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1. Truth Trees
2. The Problem with Truth Tables The problem with standard truth tables is that they grow exponentially as the number of sentence letters grows, 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!
3. Testing Sets of Sentences for Consistency
4. 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!
5. 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.
6. 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
7. Short-Cut Truth Tables and Truth Trees Truth trees are an improved version of the short-cut truth table method for determining consistency and validity
Both methods assign truth values to whole sentences and then figure out what truth values of their components produce the assigned truth valuewe are, in effect, decomposing the sentences
Both methods test to see whether it is possible to produce a correct truth value assignment to the sentence letters that gets the assigned truth value for the whole sentences
Recall the short cut truth table test for consistency
8. Short-Cut Truth Tables: Consistency
9. Short-Cut Truth Tables: Consistency
10. Short-Cut Truth Tables: Consistency
11. Short-Cut Truth Tables: Consistency
12. Short-Cut Truth Tables: Consistency
13. Short-Cut Truth Tables: Consistency
14. Short-Cut Truth Tables: Consistency
15. Short-Cut Truth Tables: Consistency
16. Short-Cut Truth Tables: Consistency
17. Short-Cut Truth Tables: Consistency
18. 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
19. Short-Cut Truth Tables: Validity
20. Short-Cut Truth Tables: Validity
21. Short-Cut Truth Tables: Validity
22. Short-Cut Truth Tables: Validity
23. Short-Cut Truth Tables: Validity
24. Short-Cut Truth Tables: Validity
25. Short-Cut Truth Tables: Summary Short-cut truth tables are a method for constructing the rows of a regular truth table that matter
We assign truth values to whole sentences and work backward until weve assigned truth values to all their parts
But there is a problem: if truth values arent forced we may have to do moreand moreand more rows (see Hurley 6.5!) so this method only works neatly on rigged problems!
Thats why we prefer the truth tree methoda fancy version of the short-cut truth table method that works better.
26. Truth Trees In doing a truth tree we start in the same way: by assigning truth values to whole sentences and then working backward until weve assigned truth values to all sentence letters.
We do this by growing a tree-structure according to tree rules which decompose sentences into their constituent sentence letters.
The tree rules represent the ways in which the sentence forms to which they apply are made trueso, e.g.
p ? q is made true by either ps being true or qs being true
? (p ? q) is made true by ? p being true and ? q being true
27. Why? p ? q is made true by either ps being true or qs being true
In both cases where p is true, p v q is true.
In both cases where q is true, p v q is true.
28. Why? ? (p ? q) is made true by ? ps being true and ? qs being true
That is, by both p and q being false
We can construct tree rules from the characteristic truth tables for the connectives in this way!
29. Truth trees are upside down
30. Tree Rules The rule for Double Negation is: rewrite, erasing two ?s
Sentences that are basically ORs are represented as branching structures
Sentences that are basically ANDs are represented by non-branching structures.
We understand conditionals and biconditionals as basically ORs and ANDs
31. Conjunction and Disjunction p qpq To make p q true, Truth has to flow through both p and q
So we represent conjunction by a non-branching rule p v q
p q To make p v q true, all we need is truth flowing through one of its parts
So we represent disjunction by a branching rule
32. Conditional and Biconditional Conditional and biconditional are actually extras in our language: we can say everything they say just in terms of conjunction, disjunction and negation.
p ? q is equivalent to either ? p OR q so we formulate the tree rule for conditional as a branching OR rule
p ? q is equivalent to either (p AND q) OR (? p AND ? q) so we formulate the tree rule for biconditional as a branching OR rule with ANDs on both branches.
To see why this is so, consider the truth tables for conditional and for biconditional
33. Truth Tables for the Connectives p ? q is true if either p is false or q is true so its logically equivalent to ? p v q
You can prove this by testing the two sentences for equivalence!
34. Truth Tables for the Connectives p ? q is true if either both p and q are true or both p and q are false.
So its equivalent to (p q) v (? p ? q)
Note: were helping ourselves to the idea that saying p is false is the same thing as saying ? pwhich is ok given the truth table for ?
35. Conditional and Biconditional p ? qp ? pq ? q To make p ? q true, Truth has to flow through either ?p or q
? p says p is false so this says what makes p ? q is p being false or q being true p ? q
? p q p ? q says either p q or ? p ? q so its a branching rule with conjunctions on both branches
Truth has to either flow through both p and q or through both ? p and ? q
36. Negation of Conditional and Biconditional ? (p ? q) p ? p? q q ? (p ? q) says p ? q is false
What makes a conditional false is true antecedent, false consequent
So we represent this as a conjunction of p and ? q ? (p ? q)p? q ? (p ? q) says that p and q have opposite truth value
Truth has to either flow through p and ? q or through ? p and q
37. Negation of Disjunction and Conjunction ? (p v q)? p? q ? (p v q) is equivalent to ? p ? q by DeMorgans Law
So we represent ? (p ? q) by this non-branching rule ? (p q)
? p ? q ? (p q) is equivalent to ? p v ? q by DeMorgans Law
So we represent ? (p q) by this branching rule
38. Double Negation ? ? pp The double negation rule is obvious!
p is equivalent to ? ? p so, a fortiori, p makes ? ? p true.
39. How to Grow a Truth Tree We use the rules to grow the tree downward.
We apply the tree rules to each sentence successively to decompose it into simpler sentences that make it true
and we decompose those sentences into even simpler sentences
until we get down to sentences that cant be decomposed any further, that is
Sentence letters and negations of sentenceletters
Then the tree is complete.
