TRUTH TABLES

What is a truth table? We start with truth-tables for the sentential connectives in SL. A truth-table shows how the truth-value of a complex WFF depends on the truth-values of its component WFFs. So what are truth-values? In SL there are only two truth-values : T (1) and F (0), which stands for truth and falsity. To say that a statement has truth-value T or 1 is just to say that it is true. To say that its truth-value is F or 0 is to say that it is false.

TRUTH-TABLES: NEGATION If  represents a WFF, ¬  is true when  is false, and it is false when  is true:  ¬  1

TRUTH-TABLES: CONJUNCTION If  and ß are WFF, (  ß) is true when  and ß are true, and it is false in the rest of cases.  ß   ß 1

TRUTH-TABLES: DISJUNCTION If  and ß are WFF, (  ß) is false when  and ß are false, and it is true in the rest of cases.  ß   ß 1

TRUTH-TABLES: IMPLICATION (CONDITIONAL) If  and ß are WFF, (  ß) is false when  is true and ß is false, and it is true in the rest of cases.  ß   ß 1

TRUTH-TABLES: BICONDITIONAL If  and ß are WFF, (  ß) is false when  and ß have a different truth-value, and it is true when they have the same truth-value.  ß   ß 1

EXERCISES Are these statements true or false? If "(P∨Q)" is true, either "P" is false or "Q" is false. If "(P^Q)" is false, then "P" is false and "Q" is also false. Whenever "(P∨Q)" is true, "(Q→P)" is also true. In order for "(P→Q)" to be true, "Q" must be true.

EXERCISES Consider this diagram :       Now determine whether the following statements are true of the diagram, using the appropriate truth-tables to interpret the connectives : All squares are red if and only if all squares are green.  If there is no red square, then there is a triangle.  Either there is a green circle, or there are no orange squares. 

Truth-table using 2 sentence letters ¬ p  q 1 In this truth-table the main connective is . The negation is the main connective in the formula ¬p. Thus, first we need to resolve the negation in ¬p. After that, we need to combine the different truth-values in this truth-table with the WFF q.

Truth-table using 2 sentence letters ¬ (p  q) 1 In this case the main connective is ¬ so we will resolve the result of this column at the end. First of all, we need to resolve this formula (p  q). After that, we will calculate the negation using the different truth-values which were the results of the formula (p^q)

Truth-table using 2 sentence letters We are going to calculate the truth-table of this formula: We need to obtain all the possible combinations (the different truth-values). “p” and “q” can have 2 different truth-values (1 and 0), so we will need to calculate 2 x 2 = 4 combinations. ( p  q )  ¬ 

Truth-table using 2 sentence letters We are going to calculate the truth-table of this formula: ( p  q )  ¬  1 First of all, we need to write the different truth-values for “p” and “q”. We will get 4 different lines.

Truth-table using 2 sentence letters We are going to calculate the truth-table of this formula: ( p  q )  ¬  1 Now, we need to solve the atomic formulas next to the negation (because we need to calculate inside the brackets).

Truth-table using 2 sentence letters We are going to calculate the truth-table of this formula: ( p  q )  ¬  1 After that, we need to calculate the connectives inside of the brackets. In this case:  and 

Truth-table using 2 sentence letters We are going to calculate the truth-table of this formula: ( p  q )  ¬  1

Truth-table using 2 sentence letters We are going to calculate the truth-table of this formula: ( p  q )  ¬  1 At the end, we must combine both formulas which are inside of brackets. So, we are going to calculate the result of the disjunction

Truth-table using 2 sentence letters We are going to calculate the truth-table of this formula: ( p  q )  ¬  1 The column which has the main connetive is the truth-table of this formula.

Truth-table using 3 sentence letters If we have 3 sentence letters in a formula, each one has 2 possible truth-values (1 and 0), so 2 x 2 x 2 = 8 combinations. We will get and calculate 8 files in this truth-table.

Truth-table using 3 sentence letters ( p  q )  ¬ r  1 We need to calculate, first of all, the formulas which are inside of brackets.

Truth-table using 3 sentence letters ( p  q )  ¬ r  1 We know that the main connective is  because it is outside of brackets. Now we need to calculate and solve the atomic formulas which have a negation (not the negation next to the brackets)

Truth-table using 3 sentence letters ( p  q )  ¬ r  1 Now we can solve the formulas which are inside of brackets.

Truth-table using 3 sentence letters ( p  q )  ¬ r  1 Now, we calculate the negation next to the brackets

Truth-table using 3 sentence letters ( p  q )  ¬ r  1 Finally, we solve the column which contains the main connective

(p  [q  ¬(r  s)])  (t  ¬u) Truth-tables Generally, if we have n sentence letters, we will need 2n files in order to get all the combinations. So, if we wanted to know the truth-table of this formula: (p  [q  ¬(r  s)])  (t  ¬u) we would need 26 = 64 files !

