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Philosophy 150 – Day 12 Using Truth Tables: Part 1

Philosophy 150 – Day 12 Using Truth Tables: Part 1. Copyright 2007 Makoto Suzuki The Figure in the Picture: Ludwig Wittgenstein (1848-1925), an Austrian philosopher, one of the two who have developed the truth table method. Aims. Discuss the uses of truth tables for:

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Philosophy 150 – Day 12 Using Truth Tables: Part 1

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  1. Philosophy 150 – Day 12Using Truth Tables: Part 1 • Copyright 2007 Makoto Suzuki The Figure in the Picture: Ludwig Wittgenstein (1848-1925), an Austrian philosopher, one of the two who have developed the truth table method.

  2. Aims • Discuss the uses of truth tables for: • Showing logical equivalence • Determining the logical status of sentences

  3. Logical Equivalence andTruth Tables • By definition, two sentences are logically equivalent if and only if they take the same truth values under all interpretations of their atoms. • We can use truth tables to show that two sentences are logically equivalent.

  4. Showing Logical Equivalence:Sample 1. p  q; (p ● q)  (p ● q) • Create the truth table of the following two sentences: • Makoto is nerdy if and only if Makoto is weird. • Makoto is nerdy and weird, or Makoto is neither nerdy nor weird. • The first sentence is symbolized p  q, and the second sentence (p ● q)  (p ● q). p q p q p  q p ● q p ● q (p ● q)  (p ● q) T T F F T T F T T F F T F F F F F T T F F F F F F F T T T F T T • Compare the column of p  q and the column of (p ● q)  (p ● q); you will see p  q and (p ● q)  (p ● q) have the same truth values under all the interpretations of their atoms.

  5. Material Equivalence • The logical equivalence of p  q with (p ● q)  (p ● q), or of p  q with (p  q) ● (q  p), is called material equivalence. • The logical equivalence of p  q with (p  q) ● (q  p) means that, for example, the following two sentences are logically equivalent: • Makoto is nerdy if and only if Makoto is weird. • Both if Makoto is nerdy then Makoto is weird, and if Makoto is weird then Makoto is nerdy. • The logical equivalence of p  q with (p  q) ● (q  p) is shown on p. 51 of the course packet.

  6. Showing Logical Equivalence:Sample 2. Associativity • Last time we studied the associativity of disjunctions, conjunctions and biconditionals: that is, • (p  q)  r is logically equivalent to p  (q  r) • (p ● q) ● r is logically equivalent to p ● (q ● r); and, • (p  q)  r is logically equivalent to p  (q  r). • The associativity of disjunctions is shown on p. 52 of the course packet. • Let’s show the associativity of conjunctions.

  7. Showing Logical Equivalence:Sample 2. (p ● q) ● r; p ● (q ● r) p q r p ● q q ● r (p ● q) ● r p ● (q ● r) T T T T T TT T T F T F FF T F T F F FF T F F F F FF F T T F T FF F T F F F FF F F T F F FF F F F F F FF • About the association of biconditionals, I encourage you to show it for yourselves.

  8. Showing Logical Equivalence:Sample 3. Commutativity • p  q is logically equivalent to q  p. • Ex.: “Makoto is nerdy or weird” is logically equivalent to “Makoto is weird or nerdy.” • p ● q is logically equivalent to q ● p. • Ex.: “Makoto is nerdy and weird” is logically equivalent to “Makoto is weird and nerdy.” • Make truth tables to show their logical equivalence.

  9. Sample 4: Distribution • Distribution: • p ● (q r) is logically equivalent to (p ● q)  (p ● r). Ex.: the sentences below are logically equivalent: “The class is both bored and either asleep or doing a side work.” “Either the class is both bored and asleep, or the class is both bored and doing a side work.” • p  (q ● r) is logically equivalent to (p  q) ● (p  r). Ex.: the sentences below are logically equivalent: “Makoto likes either baseball or both football and basketball.” “Makoto likes baseball or football, and Makoto likes baseball or basketball.” • Make truth tables to show their logical equivalence..

  10. Showing Non-Equivalence: Non-Commutativity of Conditionals • Conditionals are neither associative nor commutative: (NA) (p  q)  r is NOT logically equivalent to p  (q  r); and, (NC) p  q is NOT logically equivalent to q  p. Ex.: “If Makoto is nerdy, Makoto is weird” is not logically equivalent to “If Makoto is weird, Makoto is nerdy.” • Last time I have shown (NA). Let me now show (NC) by making a truth table.

  11. Truth Table to Show the Non-Commutativity of Conditionals p q p  q q  p T T T T T F F T F T T F F F T T This truth table shows that under two interpretations, p  q and q  p have different truth values. Hence, they are not logically equivalent.

