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Computing Truth Value

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Argument A T All cats are mammals. T No cats are reptiles. T Therefore, no reptiles are mammals. Argument B T All San Diegans are Californians. T No San Diegans are San Franciscans. F Therefore no San Franciscans are Californians. Argument C F All even numbers are primes. F No primes are odd numbers. T Therefore, no odd numbers are even numbers.

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1Validity is a necessary condition for soundness. a. V S; b. V S; c. V S; d. S V; e. S V 2Having true premises is neither necessary nor sufficient for validity. a. [(T V) (V T)]; b. [(T V) (V T)]; c. (T V); d. T V; e. [(T V) (V T)] 3An argument is valid only if its deductive. a. V D; b. V D; c. D V; d. all of the above; e. none of the above. 4You cant both have your cake and eat it. a. H E; b. H E; c. (H E); d. all of the above; e. none of the above. 5If x is an integer then x is either even or odd but not both. a. I (E O); b. I [(E O) (E O)]; c. I [(E O) B] d. all of the above; e. none of the above.

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Truth Tables Hurley 6.2 - 6.4

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Concepts Truth Function Truth Table Truth value assignment Tautologous Self-contradictory Contingent Logically equivalent Contradictory Consistent Inconsistent Valid Invalid

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How to… Use truth tables to test –Sentences for tautolgousness, self-contradiction or contingency –Pairs of sentences for equivalence or contradiction –Sets of sentences for consistency or inconsistency –Arguments for validity or invalidity Determine validity and invalidity from information about premises and conclusion

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Truth Tables for the Connectives Given the truth tables for the connectives we can compute the truth value of sentences built out of them if we know the truth values of their parts. We can do this because the connectives are truth functional! p~p TF FT pqp q TTTTTT TFFTFF FTFTTF FFFFTT

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Truth Value Assigment Each row of a truth table represents a truth value assignment: an assignment of truth values to the sentence letters. So, in the exercise where you were given truth values for the sentence letters and asked to compute the truth value of the whole sentence the directions gave you a truth value assignment. We can think of truth value assignments as a possible worlds (or really sets of possible worlds) And a complete truth table as representing all possible worlds

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Truth Table Tests Sentences Tautologous Self-contradictory Contingent Pairs of sentences Equivalent Contradictory Neither Set of sentences Consistent Inconsistent Arguments Valid Invalid

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Sentences Tautology (tautologous sentence) –Necessarily true –True in every truth value assignment Self-contradictory sentence –Necessarily false –False in every truth value assignment Contingent sentence –Neither necessarily true nor necessarily false –True in some truth value assignments, false in others Tautology (tautologous sentence) –Necessarily true –True in every truth value assignment Self-contradictory sentence –Necessarily false –False in every truth value assignment Contingent sentence –Neither necessarily true nor necessarily false –True in some truth value assignments, false in others

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Testing Sentences for Tautologousness Write the sentence ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Now we need to assign truth values to each sentence letter on each row of the column underneath it. We assign these truth values according to a standard pattern. ~P (Q P)

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Why? And what pattern? We want the truth table to display all possible truth value assignments for the sentence letters without duplicating any, so we adopt a convention to guarantee that. The column under the first sentence letter gets half true, half false; the column under the second sentence letter has half true, half false for rows where the first is true and half true, half false for rows where the first is false; the column under the third subdivides in the same way, and so on.

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Etc… The column for the first type letter is half T and half F, the second subdivides that, the third subdivides the second, and so on... 1TF1TF 2 T T T F F T F F 3 T T T T T F T F T T F F F T T F T F F F T F F F 4 T T T T T T T F T T F T T T F F T F T T T F T F T F F T T F F F F T T T F T T F F T F T F T F F F F T T F F T F F F F T F F F F We dont give you great big truth tables on tests because we have to grade them!

