University:
Harvard University
Course:
PHIL S-12 | Deductive Logic
Academic year:

2024
Views:

128

Pages:

3
Author:

laqala0uk

SENTENTIAL LOGIC PROVING THEOREMS: (Except Conditionals) Some sentences can be derived even without premises. These sentences are called theorems of the deductive system. For example, the tautology “b V ~b” is a theorem because, since it is impossible for it to be false, you can construct a deduction containing that formula on its last (and first) line. All theorems are tautologies since they are true in and of themselves and are not dependent on other sentences being true. Most theorems are not as obviously true as “b V ~ b” or “p  p.” PROVING THEOREMS THROUGH DEDUCTION To prove a theorem you must construct a deduction, with no premises, such that its last line contains the theorem (formula). 1. To get the information needed to deduce a theorem (the sentence letters that appear in the theorem) you can use two rules of sentential deduction: EMI and Addition. EMI The EMI rule of deduction allows you to write on any line in a deduction, a disjunction such that one of the disjuncts is the negation of the other. For example, you can use the EMI rule to put the formula “p V ~p” on the first line. You often start deductions (if the theorem is not a conditional), by creating an EMI line which contains some of the letters that appear in the deduction. For example, if the theorem you want to prove is: ((p V ~) V d) & (m V ~m) you would use the EMI rule to introduce the tautology p V ~p on the first line and then the EMI line again to introduce m V ~m. Addition The Addition rule of deduction allows you to create a disjunction by adding any formula as a disjunct to an existing formula. For example, you can use the Addition rule to put the formula “(p V ~p) V d” on the first line. TIP: Work backwards from what you want to deduce. If you see a letter and its negation in the theorem, you probably want to use an EMI line to introduce those letters. If you see just one letter (without its negation), you should use Addition to introduce it. 2. Once you have the sentence letters you need, you need to use rules of deduction to manipulate the formulas (by drawing inferences), until the formula you deduce is the theorem. TIP: Sometimes when you get to the end of a deduction you realize that you didn’t use the right Addition line or EMI line. CLASS DEMONSTRATION Demonstrate that the following formula is a theorem by constructing a deduction that has this formula on its last line. EXAMPLE: ((p V ~p) V d) & (m V ~m) 1. 2. 3. 4. 5. 6.

Proving Theorems

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