Truth Tables

We have learned how to take sentences in English and translate them into logical statements using letters and the symbols for the logical connectives. We have talked about when a statement made from two simple statments joined by a single logical connective is true and when it is false. Now we need a systematic way to find the truth value of more complicated statements that contain multiple logical symbols. So how do we do this?

First of all, we need some sort of order of operations: if we have a statement with multiple logical symbols in it, how do we know which one should come first? Logicians decided to use much the same order of operations as we learned in set theory.

Just like in set theory, we usually need parintheses to tell us what to do first. So here is the order in which we should do things in logic:

Logic Order of Operations
  1. We do parintheses first, working from the inside out.
  2. Within the parintheses or after the parintheses are gone, we always do negation before we do any other logical connective.
  3. If we have any other logical connectives, we must have parentheses telling us which to do first.

 

Let's look at an example:

(~p∧q)∨p

If we wanted to see what the truth value is of this statement when p=T and q=F, then we would have to go step-by-step through the statement, doing each logical quantifier one by one.


What would we do first? The parintheses.
What is there to do in the parintheses? ~p∧q

So which should we do first, the negation (~) or the or (∨)?
Because negations always go first, we begin by finding the truth value of ~p.

If p=T, then we must have ~p=F.
Now that we've done ~p, we can combine its truth value with q's truth value to find the truth value of ~p∧q.
(Remember than an "and" statment is true only when both statement on either side of it are true.)

In this case, we have a ~p on the left, which we know is false, and a q on the right, which we also know is false.
So because we don't have statements on either side of the "and" symbol that are both true, the statment ~p∧q is false.
So ~p∧q=F.


Now that we know the truth value of everything in the parintheses (~p∧q), we can join this statement with ∨p to give us the final statement (~p∧q)∨p.


When is an "or" statment true?
Whenever at least one of the individual statements on either side of it is true.
In this case, the two statements on either side of the or sign are (~p∧q) and p.

We know that ~p∧q is false, and we know that p is true.
Because at least one of these two is true, we know that our "or" statement, (~p∧q)∨p must be true.


So (~p∧q)∨p=T.

Another way to write this problem step by step would look like this:
(~p∧q)∨p
=(~T∧F)∨T
=(F∧F)∨T
=F∨T
=T

We often want to find a systematic way to find all the possible truth values of a statement. In order to do this, we create a truth table. A truth table is a table that begins with all the possible combinations of truth values for the letters in the compound statement; it then breaks the compound statment down and one step at a time determines truth values for each of the parts of the logical statement.

Let's look at an example so that this definition makes more sense:

One of the simplest truth tables we can construct is the truth table for the expression ~p.
Here's how we do it:
  1. First we begin by constructing a table with the letter p at the top of the first column:

    2. p

  2. Then we fill in all the possible truth values of p in that column. Since p can only be true or false, the column is filled in like this:

    p
    T
    F

  3. Next we put the expression ~p at the top of the next column.

    p
    ~p
    T
    F

  4. Now we look at each row; we look at what the truth value is of p in that row, and then we fill in the corresponding truth value of ~p for the given truth value of p.

    1. If we look at the first row, the truth value of p is T. So in this row, the truth value of ~p must be F.
      To see why this is true, just remember the definition of the negation ~. Think of it this way: Suppose p represents the statement, "It is raining." Then ~p represents the statement, "It is not raining." Whenever it is true that it is raining (p=T), then it must be false that it is not raining (~p=F).

      p
      ~p
      T
      F
      F
      T

    2. If we look at the second row, the truth value of p is F. So in this row, the truth value of ~p must be T.
      To see why this is true, just remember the definition of the negation ~. Think of it this way: Suppose p represents the statement, "It is raining." Then ~p represents the statement, "It is not raining." Whenever it is false that it is raining (p=F), then it must be true that it is not raining (~p=T).

      p
      ~p
      T
      F
      F
      T

So our final truth table looks like this:

p
~p
T
F
F
T

 

Now let's make a truth table for each of the major logical connectives:

 

Now, our final goal is to be able to fill in truth tables with more compound statements which have more than just one logical connective in them.
Statements like q→~s or (r∧~p)→r or (q&rarr~p)∧(p↔r) have multiple logical connectives, so we will need to do them one step at a time using the order of operations we defined at the beginning of this lecture.

