Phil 10 – Ch. 3

In chapter 3, we continue to look for ways to determine the validity of the argument. Before moving on let me remind myself again on the role of logic: “Logic does NOT answer the question of are the premises of the argument true? All logic can say is whether the inference of the conclusion from the premise is valid. Logic can never tell us if our arguments are sound.” And let me remind that whether the premise is true or not can be assumed by observation in case of science. In mathematics, however, which is called the logic fully developed, we just assume axioms and postulates to go on to prove theorem. And also remember that you cannot prove anything without accepting something.  https://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma

I can only write down summary in words. For clarification, please refer to diagrams and illustrations on the textbook! (It’s much more helpful)

  • Proof method in general
    • A way of reaching conclusion by adding lines of premises according to rules of inference
    • You keep adding new lines until you reach a conclusion statement
    • You must add new lines according to rules of inference
    • Proof is finished when you reach a conclusion
  • Rules of inference
    • Modus Ponens (MP, “method for providing”)
      • If there is a line of conditional and an another line of antecedent of that conditional, you can add the consequent of the conditional
    • Two Important remarks before moving on
      •  Justification
        • You need to write down what rules you apply to what number of premises next to the new line you add
        • In other word, you have to justify the new line you add!
      • Rules of inference apply to the whole line (not component of a line)
    • Modus Tollens (MT, “method for removing”)
    • Disjunctive Syllogism
    • Simplification
    • Conjunction
    • Disjunction Introduction (DI)

Phil 10 study note-Ch.2

Chapter 2 of formal sentential logic

  • The purpose of truth function and table
    • To find the validity of the argument easier
  • Limitation
    • 1) How do we represent other complicated statements
    • 2) What if there is no row where all premises are true?
  • Answer to limitation 1)
    • Enlarge the truth table
      • (This is easy)
    • After adding the truth columns find the row where all premises are true
    • If there is any row where all premises are true, and conclusion is false, then argument is invalid
  • Answer to limitation 2)
    • Terminology to answer limitation 2)
      • Types of statements (concerns single statement)
        • tautologies
          • statement that is always true
          • applies only to compound statement
          • nothing follows from tautology
            • if you tell your roommate “I took a shower or I didn’t”, you actually informed him nothing
        • contradictions
          • statement that is always false
          • applies only to compound statement
          • everything follows from contradiction
        • contingencies
          • atomic statements are always contingent
      • Relations between statements (concerns more than 2 statements)
        • equivalency
          • truth columns of two statements are exactly the same
        • consistency
          • there exists one row where the statements are true
          • limitation 2) is when the premises are not consistent
            • how can we tell if the argument is valid?
        • implication
          • An argument is valid iff the premises jointly imply the conclusion
            • which means there is no counter-example row
    • Answer to limitation 2)
      • Argument is valid where the premises are inconsistent
      • So before, we had to find a row where premises are all true and conclusion is also true. However, in the case where there is no such row, we can now tell the argument valid if there is no counter-example row.
      • Counter-example row
        • row where premises are all true and conclusion is false
        • when there is no counter-example row the argument is valid even if there is no true where both all premises are true and conclusion is true.
        • Basically, all you need to do to figure out validity of argument by the truth table is to find a counter-example row
          • if you find one, then the argument is invalid
          • if you don’t, then the argument is valid
          • esayyyyyyyyyy
      • The conclusion is that the test for joint implication is same as the test for validity
        • if s1, s2 jointly imply s3, then s1, s2 are premises for conclusion s3 of the valid argument.

Logically valid deductive argument is defined like this. If all the premises were true, then the conclusion has to be true. It is important to note here that it doesn’t matter whether the premises are actually true or false. With ridiculously false premises, deductive argument can still be perfectly valid. 

Aha! Even perfectly valid deductive arguments do not guarantee the truth of the conclusion. It only guarantees the truth of conclusion only if the premises were true. The conclusion derived by premises must be true if and only if the argument is both valid and sound.

When Greek philosophers reason and argue, their arguments can be perfectly valid. They have good tools of reasoning. In many cases where their premises are also true, their conclusions are true too. With this tool of logic, philosophers could figure out many truths concerning natural world.

However, when it comes to spiritual world and theology, special revelation is necessay for us to know who God is. If God does not reveal himself, our premises concerning God cannot be true. 

  • Statement Operator
    • Conjunction
      • P•Q
      • conjunct and conjunction
      • logical meaning of the operator is that both conjuncts are true
        • “however, but” are also conjunction operators
      • compound subject and compound predicate
        • can express conjunction
    • Negation
      • the negated statement
    • Disjunction
    • Conditional

That means even if but and however convey different psychological import, nevertheless they have the same logical meaning and it is that both conjuncts are true.