Philosophy - Philosophy Course Notes
Formal Logic 2 - Lecture Notes
How to Test if an Argument is Valid in Sentential Logic
Lecture 11
May 28, 2007
Once you have correctly translated an argument into SL, you will be in a much better position to evaluate whether the argument is valid or not. This lecture considers two different aspects of the validity question:
a) Give a proof that shows the conclusion follows from the premises.
b) Give a proof that shows that an argument is invalid (give a counter-proof that shows another argument in the same form that uses the same reasoning, and where the premises are true and the conclusion is false.)
To do the above proofs, you will be using a natural language deduction system. We want an efficient system (with a small number of steps) to make analyzing the validity of an argument easier. The rules that we are going to use are natural, meaning they are part of the actual kinds of rules people use when presenting arguments.
What do we want from our system of proofs:
a) Only a few rules.
b) A system that is intuitive.
I. Ways of Testing if an Argument Is Valid or Not
A. Truth Tables
1. Truth tables are mechanical and easy to do after getting some practice.
2. But you have to pay attention and follow the rules. If you do so, you do not need a lot of ingenuity to use them.
3. See the textbook for more information on using truth tables.
B. Dead Reckoning
1. Disadvantage to using truth tables is that it is not quite as mindless a process, you have to pay attention.
2. The advantages are :
i) not as boring
ii) not as long and tedious
iii) dead reckoning can manage complicated arguments without becoming hard to work with like truth tables
3. The primary purpose of dead reckoning is to give counter examples to show the invalidity of an argument.
4. In SL (because it is a weaker system), with dead reckoning you get Yes / No decision making help.
5. In stronger systems you have to make an informed guess as to whether the argument is valid or not.
II. The Proof Theory of SL
A. How to Set Out Arguments So They Are Easy To Read
1. At the top of the proof, list the premises, each on their own line, with a unique number.
2. At the bottom of the proof, list the conclusion.
3. Off to the left, number every sentence in the proof.
4. One of the rules is repetition of previous premises.
i) write a new number on the left hand side of the page.
ii) write down what you are repeating.
iii) for justification, put the line number being repeated then ", repetition"
B. Other Rules in Natural Deduction Systems
1. Come in matched pairs
2. For each connective in SL, there are two corresponding rules.
3. Ampersand Related Rules (and, but, although) (Given A&B as premise.)
a) And Introduction
If you know A is true and you know B is true, then you can introduce A&B as true.
b) And Elimination
If you know A&B is true, then you know A is true and you know B is true.
1. If Related Rules (modus ponens, conditional proofs)
a) Modus Ponens (THE fundamental rules of formal logic)
Give A => B as a premise:
If you know A is true, you also know B is true.
b) Conditional Proofs (Conditional Acceptance)
Need to indicate you are making a conditional supposition. If making an assumption for the sake of argument, must indent in the proof. Conditional proofs allow you to "make back to the left" after the supposition.
After proving B you can legitimately conclude that A => B.
2. Negation Rules (reductio and double negation elimination)
a) Reductio Ad Absurdum (reduce to absurdity)
You start by assuming what you want to prove false is true. (Indent) Then you show that absurdities follow from it and that it is therefore false.
b) Double Negation Elimination
Depends on an assumption of classical logic, that every statement is either true or false. If A is true, then not A is false.
3. Or Related Rules
a) If you know either A or either B you can write A v B.
b) Constructive Dilemma
A v B. A is good. B is good.A v B is true.
III. Derived Rules of Inference
In SL certain operations are lengthy. Sometimes best to note you've proved it in the past instead of redoing it.
In future label it by name and 'prove it' in one step.
ie. disjunctive syllogisms.
IV. Proving Invalidity in SL
The method that we will use it called dead reckoning.
A. First list across the page, all the premises, followed by the conclusion.
How can we make it so the premises are all true and the conclusion is false? (showing invalidity)
B. Assign truth values to the premises and the conclusion. (T and F)
C. Write T under each operator for the premises and F under the main operator of the conclusion.
D. Next think about what each of the sentences actually means. For each operator do something for each part.
E. Use the process in D until you can completely fill in the proof (in which case it is invalid) or you come to the point where an item is both true and false (in which case it is valid in SL)
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