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Today’s Topics
Review of Grouping and Statement Forms Truth Functions and Truth Tables Uses for Truth Tables Truth Tables and Validity
Putting Words Into Symbols
Statements are either simple (represented by a statement letter) or compound.
A compound statement is any statement containing at least one connective
In our language a Capital letter stands for an entire simple statement. A dictionary is used to indicate which letters stand for which statements.
When Symbolizing an English Sentence, Identify the Dominant Operator First, and Group AWAY from it. Paraphrasing Inward Identify the statement forms of the
component sentence(s) and repeat
How paraphrasing inward works:
If Jones wins the nomination or Dexter leaves the party, then Williams is the sure winner. (J, D, W where J = Jones wins the nomination, D = Dexter leaves the party, W=Williams wins).
The sentence is a conditional, so begin by identifying the antecedent and consequent of it.
Underline the antecedent and italicize the consequent.
You get: If Jones wins the nomination or Dexter leaves the
party, then Williams is the sure winner. Now, begin symbolizing: (Jones wins the
nomination or Dexter leaves the party) Williams is the sure winner
The antecedent is a disjunction, so show that (Jones wins the nomination ▼ Dexter leaves the
party) Williams is the sure winner Finally, replace statements with statement letters (J ▼ D) W and you are done!
Logical Operators, Truth Functions and Truth Tables
Our 5 logical operators produce statement forms that
are truth-functional Negation ~p Conjunction p q Disjunction p ▼ q Conditional p q Biconditional p q
Truth-Functions
A truth function takes one or more truth values as input and returns one truth value as the output
Truth functional operators determine the truth value of a compound statement given the truth values of the simple statements in it
Truth Tables
Chapter 7 of the text discusses truth tables.
A truth table is a complete list of all the possible permutations of truth and falsity for a set of simple statements, showing the effect of each permutation on the truth value of a compound having those simple statements as components.
Truth Tables Each permutation of truth values constitutes
one row of a truth table and the number of rows in a truth table is 2n where n equals the number of simple statements.
The truth value of any compound statement is determined by the truth values of its component sentences.
Note that I use the symbol to denote “false.”
possibility 4
possibility 3
possibility 2
possibility 1
S2S1
How many permutations are there?
A set of statements containing 2 simple statements has 4 permutations
T
T T
T
A statement with 3 simple statements has 8 permutations
Truth Tables for the Basic Truth Functional Operators
p q ~p p q p q p q p qT T T T T T
T T
T T T T
T T T
Uses for Truth Tables
• Determine the truth conditions for any compound statement
Building a Truth-Table(~R
T
T
S
4
3
T2
T1
~S)Rcase
T
T
T
T
T
Building a Truth-Table(~R
T
T
S
4
3
T2
T1
~S)Rcase
T
T
T
T
T
T
T
Building a Truth-Table[(RS)
T
T
S
4
3
T2
T1
~S]Rcase
T
T
T
T
T
T
Uses for Truth Tables
• Determine the truth conditions for any compound statement
• Determine whether a statement is a tautology, a contradiction or neither (contingent)
• Download the Handout on Truth Tables, work the problems and then discuss them on the bulletin board.
Tautologies and Contradictions
• A tautology is a statement that is true in virtue of its logical form. It is true under all possible circumstances.
• A contradiction is a statement that is false in virtue of its logical form. It is false under all possible circumstances.
Tautologies and Contradictions
• The column for the dominant operator of a tautology will contain only “Trues.” The column for the dominant operator of a contradiction will contain only “Falses.”
Truth-Table Test for tautology, contradiction, contingent statements
T
T
S
4
3
T2
T1
Rcase
Is the formula a tautology, a contradiction, or a contingent statement?
TAUTOLOGY
T
[(R S) R] S
T
T
T
T
T
T
T
Truth-Table Test for tautology, contradiction, contingent statements
T
T
S
4
3
T2
T1
Rcase
Is the formula a tautology, a contradiction, or a contingent statement?
CONTRADICTION
T
(R ~R) (S ~S)
T
T
T
Truth-Table Test for tautology, contradiction, contingent statements
T
T
S
4
3
T2
T1
Rcase
Is the formula a tautology, a contradiction, or a contingent statement?
CONTINGENT
T
[(R S) S] R
T
T
T
T
T
T
T
Uses for Truth Tables
• Determine the truth conditions for any compound statement
• Determine whether a statement is a tautology, a contradiction or neither,
• Determine whether two formulae are equivalent.