Let's do several examples so that we can see how to do these complex statements one step at a time:
  1. q→~s

    1. First we begin by writing out the table with all the possible combinations of truth values for each letter in the expression.

      Because here we have 2 letters, q and s, we will have 2 columns at the beginning of the truth table labeled q and s:

      q
      s
      T
      T
      T
      F
      F
      T
      F
      F

    2. Because the order of operations says that, unless we have parenthesis, we always do the negation first.
      So we label the first empty column ~s.
      Then, because s is the only letter in the expression ~s, we use the given truth values of s in the table to fill in the corresponding truth values for ~s:

      q
      s
      ~s
      T
      T
      F
      T
      F
      T
      F
      T
      F
      F
      F
      T

      Recall that a statement with the ~ symbol in it is only true if what follows the ~ symbol is false, and vice versa.
      • The first truth value in the ~s column is F because when s=T, ~s=F.
      • The second truth value in the ~s column is T because when s=F, ~s=T.
      • The third truth value in the ~s column is F because when s=T, ~s=F.
      • The fourth truth value in the ~s column is T because when s=F, ~s=T.

    3. Now that we have taken care of the ~ sign in the expression, all that we have left to worry about is the → symbol in q→~s.

      The symbol → always joins exactly two expressions; the two expressions it joins may themselves be more complicated statements, but we don't care how complicated they are, as long as we already have a column in the truth table with that logical expression at the top.

      In this expression, the symbol → joins the 2 expressions: q and ~s.
      Both q and ~s are at the top of columns in the truth table, so we can use these two columns to fill in the column for q→~s, just by using the definition of the → symbol:

      q
      s
      ~s
      q→~s
      T
      T
      F
      F
      T
      F
      T
      T
      F
      T
      F
      T
      F
      F
      T
      T

      Be careful! Unlike other logical connectives, the conditional does not treat the first and the second statements it joins in the same way!
      When you work with the conditional connective (→), you must mark the column corresponding to the first statement differently than the way you mark the second column.

      Be sure when you are filling in the column for the conditional statement that you are not confusing the column for the first statement with the column for the second statement!
      In these examples, we distinguish between the two by coloring the column containing the first statement yellow and the column containing the second statement aqua.

      Remember that a statement containing the symbol → is only false if the first statement (the one before the → sign) is true and the second statement (the one after the → sign) is false!
      • The first truth value in the q→~s column is F because when q=T and ~s=F, q→~s=F.
      • The second truth value in the q→~s column is T because when q=T and ~s=T, q→~s=T.
      • The third truth value in the q→~s column is F because when q=F and ~s=T, q→~s=T.
      • The fourth truth value in the q→~s column is T because when q=F and ~s=T, q→~s=T.

      4. So the final truth table for this statement will look like this:

      q
      s
      ~s
      q→~s
      T
      T
      F
      F
      T
      F
      T
      T
      F
      T
      F
      T
      F
      F
      T
      T

  2. (r∧~p)→r

    1. First we begin by writing out the table with all the possible combinations of truth values for each letter in the expression.

      Because here we have 2 letters, p and r, we will have 2 columns at the beginning of the truth table labeled p and r:

      p
      r
      T
      T
      T
      F
      F
      T
      F
      F

    2. The order of operations says that we always do the parentheses first. So we look at what is inside the parentheses: r∧~p.
      Inside the parentheses we have two symbols: ∧ and ~.
      The order of operations says that we always do negations first.
      s So we label the first empty column ~p.

      Then, because p is the only letter in the expression ~p, we use the given truth values of p in the table to fill in the corresponding truth values for ~p:

      p
      r
      ~p
      T
      T
      F
      T
      F
      F
      F
      T
      T
      F
      F
      T

      Recall that a statement with the ~ symbol in it is only true if what follows the ~ symbol is false, and vice versa.
      • The first truth value in the ~p column is F because when p=T, ~p=F.
      • The second truth value in the ~p column is T because when p=T, ~p=F.
      • The third truth value in the ~p column is F because when p=F, ~p=T.
      • The fourth truth value in the ~p column is T because when p=T, ~p=F.

    3. Now that we have taken care of the ~ sign in the expression inside the parentheses, all that we have left to worry about inside the parentheses is the ∧ symbol in r∧~p.

