yale. (2) No student at Yale has won the Nobel Prize. (3) Therefore, Bill has not won the Nobel Prize.1 + 2

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Example (independent claims):

• (1) The president is soft on the environment. (2) He has weakened clean-air regulations (3) and lifted restrictions on logging in the West.

2 3

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Example: (complex arguments)• Conclusion: (3) The idea that God is required to be the enforcer of the moral law is not

plausible. Premises: (4) In the first place, as an empirical hypothesis about the psychology of human beings, it is questionable. (5) There is no unambiguous evidence that theists are more moral than nontheists. (6) Not only have psychological studies failed to find a significant correlation between frequency of religious worship and moral conduct, but convicted criminals are much more likely to be theists than atheists. (7) Second, the threat of divine punishment cannot impose a moral obligation. (8) Might does not make right.