Equivalence
• Two formulae are said to be equivalent if, but only if, those formulae are true and false under exactly the same conditions, that is, if, but only if, the truth table columns for the dominant operators are identical.
Testing for Equivalence
To use a truth table to test two formulae for equivalence, begin by constructing truth table columns for each formula. Use one set of guide columns for both formulae.
Compare the columns for the dominant operators of the two formulae. If the truth values agree in each row, the formulae are equivalent.
• Download the Handout on Testing for Equivalence and work the problems.
Truth-Table Test for Equivalence
T
T
~ R
T
T
S
T
T
S ) ::
T
T
( R
Are the two formulas logically equivalent?
T
T
T
T
T
T
T
T
Do the formulas match in truth value? YES
YES
Truth-Table Test for Equivalence
T
T
R ~
T
T
S
T
T
S ) :: ~
T
T
( R~
Are the two formulas logically equivalent?
T
T
T
T
T
T
T
T
T
Do the formulas match in truth value? NO
NO
Truth-Table Test for Equivalence
T
T
R ~
T
T
S
T
T
S ) :: ~
T
T
( R~
Are the two formulas logically equivalent?
T
T
T
T
T
T
T
T
T
T
T
Do the formulas match in truth value? YES
YES
Truth-Table Test for Equivalence
T
T
R ~
T
T
S
T
T
S ) :: ~
T
T
( R~
Are the two formulas logically equivalent?
T
T
T
T
T
T
T
T
T
T
T
Do the formulas match in truth value? NO
NO
Truth-Table Test for Equivalence
T
T
R ~
T
T
S
T
T
S ) :: ~
T
T
( R~
Are the two formulas logically equivalent?
T
T
T
T
T
T
T
T
T
Do the formulas match in truth value? YES
YES
Do Not Distribute a Negation Across A Conjunction or a Disjunction!
not R and not Snot ( R and S)
not R or not Snot ( R or S)
Rather. . .
not R and not S
not ( R and S) not R or not S
not ( R or S)
==
Uses for Truth Tables
• Determine the truth conditions for any compound statement
• Determine whether a statement is a tautology, a contradiction or neither,
• Determine whether two formulae are equivalent.
• Determine whether an argument is valid or not.
Logical Implication
One statement logically implies another if, but only if, whenever the first is true, the second is true as well
If a statement, S1, implies S2 then the conditional (S1 S2) will be a tautology. Implication is the validity of the conditional.
Truth tables allow one to test for logical implication
Validity and Logical Implication
An argument is valid if, but only if, its premises logically imply its conclusion.
Deductive Validity
A characteristic of arguments in which the truth of the premises guarantees the truth of the conclusion.
A characteristic of arguments in which the premises logically imply the conclusion.
Truth Table Tests for Validity
Construct a column for each premise in the argument
Construct a column for the conclusion Examine each row of the truth table. Is there a
row in which all the premises are true and the conclusion is false. If so, the argument is non-valid. If not, then the argument is valid.
Download the Handout on Testing for Validity.
Truth-Table Test for Validity for R S, RSModus Ponens (MP)
R S R
T
T
S
4
3
T2
T1
SRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or non-valid?
T
T
T
T
T
T
T
VALID
NO
Truth-Table Test for Validity for R S, ~S~RModus Tollens (MT)
R S ~S
T
T
S
4
3
T2
T1
~RRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or non-valid?
T
T
T
T
T
T
T
VALID
NO
Truth-Table Test for Validity for R S, ~R~SEvil Twin of Modus Tollens
R S ~R
T
T
S
4
3
T2
T1
~SRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or non-valid?
T
T
T
T
T
T
T
NON-VALID
YES
Truth-Table Test for Validity for R S, SRFallacy of Affirming the Consequent
R S S
T
T
S
4
3
T2
T1
RRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or non-valid?
T
YES
NON-VALID
T
T
T
T
T
T
Truth-Table Test for Validity for R S, R ~S
R S R
T
T
S
4
3
T2
T1
~SRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or non-valid?
YES
NON-VALID
T
T
T
T
T
T
T
Truth-Table Test for Validity for R S, ~R SDisjunctive Syllogism (DS)
R S
~R
T
T
S
4
3
T2
T1
SRcase
Is there a case in which the premises are all true but the conclusion is false?
Is the argument form valid or non-valid?
T
T
T
T
T
T
T
VALID
NO
Key Ideas
Grouping and Meaning Paraphrasing Inward Truth Functional Operators Truth Tables• Using Truth Tables• Testing for tautologies and contradictions• Testing for equivalence Testing for validity
