A ⊃ (B ⊃ D) ~B __________________ ~R _________________ That is the important lesson that I have been trying to drill in in this section. If the killer is in the attic then he is above me. ~D 4. Although you cannot construct a proof to show that an argument is invalid, you can construct proofs to show that an argument is valid. ~C __________________ This is the rule that I introduced in the first section of this chapter. Change ), You are commenting using your Facebook account. Therefore, the killer is either _________________ or _________________. 2. A v E _________________ ~H Modus tollens, 3, 4, Example 2: The first premise is a disjunction (since the wedge is the main operator), the second premise is simply the negation of the left disjunct, “~A”, and the conclusion is the right disjunct of the original disjunction. R v S /∴ ~T 6. A __________________ (R v S) ⊃ (T ⊃ K) The next form of inference we will introduce is called “disjunctive syllogism” and it has the following form: In words, this rule states that if we have asserted a disjunction and we have asserted the negation of one of the disjuncts, then we are entitled to assert the other disjunct. This rule is incredibly powerful as it allows you to introduce new elements into a disjunction as long as we have one of its disjuncts as true. Legal. (R v S) ⊃ (T ⊃ K) If the consequent of a conditional is false, then its antecedent is also false. A ⊃ B This is so even though the antecedent of the conditional is itself complex (i.e., it is a conjunction). 3. I will introduce the 8 valid forms of inference in groups, starting with the rules that utilize the horseshoe and negation. 2. 7. It is the deduction of one proposition from another proposition. As before, it is important to realize that any inference that has the same form as conjunction is a valid inference. 3. 3. (Don’t confuse the rule called conjunction with the type of complex proposition called a conjunction.) H ⊃ N /∴ ~H 1. Here’s an example: For our purposes on this page, the visualisations for each of the rules below will not be written in this vertical fashion as they are cumbersome to format in the WordPress editor so it’ll be horizontal. A ⋅ ~D That is, line 1 is a conjunction (since the dot is the main operator of the sentence) and line 2 is inferring one of the conjuncts of that conjunction in line 1. The second premise is a conditional statement whose antecedent is the left disjunct of the disjunction in the first premise. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. ( Log Out / The next four forms of inference we will introduce utilize conjunction, disjunction and negation in different ways. The next rule we’ll introduce is called “addition.” It is not quite as “obvious” a rule as the ones we’ve introduced above. A ⊃ D Hypothetical syllogism, 3, 2 Example 1: 3. The idea of a proof is that although the inference being made in the argument is not obvious, we can break that inference down in steps, each of which is obvious. 5. H ⋅ K _________________ The next step is to apply the resolution Rule of Inference to them step by step until it cannot be applied any further. We can actually use modus ponens in the first argument of this section: 1. B ⊃ C __________________ 2. K _________________ The first premise is still a conditional statement (since the horseshoe is the main operator) and the second premise is the antecedent of that conditional statement. 4. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. 8. R v S /∴ ~T 5. That is a disjunctive syllogism. Change ), You are commenting using your Twitter account. In this case, our left disjunct in premise 1 is itself a negation, while premise 2 is simply a negation of that left disjunct.
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