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. T T F F ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. T T F F T T F F ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. T T F F T T F F T F T F Now weve assigned truth values to all the sentence letters and are ready to compute truth values for the whole sentence working from smaller to larger subformulas. ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. Compute truth values T T F F T T F F T F T F We want truth values for ~ P in the column under its main connective. Well compute them from the truth values under P given the truth table for negation. ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. Compute truth values T T F F T T F F T F T F Got it! The truth values for ~ P are in the column under its main connective. F F T T ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. Compute truth values T T F F T T F F T F T F Now we want to computer truth values for Q P so well look at the truth values for its antecedent and consequent. F F T T ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. Compute truth values T T F F T T F F T F T F Weve computed truth values for Q P and now have what we need to compute truth values for the whole sentence were testing F F T T T T F T ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. Compute truth values T T F F T T F F T F T F At last we can compute truth values for ~P (Q P)! To do that we look at the truth values for ~P and Q P, which are under their main connectives. F F T T T T F T ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. Compute truth values T T F F T T F F T F T F Now we have truth values for ~P (Q P) in the main column of the truth table--the boxed column under the main connective. F F T T T T F T T T T T ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. Compute truth values Read down the main column T T F F T T F F T F T F The truth table is complete! Now we just have to read down the main column to determine whether the sentence is tautologous, self-contradictory or contingent. F F T T T T F T T T T T ~P (Q P)

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Testing Sentences for Tautologousness Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. Compute truth values Read down main column: –Tautologous: all T –Self-contradictory: all F –Contingent: neither all T nor all F T T F F T T F F T F T F F F T T T T F T T T T T ~P (Q P)

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Done! Its a tautology! Write the sentence Determine the number of rows (for n sentence letters, 2 n rows) Identify the main connective and box the column underneath it Assign truth values to sentence letters according to pattern, duplicating columns under same sentence letters. Compute truth values Read down main column: –Tautologous: all T –Self-contradictory: all F –Contingent: neither all T nor all F T T F F T T F F T F T F F F T T T T F T T T T T Tautologous, self-contradictory or contingent? Tautologous ~P (Q P)

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Pairs of Sentences Equivalent –Necessarily have same truth value –Have same truth value in every truth value assignment Contradictory –Necessarily have opposite truth value –Have opposite truth value in every truth value assignment Neither –Neither equivalent nor contradictory Equivalent –Necessarily have same truth value –Have same truth value in every truth value assignment Contradictory –Necessarily have opposite truth value –Have opposite truth value in every truth value assignment Neither –Neither equivalent nor contradictory

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Testing Pairs of Sentences for Equivalence Write the sentences side by side with a slash between them. ~ (P Q) / ~ P ~ Q

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Testing Pairs of Sentences for Equivalence Write the sentences side by side with a slash between them. Determine the number of rows for number of sentence letters in both sentences. ~ ( P Q ) / ~ P ~ Q

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Testing Pairs of Sentences for Equivalence Write the sentences side by side with a slash between them. Determine the number of rows for sentence letters in both sentences. Identify the main connectives and box the columns underneath them. ~ ( P Q ) / ~ P ~ Q Be careful about identifying main connectives! The main connective of ~(P Q) is ~, not!

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Testing Pairs of Sentences for Equivalence Write the sentences side by side with a slash between them. Determine the number of rows for number of sentence letters in both sentences. Identify the main connectives and box the columns underneath them. Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. ~ ( P Q ) / ~ P ~ Q T T F F T T F F T T F F T T F F

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Testing Pairs of Sentences for Equivalence Write the sentences side by side with a slash between them. Determine the number of rows for number of sentence letters in both sentences. Identify the main connectives and box the columns underneath them. Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. Compute. ~ ( P Q ) / ~ P ~ Q T T F F T T F F T T F F T T F F T T T F

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Testing Pairs of Sentences for Equivalence Write the sentences side by side with a slash between them. Determine the number of rows for number of sentence letters in both sentences. Identify the main connectives and the columns underneath them. Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. Compute. ~ ( P Q ) / ~ P ~ Q T T F F T T F F T T F F T T F F T T T F F F F T

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Testing Pairs of Sentences for Equivalence Write the sentences side by side with a slash between them. Determine the number of rows for number of sentence letters in both sentences. Identify the main connectives and the columns underneath them. Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. Compute. ~ ( P Q ) / ~ P ~ Q T T F F T T F F T T F F T T F F T T T F F F F T F F T T

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Testing Pairs of Sentences for Equivalence Write the sentences side by side with a slash between them. Determine the number of rows for number of sentence letters in both sentences. Identify the main connectives and the columns underneath them. Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. Compute. ~ ( P Q ) / ~ P ~ Q T T F F T T F F T T F F T T F F T T T F F F F T F F T T T F F T

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Testing Pairs of Sentences for Equivalence Write the sentences side by side with a slash between them. Determine the number of rows for number of sentence letters in both sentences. Identify the main connectives and the columns underneath them. Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. Compute. ~ ( P Q ) / ~ P ~ Q T T F F T T F F T T F F T T F F T T T F F F F T F F T T T F F T F F F T

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Testing Pairs of Sentences for Equivalence Write the sentences side by side with a slash between them. Determine the number of rows for number of sentence letters in both sentences. Identify the main connectives and the columns underneath them. Assign truth values according to pattern, duplicating columns under same sentence letters for both sentences. Compute. Now read down the main columns row by row to determine whether the sentences are equivalent, contradictory or neither. ~ ( P Q ) / ~ P ~ Q T T F F T T F F T T F F T T F F T T T F F F F T F F T T T F F T F F F T The truth table is complete and were ready to read it to see what it tells us.