40. Growing a Truth Tree to Test Consistency Write the sentences to be tested in a vertical column: these are the initial sentences
Were looking for a truth value assignment that will make all of them true (if there is one)
So we start by considering truth value assignments that make each of them true individually
And see if we can put them together
41. Growing a Truth Tree to Test Consistency Apply tree rules to each sentence to which they apply, checking sentences when theyve had rules applied to them
We start with non-branching rules to keep the tree from getting too big.
42. Growing a Truth Tree to Test Consistency Now we apply the rule for conditional to P ? Q writing the result at the bottom the tree
The tree stops growing because no further rules can be applied.
43. Growing a Truth Tree to Test Consistency A branch or path is the result of tracing from each sentence at the bottom of the tree all the way up to the top
There are 2 (overlapping) branches on this tree: the initial sentences are on both branches.
44. Growing a Truth Tree to Test Consistency Each branch wants to represent a truth value assignment to the initial sentences which we can read off as follows:
If a sentence letter occurs on a branch, TRUE is assigned to that sentence letter; if the negation of a sentence letter occurs, FALSE is assigned to that sentence letter.
45. Growing a Truth Tree to Test Consistency On this tree, both branches assign FALSE to P and TRUE to Q
So each branch represents the same truth value assignment, viz.
The truth value assignment represented by the row of the truth table in which all sentences got true, remember
46. Testing Sets of Sentences for Consistency
47. But what if things were different? The left branch doesnt represent a truth value assignment because it assigns both TRUE and FALSE to P!
So we say that branch is closed and indicate that by putting an X at the bottom
48. Open and Closed Trees A completed tree is open if it has at least one open branch.
A completed tree is closed if it has no open branches, i.e. if all of its branches are closed.
Consistency only requires the some (i.e. at least one) truth value assignment make all the sentences true so
If the tree is open, then the initial sentences are consistent
If the tree is closed, then the initial sentences are inconsistent
49. Summing up so far So now we can do two things:
We can determine whether a set of sentences is consistent or inconsistent
Open tree consistent
Closed tree inconsistent
And if the sentences are consistent we can determine which truth value assignment(s) makes them all true by reading the the open branch(es)
But what if a set of sentences is inconsistent?
50. But what if things were different? This tree is closed so the initial sentences are inconsistent.
There is no truth value assignment that makes all initial sentences true.
51. So what should I be able to do? Know the tree rules and how how they are derived
Be able to invent a tree rule for a symbol if given its characteristic truth table
Grow a truth tree
Determine what a completed truth tree tells you about the consistency or inconsistency of initial sentences
If the initial sentences are consistent, determine which truth value assignment makes them all true
Given a completed tree, determine what its initial sentences are.
52. Growing a Truth Tree to Test Validity Write out the argument vertically, premises first and then conclusion
The truth tree test for validity is an indirect proof method (aka reductio, proof by contradiction): we want to show that its not possible for all the premises to be true and the conclusion false.
So we ask: What if the premises were true and the conclusion were false?
53. Growing a Truth Tree to Test Validity To ask that question, we negate the conclusion, grow a tree, and see what happens.
When we test an argument for validity, we call the premises + the negation of the conclusion, the sentences above, the initial sentences.
We then test these initial sentences for consistency by growing a truth tree from them.
54. Growing a Truth Tree to Test Validity We know that:
If the premises + negation of conclusion are consistent the argument is invalid.
If the premises + negation of conclusion are inconsistent the argument is valid.
So by testing these sentences for consistency, we can determine whether the argument is valid or invalid!
55. How does this show validity or invalidity?
56. How does this show validity or invalidity?
57. How does this show validity or invalidity?
58. How does this show validity or invalidity?
59. Summing Up: Testing for Validity Using the tree method, we test for validity by testing the initial sentencespremises + negation of conclusion for consistency.
If the initial sentences are consistent the argument is invalid.
If the initial sentences are inconsistent the argument is valid.
So now lets try it!
60. Growing a Truth Tree to Test Validity Were going to test these initial sentences for consistency.
If the tree closes, theyre inconsistent, so the argument is valid.
If the tree is open, theyre consistent, so the argument is invalid.
61. Growing a Truth Tree to Test Validity
62. Growing a Truth Tree to Test Validity
63. Growing a Truth Tree to Test Validity
64. Growing a Truth Tree to Test Validity
65. Growing a Truth Tree to Test Validity
66. What would an invalid argument look like?
67. Whats the conclusion? Reading from the bottom up, we look for the first sentence which wasnt the result of applying a tree rule.
That sentence is the negation of the conclusion.
So the conclusion of this argument is ? P
68. Are the initial sentences consistent or inconsistent? The initial sentences (the premises + negation of the conclusion of the argument) are consistent.
The open path represents a truth value assignment that makes all the initial sentences true.
69. What truth value assignment makes all initial sentences true? If a sentence letter appears on an open path, that truth value assignment assigns TRUE to that sentence letter.
If the negation of a sentence letter appears, it assigns FALSE to that sentence letter
70. What truth value assignment makes all initial sentences true? So, the truth value assignment that makes all initial sentences true is
P TRUE; Q TRUE; R FALSE
71. The initial sentences are consistent
72. So the argument is invalid
73. So weve saved ourselves lots of work!