      The symbol ∧ always joins exactly two expressions; the two expressions it joins may themselves be more complicated statements, but we don't care how complicated they are, as long as we already have a column in the truth table with that logical expression at the top.

      In this expression, the symbol ∧ joins the 2 expressions: r and ~p.
      Both r and ~p are at the top of columns in the truth table, so we can use these two columns to fill in the column for r∧~p, just by using the definition of the ∧ symbol:

      p
      r
      ~p
      r∧~p
      T
      T
      F
      F
      T
      F
      F
      F
      F
      T
      T
      T
      F
      F
      T
      F

      Remember that a statement containing the symbol ∧ is only true if both statements are true!
      • The first truth value in the r∧~p column is F because when r=T and ~p=F, r∧~p=F.
      • The second truth value in the r∧~p column is F because when r=F and ~p=F, r∧~p=F.
      • The third truth value in the r∧~p column is T because when r=T and ~p=T, r∧~p=T.
      • The fourth truth value in the r∧~p column is F because when r=F and ~p=T, r∧~p=F.

    4. Now that we have taken care of the parentheses, all that we have left to worry about is the symbol left outside the parentheses: is the → symbol in (r∧~p)→r.

      The symbol → always joins exactly two expressions; the two expressions it joins may themselves be more complicated statements, but we don't care how complicated they are, as long as we already have a column in the truth table with that logical expression at the top.

      In this expression, the symbol → joins the 2 expressions: r∧~p and r.
      Both r∧~p and r are at the top of columns in the truth table, so we can use these two columns to fill in the column for (r∧~p)→r, just by using the definition of the → symbol:

      p
      r
      ~p
      r∧~p
      (r∧~p)→r
      T
      T
      F
      F
      T
      T
      F
      F
      F
      T
      F
      T
      T
      T
      T
      F
      F
      T
      F
      T

      Be careful! Unlike other logical connectives, the conditional does not treat the first and the second statements it joins in the same way!
      When you work with the conditional connective (→), you must mark the column corresponding to the first statement differently than the way you mark the second column.

      Be sure when you are filling in the column for the conditional statement that you are not confusing the column for the first statement with the column for the second statement!
      In these examples, we distinguish between the two by coloring the column containing the first statement yellow and the column containing the second statement aqua.

      Remember that a statement containing the symbol → is only false if the first statement (the one before the → sign) is true and the second statement (the one after the → sign) is false!
      • The first truth value in the (r∧~p)→r column is T because when (r∧~p)=F and r=T, (r∧~p)→r=T.
      • The second truth value in the (r∧~p)→r column is T because when (r∧~p)=F and r=F, (r∧~p)→r=T.
      • The third truth value in the (r∧~p)→r column is T because when (r∧~p)=T and r=T, (r∧~p)→r=T.
      • The fourth truth value in the (r∧~p)→r column is T because when (r∧~p)=F and r=F, (r∧~p)→r=T.

      5. So the final truth table for this statement will look like this:

      p
      r
      ~p
      r∧~p
      (r∧~p)→r
      T
      T
      F
      F
      T
      T
      F
      F
      F
      T
      F
      T
      T
      T
      T
      F
      F
      T
      F
      T

  3. r∧(~p→r)

      2. First we begin by writing out the table with all the possible combinations of truth values for each letter in the expression.

      Because here we have 2 letters, p and r, we will have 2 columns at the beginning of the truth table labeled p and r:

      p
      r
      T
      T
      T
      F
      F
      T
      F
      F

    2. The order of operations says that we always do the parentheses first. So we look at what is inside the parentheses: ~p→r.
      Inside the parentheses we have two symbols: ~ and →.
      The order of operations says that we always do negations first.
      So we label the first empty column ~p.

       

      Then, because p is the only letter in the expression ~p, we use the given truth values of p in the table to fill in the corresponding truth values for ~p:

      p
      r
      ~p
      T
      T
      F
      T
      F
      F
      F
      T
      T
      F
      F
      T

      Recall that a statement with the ~ symbol in it is only true if what follows the ~ symbol is false, and vice versa.
      • The first truth value in the ~p column is F because when p=T, ~p=F.
      • The second truth value in the ~p column is T because when p=T, ~p=F.
      • The third truth value in the ~p column is F because when p=F, ~p=T.
      • The fourth truth value in the ~p column is T because when p=T, ~p=F.