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Testing Pairs of Sentences for Equivalence We compare the truth values in the main columns row by row to see whether theyre same or opposite We determine whether the sentences are equivalent, contradictory or neither as follows: –Equivalent: same in every row –Contradictory: opposite in every row –Neither: neither equivalent nor contradictory ~ ( P Q ) / ~ P ~ Q T T F F T T F F T T F F T T F F T T T F F F F T F F T T T F F T F F F T Equivalent, contradictory or neither? Equivalent

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Consistent –They can all be true together –There is some truth value assignment that makes all of the sentences true Inconsistent –Not consistent: they cant all be true together –There is no truth value assignment that makes all of the sentences true Consistent –They can all be true together –There is some truth value assignment that makes all of the sentences true Inconsistent –Not consistent: they cant all be true together –There is no truth value assignment that makes all of the sentences true Sets of Sentences

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Testing Sets of Sentences for Consistency Any one of them could win…but they cant all win.

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Testing Sets of Sentences for Consistency Do the truth table for the sentences in the usual way We want to see whether theres a truth value assignment that makes all the sentences true If there is, the set of sentences is consistent. If there isnt, the set of sentences is inconsistent. P Q / ~ P Q / Q T T F F F T F T T T F F T T F F F T T F F T T F F T F F T T T F

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Testing Sets of Sentences for Consistency We read across the main columns row by row Each row represents a truth value assignment If theres a row in which all main columns have T the set of sentences is consistent. If theres no row in which all main columns have T the set of sentences is inconsistent. P Q / ~ P Q / Q T T F F F T F T T T F F T T F F F T T F F T T F F T F F T T T F

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Testing Sets of Sentences for Consistency P Q / ~ P Q / Q T T F F F T F T T T F F T T F F F T T F F T T F F T F F T T T F Consistent or inconsistent? Consistent This row shows consistency This row doesnt show anything! We talk about sets of sentences being consistent or inconsistent. We dont talk about rows of a truth table being consistent or inconsistent--that makes no sense!

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Testing Sets of Sentences for Consistency P Q / P ~ Q / Q T T F F F T F T T T F F T T F F T T T F F T T F F F F F T T T F Consistent or inconsistent? Inconsistent Suppose things were a little different… Now theres no row where all main columns get T so this set of sentences is inconsistent!

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Arguments Valid –Its not logically possible for all the premises to be true and the conclusion false –There is no truth value assignment that makes all the premises true and the conclusion false. Invalid –Not valid. –There is some truth value assignment that makes all the premises true and the conclusion false Valid –Its not logically possible for all the premises to be true and the conclusion false –There is no truth value assignment that makes all the premises true and the conclusion false. Invalid –Not valid. –There is some truth value assignment that makes all the premises true and the conclusion false

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Testing Arguments for Validity Do the truth table with slashes between premises and a double slash between the last premise and the conclusion We want to see whether theres a truth value assignment that makes all the premises true and the conclusion false If there is, the argument is invalid. If there isnt, the argument is valid. P Q / ~ Q // P Q T T F F F F T T T T F F T T F FF T T F F TT T T F T T F F T T

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Testing Arguments for Validity T T F F F F T T T T F F T T F FF T T F F TT T T F T T F F T T We read across the main columns row by row Each row represents a truth value assignment If theres a row in which the main columns of all premises have T and the main column of the conclusion has F the argument is invalid. If theres no row in which the main columns of all premises have T and the main column of the conclusion has F the argument is valid. P Q / ~ Q // P Q

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Testing Arguments for Validity T T F F F F T T T T F F T T F FF T T F F T T T T F T T F F T T The argument is invalid because theres a row in which all premises get T and the conclusion gets F. That shows that its possible for all the premises to be true and the conclusion false--which means the argument is invalid. This row shows invalidity Valid or Invalid? invalid P Q / ~ Q // P Q

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Validity Given certain information about premises and conclusion we can sometimes determine whether an argument is valid or invalid. Suppose the conclusion of an argument is a tautology: does this show the argument is valid, is invalid or is this not enough information to determine whether its valid or invalid?

