brPage 3br Validity for argument forms Truth tables for argument forms Homework Philosophy 205 Symbolic Logic Lecture 6 Truth tables and arguments Geo64256 Pynn Northern Illinois University Spring 2012 brPage 4br Validity for argument forms Truth ta ID: 27885 Download Pdf

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brPage 3br Validity for argument forms Truth tables for argument forms Homework Philosophy 205 Symbolic Logic Lecture 6 Truth tables and arguments Geo64256 Pynn Northern Illinois University Spring 2012 brPage 4br Validity for argument forms Truth ta

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Validity for argument forms Truth tables for argument forms Homework Warm-up Create a truth table for the following formula and determine whether it is a tautology, contradiction, or contingent: (( ) & (( ))

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Validity for argument forms Truth tables for argument forms Homework Warm-up p q r (( ) & (( )) T T T T T F F T T T T T T T F T T F F T F F T T T F T F F F T T T T F F T F F F F F F T F F T F F T T T T F F F F T T T F T F T T F F F F F T T F F T T F F F F F T T F F F F T F F F F F F T F Tautology.

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Validity for argument forms Truth tables

for argument forms Homework Philosophy 205: Symbolic Logic Lecture 6: Truth tables and arguments Geoﬀ Pynn Northern Illinois University Spring 2012

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Validity for argument forms Truth tables for argument forms Homework Argument forms An argument form is a set of formulas. One of the formulas is the conclusion, and the others are the premises. Normally, the conclusion is listed last; we can also indicate that a formula is the conclusion by using the symbol Q: Why is a set of formulas an argument form , and not just an argument A: Because the statement letters are

uninterpreted Remember, we care primarily about form , not content . Once you know that a particular argument form is valid, you know that any argument with that form is valid. Deﬁnition: Interpretation ( sl An interpretation of an argument form is an assignment of a unique truth value to each statement letter in the argument form.

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Validity for argument forms Truth tables for argument forms Homework Conventions for writing argument forms Here are three diﬀerent ways of denoting the argument form consisting of the premises ( ) and and the conclusion ( ( ( It

doesn’t matter which way you use; each means the same thing.

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Validity for argument forms Truth tables for argument forms Homework Validity for argument forms An argument form is valid just in case it’s impossible for the premise formulas to be T while the conclusion formula is F. In terms of interpretations: Deﬁnition: Validity (Argument Forms) An argument form is valid if and only if there is no interpretation on which the premise formulas are T and the conclusion formula is F. Consider the following argument form: There is no interpretation on which ( ) is T but is F.

So there is no interpretation on which the premise is T and the conclusion is F. So the argument form is valid.

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Validity for argument forms Truth tables for argument forms Homework A very common valid argument form This form is called modus ponens You can see that modus ponens is valid by looking at the truth table for p q T T T T F F F T T F F T In only one row (the ﬁrst) are ( ) and both T. And, in this row, is also T. So there is no interpretation on which the premises are T but the conclusion is F.

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Validity for argument forms Truth tables for

argument forms Homework Some other simple argument forms Which of the following are valid? 1. ( Invalid 2. ( Valid 3. ( Valid 4. ( Invalid The following interpretations show that 1 and 4 are invalid: 1. [ ] = ] = 4. [ ] = ] =

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Validity for argument forms Truth tables for argument forms Homework Two unusual examples of validity First example: arguments with contradictory premises Consider the argument form: What do you notice about the premise formula? It’s a contradiction! There is no interpretation on which it is T. So there is no interpretation on which the premise is T but

the conclusion is F. Hence the argument form is valid, by our deﬁnition of validity for argument forms. The general lesson: Any argument form with a contradictory premise is valid.

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Validity for argument forms Truth tables for argument forms Homework Two unusual examples of validity Second example: arguments with tautological conclusions Now consider: ∨¬ The conclusion of this argument is a tautology: there is no interpretation on which ( ∨¬ ) is F. So there is no interpretation on which the premises are T but the conclusion is F. Hence the argument form is

valid, by our deﬁnition. The general lesson: Any argument whose conclusion is a tautology is valid.

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Validity for argument forms Truth tables for argument forms Homework More complicated forms Not all argument forms are simple. (In fact, most aren’t!) Consider the following example from your book: (( Unless you are a logic wiz, you probably can’t ﬁgure out whether this argument form is valid just by looking at it and thinking about the meanings of the connectives. So how can you tell whether this argument is valid? What are you going to do??? You’re going to make

a truth table!!!

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Validity for argument forms Truth tables for argument forms Homework Truth tables for argument forms A truth table for an argument form is just like a truth table for a formula, only instead of containing just one formula, it contains all the formulas in the argument form: p q ) p T T T T F T F T F F F F Each row represents one interpretation of the argument form. So the table gives us the truth values of all of the formulas in the argument form on every interpretation. What do you notice about the ﬁrst row? Both premise formulas are T, and the

conclusion formula is F! So the argument form is invalid.

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Validity for argument forms Truth tables for argument forms Homework Truth tables for argument forms Construction instructions You construct a truth table for an argument form just as you would a truth table for a formula. First, write down all the statement letters in all the formulas on the left: p q Then write each formula to the right of the statement letters, being sure to put space in between them. You might ﬁnd it helpful to put a ” in front of the conclusion formula, though this is not strictly

necessary: p q

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Validity for argument forms Truth tables for argument forms Homework Truth tables for argument forms Construction instructions Write the interpretations under the statement letters on the left: p q ) p T T T F F T F F Then place the corresponding truth-value under each statement letter in the formulas on each row: p q ) p T T T T T T T F T F T F F T F T F T F F F F F F

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Validity for argument forms Truth tables for argument forms Homework Truth tables for argument forms Construction instructions Then complete the table for each formula: p q ) p

T T T T F T F T F F F F And you’re done! The ﬁnal column (which I’ve indicated in red) for each formula gives the truth values for that formula on each interpretation represented by the four rows.

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Validity for argument forms Truth tables for argument forms Homework Truth tables for argument forms Using what you’ve done Now that you have the truth table, you can look at it to see whether the argument form is valid: p q ) p T T T T F T F T F F F F Remember, an argument form is valid when there are no interpretations where the premise formulas are T and the conclusion

formula is F. So in checking for validity, you are searching for rows where the premises are all T and the conclusion is F. If there is at least one such row, the argument is invalid. Otherwise, it’s valid.

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Validity for argument forms Truth tables for argument forms Homework Truth tables for argument forms Using what you’ve done So is this argument valid or invalid? p q ) p T T T T F T F T F F F F It’s invalid: the ﬁrst row’s premises are T and its conclusion is F. p q ) p T T T

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Validity for argument forms Truth tables for argument forms Homework

Truth table testing for argument form validity Summing up So, to use a truth table to determine whether an argument form is valid you: Create a truth table for the argument form. Look for a row in which each premise is T but the conclusion is F. If there is such a row, then the argument is invalid. If there is no such row, then the argument is valid. (Provided you haven’t made a mistake in creating the truth table.)

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Validity for argument forms Truth tables for argument forms Homework Truth tables for argument forms More practice Determine whether the following argument forms

are valid: 1. 2. ( (( Both are valid.

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Validity for argument forms Truth tables for argument forms Homework Homework practice Determine whether the following argument is valid: Unless I’m mistaken, I’m a fool. But if I am a fool, I must be mistaken. Therefore, I’m no fool. I am mistaken I am a fool f m ) ( T T ( ( T F ( ( F T ( ( F F ( ( Invalid.

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Validity for argument forms Truth tables for argument forms Homework Homework assignment Bonevac, section 2.8, 1-15

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