    3. Now that we have taken care of the ~ sign in the expression inside the parentheses, all that we have left to worry about inside the parentheses is the → symbol in ~p→r.

      The symbol → always joins exactly two expressions; the two expressions it joins may themselves be more complicated statements, but we don't care how complicated they are, as long as we already have a column in the truth table with that logical expression at the top.

      In this expression, the symbol → joins the 2 expressions: ~p and r.
      Both ~p and r are at the top of columns in the truth table, so we can use these two columns to fill in the column for ~p→r, just by using the definition of the → symbol:

      p
      r
      ~p
      ~p→r
      T
      T
      F
      T
      T
      F
      F
      T
      F
      T
      T
      T
      F
      F
      T
      F

      Be careful! Unlike other logical connectives, the conditional does not treat the first and the second statements it joins in the same way!
      When you work with the conditional connective (→), you must mark the column corresponding to the first statement differently than the way you mark the second column.

      Be sure when you are filling in the column for the conditional statement that you are not confusing the column for the first statement with the column for the second statement!
      In these examples, we distinguish between the two by coloring the column containing the first statement yellow and the column containing the second statement aqua.

      Remember that a statement containing the symbol → is only false if the first statement (the one before the → sign) is true and the second statement (the one after the → sign) is false!
      • The first truth value in the ~p→r column is T because when ~p=F and r=T, ~p→r=T.
      • The second truth value in the ~p→r column is T because when ~p=F and r=F, ~p→r=T.
      • The third truth value in the ~p→r column is T because when ~p=T and r=T, ~p→r=T.
      • The fourth truth value in the ~p→r column is F because when ~p=T and r=F, ~p→r=F.

    4. Now that we have taken care of the parentheses, all that we have left to worry about is the symbol left outside the parentheses: is the ∧ symbol in r∧(~p→r).

      The symbol ∧ always joins exactly two expressions; the two expressions it joins may themselves be more complicated statements, but we don't care how complicated they are, as long as we already have a column in the truth table with that logical expression at the top.

      In this expression, the symbol ∧ joins the 2 expressions: r and ~p→r.
      Both r and ~p→r are at the top of columns in the truth table, so we can use these two columns to fill in the column for r∧(~p→r), just by using the definition of the ∧ symbol:

      p
      		 r
      ~p
      ~p→r
      r∧(~p→r)
      T
      		 T
      F
      T
      T
      T
      		 F
      F
      T
      F
      F
      		 T
      T
      T
      T
      F
      		 F
      T
      F
      F

      Remember that a statement containing the symbol ∧ is only true if both statements are true!
      • The first truth value in the r∧(~p→r) column is T because when r=T and ~p→r=T, r∧(~p→r)=T.
      • The second truth value in the r∧(~p→r) column is F because when r=F and ~p→r=T, r∧(~p→r)=F.
      • The third truth value in the r∧(~p→r) column is T because when r=T and ~p→r=T, r∧(~p→r)=T.
      • The fourth truth value in the r∧(~p→r) column is F because when r=F and ~p→r=F, r∧(~p→r)=F.

      So the final truth table for this statement will look like this:

      p
      		 r
      ~p
      ~p→r
      r∧(~p→r)
      T
      		 T
      F
      T
      T
      T
      		 F
      F
      T
      F
      F
      		 T
      T
      T
      T
      F
      		 F
      T
      F
      F

  4. ~(p∨q)↔r

      2. First we begin by writing out the table with all the possible combinations of truth values for each letter in the expression.

      Because here we have 3 letters, p, q and r, we will have 3 columns at the beginning of the truth table labeled p, q and r:

      p
      q
      r
      T
      T
      T
      T
      T
      F
      T
      F
      T
      T
      F
      F
      F
      T
      T
      F
      T
      F
      F
      F
      T
      F
      F
      F

    2. The order of operations says that we always do the parentheses first. So we look at what is inside the parentheses: p∨q.
      Inside the parentheses we have one symbol: ∨.

      The symbol ∨ always joins exactly two expressions; the two expressions it joins may themselves be more complicated statements, but we don't care how complicated they are, as long as we already have a column in the truth table with that logical expression at the top.