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Conclusion is a tautology P 1 / P 2 /... P n // C TTTT…TTTT… Conclusion is true in every row Must be validMust be invalidCan be valid or invalid

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Conclusion is a tautology P 1 / P 2 /... P n // C TTTT…TTTT… Conclusion is true in every row Must be validMust be invalidCan be valid or invalid Theres no row in which the conclusion is false so Theres no row in which all the premises are true and the conclusion is false so The argument must be valid.

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A tautology follows from anything We can prove a tautology from any set of premises even if they have nothing to do with the tautology and Even from the empty set of premises, i.e. a tautology can be proved from nothing at all! And in doing proofs, thats how well prove a sentence is tautologous!

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Premises are inconsistent P 1 / P 2 /... P n // C Must be validMust be invalidCan be valid or invalid TT T Theres NO row like this

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Premises are inconsistent P 1 / P 2 /... P n // C Must be validMust be invalidCan be valid or invalid Theres no row in which all the premises are true so Theres no row in which all the premises are true and the conclusion is false so The argument must be valid. TT T So no row like THIS F

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Ex contradictione quod libet! Translation: From a contradiction anything follows. If the premises are inconsistent then anything can be proved from them! This means that if a formal system includes an inconsistency we can prove any darn thing and that is BAD!

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Premises + negation of conclusion inconsistent P 1 / P 2 /... P n // ~ C Must be validMust be invalidCan be valid or invalid TT T T Theres NO row like this

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Premises + negation of conclusion inconsistent P 1 / P 2 /... P n // ~ C Must be validMust be invalidCan be valid or invalid Theres no row in which all the premises and the negation of the conclusion are all true so Theres no row in which all the premises are true and the conclusion itself is false (by definition of negation!) so The argument must be valid. TT T T F So NO row like this

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Reductio ad Absurdem In reductio arguments (a.k.a indirect proof, proof by contradiction) we exploit the fact that from inconsistent premises anything followsincluding a contradiction. We show that the premises + negation of conclusion of an argument are inconsistent by deriving a contradiction from them And hence that the argument is valid!

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The Problem with Truth Tables The problem with standard truth tables is that they grow exponentially as the number of sentence letters increases, so… Most of our work is wasted because most of the Ts and Fs we plug in dont show anything! Testing for consistency, for example, only the presence or absence of an all T row is relevant!

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Testing Sets of Sentences for Consistency P Q / P Q / Q T T F F F T F T T T F F T T F F F T T F F T T F F T F F T T T F Consistent or inconsistent? Consistent This row shows consistency This row doesnt show anything ! Only the pink row matters! Is there some way we could have saved ourselves the trouble of filling in all the other rows?

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What we need To short-cut the truth table test for consistency we need a procedure that will do two things: –Construct a truth value assignment in which all sentences are true, if there is one and –Show conclusively that there is no truth value assignment that makes all sentences true if there isnt one Short-cut truth tables (Hurley 6.5) do both these jobs. Truth trees do them better!

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Short-cut Truth Tables Short-cut truth tables provide a quick and dirty way of testing for consistency and validity. Instead of assigning truth values to sentence letters and calculating the truth value of whole sentences from there We assign truth values to whole sentences and attempt to construct a truth value assignment that will produce that result. Short-cut truth tables are assbackwards

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Short-Cut Truth Tables: Consistency A set of sentences is consistent if there is some truth value assignment that makes all the sentences true To test for consistency we write the sentences on a single line with slashes between them We assign true to each of the sentences by writing T under its main connective And attempt to construct a truth value assignment that gets that result –If thats possible, the set of sentences is consistent –If its not possible, the set of sentences is inconsistent

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Short-Cut Truth Tables: Consistency A B / B (C A) / C B / A TTTT Write the sentences on one line with slashes between them Assign true to each sentence by writing T under its main connective Write the sentences on one line with slashes between them Assign true to each sentence by writing T under its main connective

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Short-Cut Truth Tables: Consistency TTTT Assign forced truth values. We start with the last sentence because assigning true to the other sentences doesnt force truth values on their parts. Assign forced truth values. We start with the last sentence because assigning true to the other sentences doesnt force truth values on their parts. F Since A is true, A must be false so this truth value isforced on A A B / B (C A) / C B / A

