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Truth TablesHurley 6.2 - 6.4
Concepts
• Truth Function
• Truth Table
• Truth value assignment
• Tautologous
• Self-contradictory
• Contingent
• Logically equivalent
• Contradictory
• Consistent
• Inconsistent
• Valid
• Invalid
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
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
T F
F T
p q p • q p ∨ q p ⊃ q p ≡q
T T T T T T
T F F T F F
F T F T T F
F F F F T T
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
Truth Table Tests
Sentences
• Tautologous
• Self-contradictory
• Contingent
Pairs of sentences
• Equivalent
• Contradictory
• Neither
Set of sentences
• Consistent
• Inconsistent
Arguments
• Valid
• Invalid
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
Testing Sentences for Tautologousness
• Write the sentence
~ P ∨ (Q ⊃ P)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows
(for n sentence letters, 2n rows)~ P ∨ (Q ⊃ P)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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)
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.
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...
1
TF
2
T TT FF TF F
3
T T TT T FT F TT F FF T TF T FF F TF F F
4
T T T TT T T FT T F TT T F FT F T TT F T FT F F TT F F FF T T TF T T FF T F TF T F FF F T TF F T FF F F TF F F F
We don’t give yougreat big truth tableson tests because wehave to grade them!
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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 we’ve 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)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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. We’ll compute them from the truth values under P given the truth table for negation.
~ P ∨ (Q ⊃ P)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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 we’ll look at the truth values for its antecedent and consequent.
F
F
T
T
~ P ∨ (Q ⊃ P)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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’ve computed truth values for Q ⊃ P and now have what we need to compute truth values for the whole sentence we’re testing
F
F
T
T
T
T
F
T
~ P ∨ (Q ⊃ P)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows (for
n sentence letters, 2n 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)
Testing Sentences for Tautologousness
• Write the sentence• Determine the number of rows
(for n sentence letters, 2n 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)
Done! It’s a tautology!
• Write the sentence• Determine the number of rows
(for n sentence letters, 2n 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)
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
Testing Pairs of Sentences for Equivalence
• Write the sentences side by side with a slash between them. ~ (P Q) / ~ P ~ Q
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
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 “~(PQ)” is “~”, not “”!
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
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
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
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
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
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
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 we’re ready to read it to see what it tells us.
Testing Pairs of Sentences for Equivalence
• We compare the truth values in
the main columns row by row to
see whether they’re 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
• Consistent
– They can all be true together
– There is some truth value assignment that makes all of the sentences true
• Inconsistent
– Not consistent: they can’t 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 can’t all be true together
– There is no truth value assignment that makes all of the sentences true
Sets of Sentences
Testing Sets of Sentences for Consistency
Any one of them could win…but they can’t all win.
Testing Sets of Sentences for Consistency
• Do the truth table for the sentences in the usual way
• We want to see whether there’s a truth value assignment that
makes all the sentences true
• If there is, the set of sentences is consistent.
• If there isn’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
Testing Sets of Sentences for Consistency
• We read across the main columns row by row
• Each row represents a truth value assignment
• If there’s a row in which all main columns have T the set of sentences
is consistent.
• If there’s 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
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 rowshows
consistency
This rowdoesn’tshow
anything!
We talk about sets of sentences being consistent or inconsistent. We don’t talk about rows of a truth table being consistent or inconsistent--that makes no sense!
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 there’s no row where all main columns get T so this set of sentences is inconsistent!
Arguments
• Valid
– It’s 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
– It’s 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
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 there’s a truth value assignment that makes
all the premises true and the conclusion false
• If there is, the argument is invalid.
• If there isn’t, the argument is valid.
P Q / ~ Q // P ⊃ ∨ Q
T
T
F
F
F
F
T
T
T
T
F
F
T
T
F
F F
T
T
F
F
TT
T
T
F
T
T
F
F
T
T
Testing Arguments for Validity
T
T
F
F
F
F
T
T
T
T
F
F
T
T
F
F F
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 there’s 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 there’s 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
Testing Arguments for Validity
T
T
F
F
F
F
T
T
T
T
F
F
T
T
F
F F
T
T
F
F
TT
T
T
F
T
T
F
F
T
T
• The argument is invalid because there’s a row in which all premises get T and
the conclusion gets F.
• That shows that it’s possible for all the premises to be true and the conclusion
false--which means the argument is invalid.
This rowshows
invalidity
Valid or Invalid? invalid
P Q / ~ Q // P ⊃ ∨ Q
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 it’s valid or invalid?
Conclusion is a tautology
P1 / P2 / . . . Pn // C
TTTT…
Conclusion is true in every row
Must be valid Must be invalid Can be valid or invalid
Conclusion is a tautology
P1 / P2 / . . . Pn // C
TTTT…
Conclusion is true in every row
Must be valid Must be invalid Can be valid or invalid
There’s no row in which the conclusion is false so
There’s no row in which all the premises are true and the conclusion is false so
The argument must be valid.
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, that’s how we’ll prove a sentence is tautologous!
Premises are inconsistent
P1 / P2 / . . . Pn // C
Must be valid Must be invalid Can be valid or invalid
T T TThere’s NO row like this
Premises are inconsistent
P1 / P2 / . . . Pn // C
Must be valid Must be invalid Can be valid or invalid
There’s no row in which all the premises are true so
There’s no row in which all the premises are true and the conclusion is false so
The argument must be valid.
T T TSo no row like THIS F
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!
Premises + negation of conclusion inconsistent
P1 / P2 / . . . Pn // ~ C
Must be valid Must be invalid Can be valid or invalid
T T T TThere’s NO row like this
Premises + negation of conclusion inconsistent
P1 / P2 / . . . Pn // ~ C
Must be valid Must be invalid Can be valid or invalid
There’s no row in which all the premises and the negation of the conclusion are all true so
There’s 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.