  12. Important Logical Equivalences Concerning the Conditional: Part 1 • However, there are interesting logical equivalences concerning conditionals. • Material Implication: p  qis logically equivalent to p  q. • Ex.: the two sentences below are logically equivalent: “If Makoto is nerdy, Makoto is weird.” “Either Makoto is not nerdy or Makoto is weird.” • Transposition: p  q is logically equivalent to q p. • Ex.: the two sentences below are logically equivalent: “If Makoto is nerdy, then Makoto is weird. “If Makoto is not weird, then Makoto is not nerdy.”

  13. Showing Logical Equivalence:Sample 3. Material Implication p q p  q p p  q T T T F T T F F F F F T T T T F F T T T • Material implication shows the nature of material conditional: it is true if and only if either the antecedent is false (i.e., its negation is true) or the consequent is true.

  14. Showing Logical Equivalence:Sample 4. Transposition p q p  q q p q  p T T T F F T T F F T F F F T T F T T F F T T T T • Many of you seem puzzled why we should symbolize the form of English sentences “A only if B” as A  B, not B  A. • Transposition (together with the non-commutability of conditionals) explains this. Let me show how it does.

  15. Important Logical Equivalences Involving the Negation • There are also logical equivalences involving the negation. • Double Negation: • p is logically equivalent to p. • De Morgan’s Laws: • 1. p ● q is logically equivalent to (p  q). • This is the reason why we can symbolize “Makoto is neither popular nor cool” in either way. • 2. (p ● q) is logically equivalent to p  q. • 3. p ● q is logically equivalent to (p  q). • 4. p  q is logically equivalent to (p ● q). • In Exercise, you are asked to show the third of De Morgan’s Laws. I will in effect show the first later. I again encourage you to show the rest for yourselves.

  16. The Figure in the Picture: Augustus De Morgan (1806-1871), a logician who formulated what we call De Morgan’s Laws. Exercise 1 of Day 12

  17. The Logical Status of Sentences:Tautology, Self-Contradiction & Contingency • Some compound sentences are true (or false) on the basis of their truth-functional structure alone. • This means that they are true (or false) under every interpretation of their atoms. • A sentence that is true under every interpretation of its atoms is called a tautology. • A sentence that is false under every interpretation of its atoms is called a (self-)contradiction. • A sentence that is neither a tautology nor a self-contradiction is called a contingency or a contingent sentence. • These logical statuses of sentences (tautology, self-contradiction or contingency) are mutually exclusive.

  18. The Logical Status of Sentences: Continued • Tautologies and self-contradictions are respectively true or false under every interpretation of their atoms. • This means that the truth values of tautologies and self- contradictions do not depend on the actual states of affairs in the world. • In other words, we do not have to perform empirical or scientific studies to determine their truth or falsity. We can ascertain their truth values with logical ingenuity alone. • Ex.: “If it is snowing, then it is snowing” has a form A  A and is a tautology. • Its truth does not depend on what the weather is, so we do not have to gather that empirical information. • The truth values of contingent sentences depend on the actual states of affairs. We need to perform empirical or scientific studies to determine their truth or falsity.

  19. Using Truth Tables to Determine the Logical Status of a Sentence • Truth tables can be used to determine the logical status of a sentence. • To do so, just construct a truth table for the sentence under question. • The sentence is a tautology if and only if the column below it consists entirelyof Ts. • The sentence is a self-contradiction if and only if the column below it consists entirely of Fs. • The sentence is contingent if and only if both T and F are present in the column below that.

  20. A Truth Table of a Tautology:(p ● q)  (p  q) p q p q p ● q p  q (p  q) (p ● q)  (p  q) T T F F F T FT T F F T F T F T F T T F F T FT F F T T T F TT • In addition to the last column, look at the columns under p ● q and (p  q). They tell us that these two sentences are logically equivalent: thus, this truth table also shows the first of De Morgan’s Laws.

  21. A Truth Table of a Contingency:((p q) ● p)  q p q p q p  q ((p  q) ● q) ((p q) ● p)  q T T F F T FT T F F T F FT F T T F T TF F F T T T TT

  22. A Truth Table of a Self-Contradiction:(((p q) ● p)  q) p q p  q (p  q) ● p ((p q) ● p)  q ((p q) ● p)  q T T T T TF T F F F TF F T T F TF F F T F TF • In addition to the last column, look at the column below ((p q) ● p)  q. This sentence is a tautology because its column consists entirely of Ts.

  23. Exercise 2 of Day 12

  24. 2nd Midterm (on Wednesday) • Part 1: Argument Form For Each of Five Arguments, • Construct Dictionaries • Symbolize Arguments • Figure Out the Form of the Arguments (Modus Ponens, Disjunctive Syllogism etc.) • State whether the Form is valid • Part 2: Truth Tables • Determine the Logical Status of Sentences (2Q) • Determine whether Pairs of Sentences are Logically Equivalent (2Q) • Determine whether Argument Forms are valid (2Q)

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