Truth-table using 4 sentence letters ( p  q )  [(  r s )] 1

Truth-table using 4 sentence letters ( p  q )  ¬ r  1 1ª 2ª 3ª 4ª 5ª 6ª 7ª 8ª

Truth-table using 4 sentence letters ( p  q )  ¬ r  1 1ª 2ª 3ª 4ª 5ª 6ª 7ª 8ª The first line shows us that if p, q and r have as a truth-value 1, then the formula will have the truth-value of 0. The rest of lines show that any other combination for p, q and r will have a truth-table which result is 1.

Truth-table using 4 sentence letters ( p  q )  ¬ r  1 1ª 2ª 3ª 4ª 5ª 6ª 7ª 8ª

TAUTOLOGY, CONTRADICTION, CONTINGENCY

Is this set of formulas satisfiable? p  q ¬(r  ¬q) ¬p  r ( p  q ) ¬ r  1  1 ( ¬ p  r ) 1 1 In the fifth line there is a set of formulas which is satisfiable

Is this set of formulas satisfiable? p  q ¬(q  ¬r) r  ¬p ( p  q ) ¬  r 1  ( r  ¬ p ) 1 1 It is not satisfiable because there is not any 1 in any line (we cannot find a main connective with a true truth-value)

1) Logical consequence 2) Logical truth 3) Logical equivalence

LOGICAL CONSEQUENCE Logical consequence is a fundamental concept in logic, which describes the relationship between statements that holds true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusions entail from the premises, because the conclusions are consequences of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises?

LOGICAL CONSEQUENCE For a valid argument, the conclusion is followed by the premises. An argument is valid in these cases: Premises are true and conclusion is true Premises are false and conclusion is false Premises are false and conclusion is true. In these situations, we can say that the conclusion of an argument is a logical consequence of its premises. Therefore, a formula ß will NOT be a logical consequence of a formula  if  is true (premises) and ß is false (conclusion).

LOGICAL CONSEQUENCE WE WILL USE THIS CONNECTIVE (THE IMPLICATION) to write the whole argument (premises and conclusion) 

is ß a logical consequence of  ?   p  q ß  p  q is ß a logical consequence of  ? Yes, it is a logical consequence because there is not any case in which  is true and ß is false. So, there is not any case where the conditional is false. (p  q)   1

Is ß a logical consequence of 1 and 2? 1  (p  q) 2  (r  ¬p) ß  r  q Is ß a logical consequence of 1 and 2? [(p  q)  (r  ¬ p)]  1 Yes, because the conditional is always true (1)

EXAMPLE (EXERCISE): LOGICAL CONSEQUENCE A necessary condition for humanity to be free is that human beings do not have an essence. If God created humans, then we must have an essence. Obviously, humans are free. Therefore, God did not create humans. 1º) First we need to select the propositions and sentences and translate them into logical language: p  humans are free q  humnas have an essence r  God created humans Premises: (p  ¬q) ; (r  q) ; p Conclusion:  ¬r

EXAMPLE (EXERCISE): LOGICAL CONSEQUENCE A necessary condition for humanity to be free is that human beings do not have an essence. If God created humans, then we must have an essence. Obviously, humans are free. Therefore, God did not create humans. 2º) Let’s write the implication (logical consequence formula): [(p  ¬q)  (r  q)  p]  ¬r

EXAMPLE (EXERCISE): LOGICAL CONSEQUENCE A necessary condition for humanity to be free is that human beings do not have an essence. If God created humans, then we must have an essence. Obviously, humans are free. Therefore, God did not create humans. 3º) We calcualte the formulas and the conditional: [(p  ¬ q)  (r p] r 1

EXAMPLE (EXERCISE): LOGICAL CONSEQUENCE A necessary condition for humanity to be free is that human beings do not have an essence. If God created humans, then we must have an essence. Obviously, humans are free. Therefore, God did not create humans. The formula [(p  ¬q)  (r  q)  p]  ¬r , is a tautology. Therefore, this argument is VALID.

LOGICAL TRUTH We already have the notion of logical consequence. A sentence is a logical consequence of a set of sentences if it is impossible for that sentence to be false when all the sentences in the set are true. We will define logical truth in terms of logical consequence. Suppose a given sentence is a logical consequence of every set of sentences. That means that it is impossible for that sentence to be false – it comes out true in every possible circumstance. A sentence is a logical truth if it is a logical consequence of every set of sentences. A sentence is also a logical truth if it is a tautology.

LOGICAL EQUIVALENCE Sentences that have the same truth value in every possible circumstance are logically equivalent. In logic, statements p and q are logically equivalent if they have the same logical content. Propositions r and s are logically equivalent if the statement r ↔ s is a tautology.

LOGICAL EQUIVALENCE