T T T T FSo NO row like this
Reductio ad Absurdem
• In reductio arguments (a.k.a indirect proof, proof by contradiction) we exploit the fact that from inconsistent premises anything follows—including 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!
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 don’t show anything!
• Testing for consistency, for example, only the presence or absence of an all T row is relevant!
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 rowshows
consistency
This rowdoesn’tshow
anything!
Only the pink row matters! Is there some way we could have saved ourselves the trouble of filling in all the other rows?
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 isn’t one
• Short-cut truth tables (Hurley 6.5) do both these jobs.
• Truth trees do them better!
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
Short-cut truth tables are
assbackwards
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 that’s possible, the set of sentences is consistent
– If it’s not possible, the set of sentences is inconsistent
Short-Cut Truth Tables: Consistency
A ∨ B / B (C ⊃ ∨ A) / C B / A ⊃ ∼ ∼ T TT T
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
Short-Cut Truth Tables: Consistency
T TT T
Assign “forced” truth values.
We start with the last sentence because assigning true to the other sentences doesn’t “force” truth values on their parts.
Assign “forced” truth values.
We start with the last sentence because assigning true to the other sentences doesn’t “force” truth values on their parts.
F
Since A is ∼true, A must be
false so this truth value is “forced” on A
Since A is ∼true, A must be
false so this truth value is “forced” on A
A ∨ B / B (C ⊃ ∨ A) / C B / A ⊃ ∼ ∼
Short-Cut Truth Tables: Consistency
T TT T
Now that we’ve assigned a truth value to A, other truth values are forced by that:
All the other A’s must be false too!
Now that we’ve assigned a truth value to A, other truth values are forced by that:
All the other A’s must be false too!
FF FA ∨ B / B (C ⊃ ∨ A) / C B / A ⊃ ∼ ∼
Short-Cut Truth Tables: Consistency
T T T T
This forces more truth values:
Since A is false, to make the first sentence true we have to assign true to B—which makes all the B’s true.
This forces more truth values:
Since A is false, to make the first sentence true we have to assign true to B—which makes all the B’s true.
FF F TT TA ∨ B / B (C ⊃ ∨ A) / C B / A ⊃ ∼ ∼
Short-Cut Truth Tables: Consistency
T T T T
Since B is true, B must be false—so yet ∼another truth value is forced
Since B is true, B must be false—so yet ∼another truth value is forced
FF F TT T FA ∨ B / B (C ⊃ ∨ A) / C B / A ⊃ ∼ ∼
Short-Cut Truth Tables: Consistency
T T T T
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 C’s have to be false
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 C’s have to be false
FF F TT T FFFA ∨ B / B (C ⊃ ∨ A) / C B / A ⊃ ∼ ∼
Short-Cut Truth Tables: Consistency
T T T T
Now we can complete the truth value assignment—and there’s only one way to do it: by assigning false to C A, since both of its parts are false.
Now we can complete the truth value assignment—and there’s only one way to do it: by assigning false to C A, since both of its parts are false.
FF F TT T FFF FA ∨ B / B (C ⊃ ∨ A) / C B / A ⊃ ∼ ∼
Short-Cut Truth Tables: Consistency
T T T T
But this isn’t 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 there’s no way to avoid this since all truth values were forced!
But this isn’t 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 there’s no way to avoid this since all truth values were forced!
FF F TT T FFF FA ∨ B / B (C ⊃ ∨ A) / C B / A ⊃ ∼ ∼
Short-Cut Truth Tables: Consistency
T T T T
This shows that there’s no truth value assignment that makes all sentences true
Therefore that this set of sentences is inconsistent.
This shows that there’s no truth value assignment that makes all sentences true
Therefore that this set of sentences is inconsistent.
FF F TT T FFF FA ∨ B / B (C ⊃ ∨ A) / C B / A ⊃ ∼ ∼
Short-Cut Truth Tables: Consistency
T T T T
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!
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!
FF F TT T FTT TA ∨ B / B (C ⊃ ∨ A) / C B / A ⊃ ∼ ∼
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 that’s possible, the argument is invalid
– If it’s not possible, the argument is valid
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 it’s main connective.
We’re 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 it’s main connective.
We’re seeing if we can show invalidity.
T FT
Short-Cut Truth Tables: Validity
Making the conclusion, C A, false forces C to be true and A to be false since that’s the only case in which a conditional is false.
Making the conclusion, C A, false forces C to be true and A to be false since that’s the only case in which a conditional is false.
T FT T F∼A ⊃ (B ∨ C) / B ∼ // C A⊃
Short-Cut Truth Tables: Validity
This forces truth values on all the other C’s and A’s: all the C’s get true and and the A’s get false
This forces truth values on all the other C’s and A’s: all the C’s get true and and the A’s get false
T FT T FTF∼A ⊃ (B ∨ C) / B ∼ // C A⊃
Short-Cut Truth Tables: Validity
There are more forced truth values: since B is true, B must be false, so we assign ∼
‘F’ to all the B’s
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 assign ∼
‘F’ to all the B’s
And now that we know A is false, A must ∼be true.
T FT T FTF FT F∼A ⊃ (B ∨ C) / B ∼ // C A⊃
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 consequent—and that’s 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 consequent—and that’s ok. The other sentences are ok too.
T FT T FTF FT F T∼A ⊃ (B ∨ C) / B ∼ // C A⊃
Short-Cut Truth Tables: Validity
Since everything’s 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 everything’s 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.
T FT T FTF FT F T∼A ⊃ (B ∨ C) / B ∼ // C A⊃
So we’ve saved ourselveslots of work…
So we’ve saved ourselveslots of work…
And can go home and relax!And can go home and relax!
