Introduction to Logic Lecture 13 An Introduction to Truth Tables By David Kelsey

Constructing a truth table When constructing a truth table, always work from the simplest to the most complex: Consider the claim ‘If Bolton won, then Arsenal didn’t.’ – We will let ‘B’ stand for ‘Bolton won’, and ‘A’ for ‘Arsenal won’. (Notice A doesn’t stand for Arsenal didn’t win as this claim contains a negation.) – Thus, we symbolize our claim like this: B  ~A – We can now construct the truth table for our claim, working from simplest to most compound:

The truth table for B  ~A So our claim is: B  ~A – We start with our claim variables: A B T T F F T F Then we add the consequent of the conditional: A B ~A T T F T F F F T T F F T Now that we have added all of the simple parts of our claim to the truth table, we can add the claim itself: A B ~A B  ~A T T F F T F F T F T T T F F T T

Truth tables: rows and columns A truth table is composed of rows and columns. Rows are any set of truth values running left to right from one side of the table to the other. Columns are any vertical set of truth values running top to bottom. For example: P Q PvQ T T T T F T F T T F F F Examples of columns are bold.

Finding the number of rows The number of rows in a truth table: – For claims with only one claim variable, there are only two possibilities: the claim is true and the claim is false P T F – For claims with 2 claim variables we have 4 possibilities: both claims are true, both are false, one is true and the other false and vice versa: P Q T T T F F T F F

A formula for finding the number of rows A formula for finding the number of rows: – So for claims with 2 variables we write 4 rows. – For claims with 3 claim variables we have 8 possibilities, so we write 8 rows. – There is a formula for determining the number of rows in a truth table. The number of rows R is equal to 2 to the nth power, where n is equal to the number of claim variables the claim contains. Thus, the truth table for a claim with 3 variables has 8 rows & the table for a claim with 4 variables has 16 rows.

Constructing a truth table for a claim To begin your construction of a truth table for a claim: – first write down all of the claim variables, in alphabetical order, working from left to right. – Then list all parts of the claim, working from simplest to most compound, making sure to finish with the claim itself in the rightmost position. – Now write truth values, Ts and Fs, under the claim variables. Make sure to create a row for every possible combination of truth values. – There is a rule for writing truth values under the claim variables…

Constructing your reference columns There is a rule for writing truth values in the columns under the claim variables. – These columns are called reference columns. Here it is: – Of the reference columns, start with the rightmost one and, moving downward, alternate T then F, T then F until reaching the last row of the table. – Then move to the next rightmost reference column and alternate pairs of Ts and Fs. – Then to the next rightmost column and alternate sets of 4 Ts and 4 Fs – The leftmost reference column will always be half Ts and then half Fs.

Claims are truth functional The truth value of any claim is dependent upon the truth values of the parts of that claim. Even further though, if the parts of a claim are themselves compound claims, then their truth values will depend upon the truth values of their parts, and so on and so on until we get to the claim’s claim variables standing alone. Consider, for example, the claim P  (Qv~Q) We first write our reference columns: P Q T T F F T F Next, add any of the claim’s simpler parts to the truth table…

The truth table for the claim: P  (Qv~Q) We are working on P  (Qv~Q) if you will remember. After we have our reference columns we add the consequent of the conditional claim and any of its simpler parts. P Q ~Q Qv~Q T T F T T F T T F T F F T T Now that we have written all of the claims simple parts we can add the claim itself P Q ~Q Qv~Q P  (Qv~Q) T T F T T T F T T T F T F T T F F T T T

Parenthesis Besides the truth functional symbols ‘~’, ‘v’, ‘&’ and ‘  ’ we also have ‘(‘ and ‘)’. Parenthesis can effect the meaning of a claim, consider for example: P  (Q&R) and (P  Q)&R The first claim is a conditional with a conjunction as its consequent while the second is a conjunction with a conditional as a conjunct. P Q R P  (Q&R) (P  Q)&R T T T T T T T F F F T F T F F T F F F F F T T T T F T F T F F F T T T F F F T F Note the differences between the two columns: the second has Fs in the 6th and 8th rows while the first does not.

Truth functional equivalence To say that 2 claims are truth functionally equivalent is to say that they have the same meaning. Two claims are truth functionally equivalent if and only if the column of truth values under each claim is exactly the same as the other. Thus, claims are truth functionally equivalent just when if one claim is true so is the other and if one claim is false so is the other.

Conditionals again If the word ‘if’ occurs in a claim this introduces the antecedent of a conditional. Consider the claim ‘Bolton won if Sunderland lost.’ Because if introduces the antecedent of a conditional our claim can be reformulated as this: – If Sunderland lost then Bolton won. If the phrase ‘only if’ occurs in a claim this introduces the consequent of a conditional claim. Consider the claim ‘Sunderland will beat Bolton only if Bolton has a very bad day.’ Because only if introduces the consequent of a conditional our claim can be reformulated as this: – if Sunderland beats Bolton then Bolton has a very bad day.

Bi-conditional claims Consider this claim: You will pass the final exam if and only if you study hard and take copious notes. Given the word ‘if’ introduces the antecedent of a conditional claim and that the phrase ‘only if’ introduces the consequent of a conditional it follows that: – ‘if and only if’ introduces both the antecedent of a conditional claim and the consequent of another conditional claim. – Thus, the example above is logically equivalent to the claim ‘if you pass the final exam then you studied hard and took copious notes & if you studied hard and took copious notes then you passed the final exam.

Truth tables for Bi-conditional claims We symbolize the phrase ‘if and only if’ with a 2 sided arrow, I.e. ‘  ” Thus, P  Q is logically equivalent to the claim P  Q & Q  P. A claim that has the  as its main logical operator is a Bi-conditional claim. The truth table for the bi-conditional is as follows: P Q P  Q T T T T F F F T F F F T

Necessary and Sufficient Conditions If oxygen is a necessary condition of combustion, then we cannot have combustion without oxygen. So if we have combustion we must have oxygen as well. To say that oxygen is a necessary condition of combustion though is to make the logically equivalent claim that if there is combustion then there is oxygen. If being born in the United States is a sufficient condition for US citizenship, then being born in the US is all that one needs to be a US citizen. But to say that being born in the US is a sufficient condition for US citizenship, is to make the equivalent claim that If one is born in the US then she is a US citizen.

Necessary and Sufficient Conditions If being X is both a necessary and a sufficient condition for being Y then both: 1) one cannot possess Y without her also possessing X & 2) possessing X is all it takes to possess Y. Thus, to say that X is a necessary and sufficient condition of Y is to make this logically equivalent claim: If something is an X then it is a Y & if something is a Y then it is an X. Thus, the necessary and sufficient condition is both the antecedent of one conditional and the consequent of another.

Unless The word ‘unless’ introduces a disjunction. Thus, ‘unless’ has the same meaning as ‘or’. Thus, we use the symbol ‘v’ to stand for ‘unless’. Consider, for instance, the claim: Arsenal won unless Manchester did. This is logically equivalent to the claim either Arsenal won or Manchester did.

Either The word ‘either’ tells us when a disjunction begins. So either introduces the first disjunct of a disjunction. Consider the claim ‘Either P and Q or R’: – Here either introduces the first disjunct P and Q of the disjunction P and Q or R. – We can symbolize the above claim as this: (P&Q)vR