      In this expression, the symbol ∨ joins the 2 expressions: p and q.
      Both p and q are at the top of columns in the truth table, so we can use these two columns to fill in the column for p∨q, just by using the definition of the ∨ symbol:

      p
      q
      r
      p∨q
      T
      T
      T
      T
      T
      T
      F
      T
      T
      F
      T
      T
      T
      F
      F
      T
      F
      T
      T
      T
      F
      T
      F
      T
      F
      F
      T
      F
      F
      F
      F
      F

      Remember that a statement containing the symbol ∨ is true if at least one of the statements is true!
      • The first truth value in the p∨q column is T because when p=T and q=T, p∨q=T.
      • The second truth value in the p∨q column is T because when p=T and q=T, p∨q=T.
      • The third truth value in the p∨q column is T because when p=T and q=F, p∨q=T.
      • The fourth truth value in the p∨q column is T because when p=T and q=F, p∨q=T.
      • The fifth truth value in the p∨q column is T because when p=F and q=T, p∨q=T.
      • The sixth truth value in the p∨q column is T because when p=F and q=T, p∨q=T.
      • The seventh truth value in the p∨q column is F because when p=F and q=F, p∨q=F.
      • The eighth truth value in the p∨q column is F because when p=F and q=F, p∨q=F.

    3. Now that we have taken care of everything inside the parentheses, we can take care of the ~ symbol that is outside the parentheses: ~(p∨q).

      Because p∨q is the statement in the expression ~(p∨q), we use the given truth values of p∨q in the table to fill in the corresponding truth values for ~(p∨q):

      p
      q
      r
      p∨q
      ~(p∨q)
      T
      T
      T
      T
      F
      T
      T
      F
      T
      F
      T
      F
      T
      T
      F
      T
      F
      F
      T
      F
      F
      T
      T
      T
      F
      F
      T
      F
      T
      F
      F
      F
      T
      F
      T
      F
      F
      F
      F
      T

      Recall that a statement with the ~ symbol in it is only true if what follows the ~ symbol is false, and vice versa.
      • The first truth value in the ~(p∨q) column is F because when p∨q=T, ~(p∨q)=F.
      • The second truth value in the ~(p∨q) column is F because when p∨q=T, ~(p∨q)=F.
      • The third truth value in the ~(p∨q) column is F because when p∨q=T, ~(p∨q)=F.
      • The fourth truth value in the ~(p∨q) column is F because when p∨q=T, ~(p∨q)=F.
      • The fifth truth value in the ~(p∨q) column is F because when p∨q=T, ~(p∨q)=F.
      • The sixth truth value in the ~(p∨q) column is F because when p∨q=T, ~(p∨q)=F.
      • The seventh truth value in the ~(p∨q) column is T because when p∨q=F, ~(p∨q)=T.
      • The eighth truth value in the ~(p∨q) column is T because when p∨q=F, ~(p∨q)=T.

    4. Now that we have taken care of the parentheses, all that we have left to worry about is the symbol left outside the parentheses: the ↔ symbol in ~(p∨q)↔r.

      The symbol ↔ always joins exactly two expressions; the two expressions it joins may themselves be more complicated statements, but we don't care how complicated they are, as long as we already have a column in the truth table with that logical expression at the top.

      In this expression, the symbol ↔ joins the 2 expressions: ~(p∨q) and r.
      Both ~(p∨q) and r are at the top of columns in the truth table, so we can use these two columns to fill in the column for ~(p∨q)↔r, just by using the definition of the ↔ symbol:

      p
      q
      r
      p∨q
      ~(p∨q)
      ~(p∨q)↔r
      T
      T
      T
      T
      F
      F
      T
      T
      F
      T
      F
      T
      T
      F
      T
      T
      F
      F
      T
      F
      F
      T
      F
      T
      F
      T
      T
      T
      F
      F
      F
      T
      F
      T
      F
      T
      F
      F
      T
      F
      T
      T
      F
      F
      F
      F
      T
      F