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Short-Cut Truth Tables: Consistency TTTT Now that weve assigned a truth value to A, other truth values are forced by that: All the other As must be false too! Now that weve assigned a truth value to A, other truth values are forced by that: All the other As must be false too! F FF A B / B (C A) / C B / A

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Short-Cut Truth Tables: Consistency TTTT This forces more truth values: Since A is false, to make the first sentence true we have to assign true to Bwhich makes all the Bs true. This forces more truth values: Since A is false, to make the first sentence true we have to assign true to Bwhich makes all the Bs true. F FF T T T A B / B (C A) / C B / A

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Short-Cut Truth Tables: Consistency TTT T Since B is true, B must be falseso yet another truth value is forced Since B is true, B must be falseso yet another truth value is forced F FF T T T F A B / B (C A) / C B / A

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Short-Cut Truth Tables: Consistency TTTT Since B is false, C must be false in order to make the conditional, C B, true--so we have another forced truth value: all Cs have to be false F FF T T T F F F A B / B (C A) / C B / A

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Short-Cut Truth Tables: Consistency TTTT Now we can complete the truth value assignmentand theres only one way to do it: by assigning false to C A, since both of its parts are false. F FF T T T F F FF A B / B (C A) / C B / A

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Short-Cut Truth Tables: Consistency TTTT But this isnt a possible truth value assignment because it says that the conditional, B (C A), is true even though its antecedent is true and its consequent false. And theres no way to avoid this since all truth values were forced! But this isnt a possible truth value assignment because it says that the conditional, B (C A), is true even though its antecedent is true and its consequent false. And theres no way to avoid this since all truth values were forced! F FF T T T F F FF A B / B (C A) / C B / A

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Short-Cut Truth Tables: Consistency TTTT This shows that theres no truth value assignment that makes all sentences true Therefore that this set of sentences is inconsistent. This shows that theres no truth value assignment that makes all sentences true Therefore that this set of sentences is inconsistent. F FF T T T F F FF A B / B (C A) / C B / A

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Short-Cut Truth Tables: Consistency TTTT Note: if you assigned truth values in a different order the problem will pop up in a different place (see Hurley p. 40)but it will pop up somewhere, like a lump under the carpet! F FF T T T F T TT A B / B (C A) / C B / A

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Short-Cut Truth Tables: Validity An argument is valid if there is no truth value assignment that makes all its premises true and its conclusion false. To test for validity we write the argument on a single line with slashes between the premises and a double slash between the last premise and the conclusion We assign true to each of the premises by writing T under its main connective, and false to the conclusion by writing F under its main connective And attempt to construct a truth value assignment that gets that result –If thats possible, the argument is invalid –If its not possible, the argument is valid

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Short-Cut Truth Tables: Validity A (B C) / B // C A We assign true to each of the premises by writing T under its main connective and assign false to the conclusion by writing F under its main connective. Were seeing if we can show invalidity. We assign true to each of the premises by writing T under its main connective and assign false to the conclusion by writing F under its main connective. Were seeing if we can show invalidity. TF T

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Short-Cut Truth Tables: Validity Making the conclusion, C A, false forces C to be true and A to be false since thats the only case in which a conditional is false. TF TT F A (B C) / B // C A

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Short-Cut Truth Tables: Validity This forces truth values on all the other Cs and As: all the Cs get true and and the As get false TF TT F TF A (B C) / B // C A

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Short-Cut Truth Tables: Validity There are more forced truth values: since B is true, B must be false, so we assignF to all the Bs And now that we know A is false, A must be true. There are more forced truth values: since B is true, B must be false, so we assignF to all the Bs And now that we know A is false, A must be true. TF TT F TFFT F A (B C) / B // C A

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Short-Cut Truth Tables: Validity Now we can complete the table by filling in the truth value for the first premise. So the first premise is a true conditional with a true antecedent and true consequentand thats ok. The other sentences are ok too. Now we can complete the table by filling in the truth value for the first premise. So the first premise is a true conditional with a true antecedent and true consequentand thats ok. The other sentences are ok too. TF TT F TFFT F T A (B C) / B // C A

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Short-Cut Truth Tables: Validity Since everythings ok, this is a possible truth value assignment Since this truth value assignment makes all the premises true and the conclusion false the argument is shown to be invalid. Since everythings ok, this is a possible truth value assignment Since this truth value assignment makes all the premises true and the conclusion false the argument is shown to be invalid. TF TT F TFFT F T A (B C) / B // C A

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So weve saved ourselves lots of work… And can go home and relax!

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