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Probability Theory. Validity. A bar is obeying the law when it has the following property: If any of the patrons are below the age of 18, then that person is not drinking alcohol. Legal or Illegal?. Ignore People over 18. Make Sure No One Else Has Alcohol. Illegal !. Legal or Illegal?.
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A bar is obeying the law when it has the following property: If any of the patrons are below the age of 18, then that person is not drinking alcohol.
A bar is obeying the law when it has the following property: If any of the patrons are below the age of 18, then that person is not drinking alcohol. An argument is valid when it has the following property: If any possibility (evaluation) makes the premises true, then the conclusion is not false.
Make Sure No Remaining Possibilities Make the Conclusion False
Make Sure No Remaining Possibilities Make the Conclusion False
Inductive Arguments An inductive argument tries to show that its conclusion is supported by its premises. In other words, it tries to show that the truth of its premises makes it more likely that its conclusion will be true.
Inductive Strength The quality of an inductive argument is measured by its strength – the degree to which its premises raise the probability of its conclusion. If they don’t raise the probability very much, the argument is not very strong. If they do, the argument is strong.
Additional Evidence Even strong inductive arguments with true premises can be shown to be bad arguments with the addition of more evidence.
Inductive Syllogism In standard form: 1) Most bankers are rich. 2) Bill is a banker. 3) Bill is rich. The general form of the argument I just gave is: 1) Most X’s are Y. 2) A is an X. C) A is Y. This type of argument is called an inductive syllogism.
Evaluating Inductive Syllogisms The strength of an inductive syllogism depends primarily on the strength of the generalization. But our assessment of the argument also has to do with the amount of available evidence that has been taken into account.
Inductive Generalization The argument form of an inductive generalization is: 1) Most of the observed sample of X’s are Y. C) Most X’s are Y.
Samples Ideally, when we are trying to find out whether a large percentage of a group has a certain property, we would check every member of the group. But for a lot of groups, that’s just not possible – there are too many to check. Instead, we look at a sample, or a subset of the group.
Representative Samples The success of an inductive generalization depends on how good the match is between the sample and the entire group. If our sample of bankers is 90% rich, but bankers on a whole are only 30% rich, our argument will not be a good one.
Sentential Logic: Vocabulary Sentence letters: A, B, C,… Logical connectives: ~, &, v, →, ↔ Punctuation: ), (
Sentential Logic: Grammar • All sentence letters are WFFs. • If φ is a WFF, then ~φ is a WFF. • If φ and ψ are WFFs, then (φ & ψ), (φ v ψ), (φ → ψ), (φ ↔ ψ) are also WFFs. • Nothing else is a WFF.
Sentential Logic: Examples • (Q & R) • ((Q & R) v P) • ~((Q & ~R) v P) • (S → ~((Q & ~R) v P)) • ~((~P ↔ S) → ~((Q & ~R) v P))
Sentential Logic: Outside Parentheses • (Q & R) • ((Q & R) v P) • ~((Q & ~R) v P) • (S → ~((Q & ~R) v P)) • ~((~P ↔ S) → ~((Q & ~R) v P))
Convention: Omit Outside Parentheses • Q & R • (Q & R) v P • ~((Q & ~R) v P) • S → ~((Q & ~R) v P) • ~((~P ↔ S) → ~((Q & ~R) v P))
Note on Negation Can’t omit parentheses when negation is main connective. Example: 1. ~((Q & ~R) v P) • Is false whenever P is true. But (2) is true whenever P is true: 2. ~(Q & ~R) v P
Probability Theory: Vocabulary Sentence letters: A, B, C,… Logical connectives: ~, &, v Punctuation: ), ( Real numbers between 0 and 1: 0.5, 0.362, π/4… Probability function symbol: Pr Equality sign: =
Functions A function is a relation between the members of two sets X and Y, called its domain and its range. The function takes each member of its domain and relates it to exactly one member of its range.
Numerical Functions Most functions that you’ve learned about are numerical functions: their domains are numbers or pairs of numbers, and their range is also numbers. Addition: +: 2,2 4 +: 16,9 25 +: 9,16 25 +: 0.5,0.25 0.75 +: 7,-18 -11
Truth-Functions But we also learned about truth-functions in class too. ~:T F ~:F T &:T,T T &:T,F F &:F,T F &:F,F F
Pr “Pr” is the symbol for a probability function. It’s a function from SL formulas to real numbers in [0, 1]. There are lots of probability functions, but they all follow these rules: Rule 1: 0 ≤ Pr(φ) ≤ 1 Rule 2: If φ is a tautology, then Pr(φ) = 1 Rule 3: If φ and ψare mutually exclusive, then Pr(φ v ψ) = Pr(φ) +Pr(ψ)
Probabilities of Complex Sentences Just as in Sentential Logic, where you can calculate the truth-values of complex sentences if you know the truth-values of their parts, we can calculate the probabilities of complex sentences from the probabilities of their parts.
Probability of Negation If you know the Pr(φ), you can calculate Pr(~φ): Pr(~φ) =1 – Pr(φ) If the probability that it will rain tomorrow is 20%, then the probability that it will not rain is 1 – 20% = 80%. If the probability that the die will land 4 is 1/6, then the probability that it will not land 4 is 1 – 1/6 = 5/6.
Probability of Disjunction There are two ways we need to use to calculate the probability of a disjunction (φ v ψ). The first way we use if φ and ψ are mutually exclusive: they can’t both be true together. Then the rule is: Pr(φ v ψ) = Pr(φ) + Pr(ψ)
Examples of Mutually Exclusive Possibilities When you roll a fair 6-sided die, the probability it will land on any one side is 1/6. It cannot land on two sides at the same time.
Examples of Mutually Exclusive Possibilities Landing 4 on one roll and landing 6 on the same roll are mutually exclusive possibilities. So the probability that on one roll it will land 4 or it will land 6 is 1/6 + 1/6 or 2/6.