      Remember that a statement containing the symbol ↔ is true if both statements are true or both statements are false!
      • The first truth value in the ~(p∨q)↔r column is F because when r=T and ~(p∨q)=F, ~(p∨q)↔r=F.
      • The second truth value in the ~(p∨q)↔r column is T because when r=F and ~(p∨q)=F, ~(p∨q)↔r=T.
      • The third truth value in the ~(p∨q)↔r column is F because when r=T and ~(p∨q)=F, ~(p∨q)↔r=F.
      • The fourth truth value in the ~(p∨q)↔r column is T because when r=F and ~(p∨q)=F, ~(p∨q)↔r=T.
      • The fifth truth value in the ~(p∨q)↔r column is F because when r=T and ~(p∨q)=F, ~(p∨q)↔r=F.
      • The sixth truth value in the ~(p∨q)↔r column is T because when r=F and ~(p∨q)=F, ~(p∨q)↔r=T.
      • The seventh truth value in the ~(p∨q)↔r column is T because when r=T and ~(p∨q)=T, ~(p∨q)↔r=T.
      • The eighth truth value in the ~(p∨q)↔r column is F because when r=F and ~(p∨q)=T, ~(p∨q)↔r=F.

      So the final truth table for this statement will look like this:

      p
      q
      r
      p∨q
      ~(p∨q)
      ~(p∨q)↔r
      T
      T
      T
      T
      F
      F
      T
      T
      F
      T
      F
      T
      T
      F
      T
      T
      F
      F
      T
      F
      F
      T
      F
      T
      F
      T
      T
      T
      F
      F
      F
      T
      F
      T
      F
      T
      F
      F
      T
      F
      T
      T
      F
      F
      F
      F
      T
      F

  5. (~r∧(p→~q))→p

      2. First we begin by writing out the table with all the possible combinations of truth values for each letter in the expression.

      Because here we have 3 letters, p, q and r, we will have 3 columns at the beginning of the truth table labeled p, q and r:

      p
      q
      r
      T
      T
      T
      T
      T
      F
      T
      F
      T
      T
      F
      F
      F
      T
      T
      F
      T
      F
      F
      F
      T
      F
      F
      F

    2. The order of operations says that we always do the parentheses first. So we look at what is inside the parentheses: p→~q.
      Inside the parentheses we have two symbols: → and ~.

      The order of operations says that we always do negations first.
      So we label the first empty column ~q.

       

      Because q is the statement in the expression ~q, we use the given truth values of q in the table to fill in the corresponding truth values for ~q:

      p
      q
      r
      ~q
      T
      T
      T
      F
      T
      T
      F
      F
      T
      F
      T
      T
      T
      F
      F
      T
      F
      T
      T
      F
      F
      T
      F
      F
      F
      F
      T
      T
      F
      F
      F
      T

      Recall that a statement with the ~ symbol in it is only true if what follows the ~ symbol is false, and vice versa.
      • The first truth value in the ~q column is F because when q=T, ~q=F.
      • The second truth value in the ~q column is F because when q=T, ~q=F.
      • The third truth value in the ~q column is T because when q=F, ~q=T.
      • The fourth truth value in the ~q column is T because when q=F, ~q=T.
      • The fifth truth value in the ~q column is F because when q=T, ~q=F.
      • The sixth truth value in the ~q column is F because when q=T, ~q=F.
      • The seventh truth value in the ~q column is T because when q=F, ~q=T.
      • The eighth truth value in the ~q column is T because when q=F, ~q=T.

    3. Now that we have taken care of the negation inside the innermost set of parentheses, we can take care of the → symbol, the only other remaining symbol in the expression inside the parentheses: p→~q.

      The symbol → always joins exactly two expressions; the two expressions it joins may themselves be more complicated statements, but we don't care how complicated they are, as long as we already have a column in the truth table with that logical expression at the top.

      In this expression, the symbol → joins the 2 expressions: p and ~q.
      Both p and ~q are at the top of columns in the truth table, so we can use these two columns to fill in the column for p→~q, just by using the definition of the → symbol:

      p
      q
      r
      ~q
      p→~q
      T
      T
      T
      F
      F
      T
      T
      F
      F
      F
      T
      F
      T
      T
      T
      T
      F
      F
      T
      T
      F
      T
      T
      F
      T
      F
      T
      F
      F
      T
      F
      F
      T
      T
      T
      F
      F
      F
      T
      T

      Be careful! Unlike other logical connectives, the conditional does not treat the first and the second statements it joins in the same way!
      When you work with the conditional connective (→), you must mark the column corresponding to the first statement differently than the way you mark the second column.

      Be sure when you are filling in the column for the conditional statement that you are not confusing the column for the first statement with the column for the second statement!
      In these examples, we distinguish between the two by coloring the column containing the first statement yellow and the column containing the second statement aqua.

      Remember that a statement containing the symbol → is only false if the first statement (the one before the → sign) is true and the second statement (the one after the → sign) is false!
      • The first truth value in the p→~q column is F because when p=T and ~q=F, p→~q=F.
      • The second truth value in the p→~q column is F because when p=T and ~q=F, p→~q=F.
      • The third truth value in the p→~q column is T because when p=T and ~q=T, p→~q=T.
      • The fourth truth value in the p→~q column is T because when p=T and ~q=T, p→~q=T.
      • The fifth truth value in the p→~q column is T because when p=F and ~q=F, p→~q=T.
      • The sixth truth value in the p→~q column is T because when p=F and ~q=F, p→~q=T.
      • The seventh truth value in the p→~q column is T because when p=F and ~q=T, p→~q=T.
      • The eighth truth value in the p→~q column is T because when p=F and ~q=T, p→~q=T.

    4. Now that we have taken care of the innermost set of parentheses, we need to look at what is in the next set of parentheses: ~r∧(p→~q).
      There are 2 symbols left inside this larger set of parentheses: the symbol ~ and the symbol ∧.

      The order of operations says that we always do negations first.
      So we label the first empty column ~r.

       

      Because r is the statement in the expression ~r, we use the given truth values of r in the table to fill in the corresponding truth values for ~r:

      p
      q
      r
      ~q
      p→~q
      ~r
      T
      T
      T
      F
      F
      F
      T
      T
      F
      F
      F
      T
      T
      F
      T
      T
      T
      F
      T
      F
      F
      T
      T
      T
      F
      T
      T
      F
      T
      F
      F
      T
      F
      F
      T
      T
      F
      F
      T
      T
      T
      F
      F
      F
      F
      T
      T
      T

      Recall that a statement with the ~ symbol in it is only true if what follows the ~ symbol is false, and vice versa.
      • The first truth value in the ~r column is F because when r=T, ~r=F.
      • The second truth value in the ~r column is T because when r=F, ~r=T.
      • The third truth value in the ~r column is F because when r=T, ~r=F.
      • The fourth truth value in the ~r column is T because when r=F, ~r=T.
      • The fifth truth value in the ~r column is F because when r=T, ~r=F.
      • The sixth truth value in the ~r column is T because when r=F, ~r=T.
      • The seventh truth value in the ~r column is F because when r=T, ~r=F.
      • The eighth truth value in the ~r column is T because when r=F, ~r=T.

    5. Now that we have taken care of the ~ sign in the expression inside the parentheses, all that we have left to worry about inside the larger parentheses is the ∧ symbol in ~r∧(p→~q).

      The symbol ∧ always joins exactly two expressions; the two expressions it joins may themselves be more complicated statements, but we don't care how complicated they are, as long as we already have a column in the truth table with that logical expression at the top.

      In this expression, the symbol ∧ joins the 2 expressions: ~r and p→~q.
      Both ~r and p→~q are at the top of columns in the truth table, so we can use these two columns to fill in the column for ~r∧(p→~q), just by using the definition of the ∧ symbol:

      p
      q
      r
      ~q
      p→~q
      ~r
      		 ~r∧(p→~q)
      T
      T
      T
      F
      F
      F
      F
      T
      T
      F
      F
      F
      T
      F
      T
      F
      T
      T
      T
      F
      F
      T
      F
      F
      T
      T
      T
      T
      F
      T
      T
      F
      T
      F
      F
      F
      T
      F
      F
      T
      T
      T
      F
      F
      T
      T
      T
      F
      F
      F
      F
      F
      T
      T
      T
      T

      Remember that a statement containing the symbol ∧ is only true if both statements are true!
      • The first truth value in the ~r∧(p→~q) column is F because when ~r=F and p→~q=F, ~r∧(p→~q)=F.
      • The second truth value in the ~r∧(p→~q) column is F because when ~r=F and p→~q=T, ~r∧(p→~q)=F.
      • The third truth value in the ~r∧(p→~q) column is F because when ~r=T and p→~q=F, ~r∧(p→~q)=F.
      • The fourth truth value in the ~r∧(p→~q) column is T because when ~r=T and p→~q=T, ~r∧(p→~q)=T.
      • The fifth truth value in the ~r∧(p→~q) column is F because when ~r=T and p→~q=F, ~r∧(p→~q)=F.
      • The sixth truth value in the ~r∧(p→~q) column is T because when ~r=T and p→~q=T, ~r∧(p→~q)=T.
      • The seventh truth value in the ~r∧(p→~q) column is F because when ~r=T and p→~q=F, ~r∧(p→~q)=F.
      • The eighth truth value in the ~r∧(p→~q) column is T because when ~r=T and p→~q=T, ~r∧(p→~q)=T.

    6. Now that we have finished everything inside the parentheses, we are left with one more symbol: →, which joins the two expressions ~r∧(p→~q) and p.

      When we join these two expressions together with the → symbol leaves us with the final expression: (~r∧(p→~q))→p.

      The symbol → always joins exactly two expressions; the two expressions it joins may themselves be more complicated statements, but we don't care how complicated they are, as long as we already have a column in the truth table with that logical expression at the top.

      In this expression, the symbol → joins the 2 expressions: ~r∧(p→~q) and P.
      Both ~r∧(p→~q) and P are at the top of columns in the truth table, so we can use these two columns to fill in the column for (~r∧(p→~q))→P, just by using the definition of the → symbol:

      p
      q
      r
      ~q
      p→~q
      ~r
      		 ~r∧(p→~q) (~r∧(p→~q))→p
      T
      T
      T
      F
      F
      F
      F
      T
      T
      T
      F
      F
      F
      T
      F
      T
      T
      F
      T
      T
      T
      F
      F
      T
      T
      F
      F
      T
      T
      T
      T
      T
      F
      T
      T
      F
      T
      F
      F
      T
      F
      T
      F
      F
      T
      T
      T
      F
      F
      F
      T
      T
      T
      F
      F
      T
      F
      F
      F
      T
      T
      T
      T
      F

      Be careful! Unlike other logical connectives, the conditional does not treat the first and the second statements it joins in the same way!
      When you work with the conditional connective (→), you must mark the column corresponding to the first statement differently than the way you mark the second column.

      Be sure when you are filling in the column for the conditional statement that you are not confusing the column for the first statement with the column for the second statement!
      In these examples, we distinguish between the two by coloring the column containing the first statement yellow and the column containing the second statement aqua.

      Remember that a statement containing the symbol → is only false if the first statement (the one before the → sign) is true and the second statement (the one after the → sign) is false!
      • The first truth value in the (~r∧(p→~q))→P column is T because when (~r∧(p→~q))=F and P=T, (~r∧(p→~q))→P=T.
      • The second truth value in the (~r∧(p→~q))→P column is T because when (~r∧(p→~q))=F and P=T, (~r∧(p→~q))→P=T.
      • The third truth value in the (~r∧(p→~q))→P column is T because when (~r∧(p→~q))=F and P=T, (~r∧(p→~q))→P=T.
      • The fourth truth value in the (~r∧(p→~q))→P column is T because when (~r∧(p→~q))=T and P=T, (~r∧(p→~q))→P=T.
      • The fifth truth value in the (~r∧(p→~q))→P column is T because when (~r∧(p→~q))=F and P=F, (~r∧(p→~q))→P=T.
      • The sixth truth value in the (~r∧(p→~q))→P column is F because when (~r∧(p→~q))=T and P=F, (~r∧(p→~q))→P=F.
      • The seventh truth value in the (~r∧(p→~q))→P column is T because when (~r∧(p→~q))=F and P=F, (~r∧(p→~q))→P=T.
      • The eighth truth value in the (~r∧(p→~q))→P column is F because when (~r∧(p→~q))=T and P=F, (~r∧(p→~q))→P=F.

      So the final truth table for this statement will look like this:

      p
      q
      r
      ~q
      p→~q
      ~r
      		 ~r∧(p→~q) (~r∧(p→~q))→p
      T
      T
      T
      F
      F
      F
      F
      T
      T
      T
      F
      F
      F
      T
      F
      T
      T
      F
      T
      T
      T
      F
      F
      T
      T
      F
      F
      T
      T
      T
      T
      T
      F
      T
      T
      F
      T
      F
      F
      T
      F
      T
      F
      F
      T
      T
      T
      F
      F
      F
      T
      T
      T
      F
      F
      T
      F
      F
      F
      T
      T
      T
      T
      F

Are you ready to try making your own truth tables now?

Now return to Blackboard to answer the Logic Lecture questions 2: Creating Truth Tables!