Truth Table

Construct Full Truth Tables for the arguments. Indicate whether the argument is valid or invalid. These are the sequents for the arguments in ch 10 ex 2-10:
2. TvP, P + -T
3. in book
4. (C?-F)&-F + C
5. P?W, -(P&W) + -(PvW)
6. E?P, (E&P)?T, T + E
7. S?M, R?M, R + S
8. A?C, C?A, (A&C)?E + (AvC)?E
9. [Nv-(NvP)]?-R, P + R
10. Rv-R, R?(F&S), -R?-(FvS) + -(F&-S)

Ch. 10 Truth Tables
? Method for determining whether an argument is valid or invalid
? Truth Tables provide a listing of all possible truth combinations of the
constituents of an argument.
? To complete a Truth Table, you must determine the truth values of
each of the statements that make up the argument under consideration.
? Basic Truth Table and the connectives: read this in terms of rows and
columns. The very top row gives you the guide columns (on the left of
the double lined column) and then the sentences we are interested in
(e.g. –P, P&Q, etc.). These sentences have an F (for false) or a T (for
true). Whether there is an F or a T on each row under that sentence is
determined by two things: the truth value of the simple statements in
the guide column for that row and how that particular connective (e.g.
the dash, the ampersand, etc.) works. This table is a description of
how the truth value of each of our connectives works.
Guide Columns
P Q – P P & Q P v Q P ? Q P ? Q
T T F T T T T
F T T F T T F
T F F F T F F
F F T

F F T T
When is it true?
You have to memorize the truth functions of each
connective. It is the truth function of each connective that
will enable you to determine the truth value of each
statement composed of those connectives.
Dash
? The winter is HERE.
? If H is true, then – H is false.
? But, if H is false, then – H is true.
Ampersand
? The winter is HERE and it is COLD.
? When is the sentence H & C true?
? Only when it’s true that the winter is here and that it
is cold.
Or
? The winter is HERE or it is COLD.
? H v C is false when both disjuncts are false,
? Otherwise it’s true.
Arrow
? Only false when the antecedent is true and
consequent false.
? Otherwise, it’s true.
Biconditional
? True when both sides of the biconditional have the
same truth value.
Setting up a Truth Table
? Determine the number of simple constituents in the
sequent you are evaluating, e.g. A, B, etc.
? Your table will have the following number of rows:
2 to the nth power (where n is the number of simple
constituents).
? Your table will have the same number of columns
as your sequent has simple constituents plus
premises and conclusion.
? The columns of the simple constituents are referred
to as the guide columns.
? After placing each of the sequent’s premises,
conclusion and simple constituents into a separate
column, you will begin to insert T’s (for true) and
F’s (for false) beneath the statements to set out the
possible truth combinations.
? Begin doing this by taking the total number of rows
you will have and dividing that in half. The first
half of the far right hand guide column should
receive all T’s, and the second half of that column
should receive all F’s.
? Move to the column on the immediate left of the
previous one and place T’s in half of the number of
rows that you placed T’s in the one immediately
before it. Then place that same amount of F’s in the
column. Continue to do this until you have
completed the values for the guide columns.
? With the truth values in hand for the simple
constituents, fill out the truth values for the complex
statements in each row accordingly.
? When the table is complete, examine for
validity/invalidity.
2 Letter Truth Tables
A B
T
F
T
F
T
T
F
F
3 Letter Truth Tables
A B C
T
F
T
F
T
F
T
F
T
T
F
F
T
T
F
F
T
T
T
T
F
F
F
F
Set up a table for this formula, just to get the hang of it.
(A & B) v –A
*
Notice the asterisk under the wedge. This indicates the
main connective of the sentence. The T’s and F’s in this
column (under the ‘v’) tell you the truth value of that whole
sentence for each row.
To find the truth value for the sentence for each row, you
have to start with the smallest parts of the sentence and
work your way to the main connective. First, you fill out
the T’s and F’s for the simple statements in the guide
columns (i.e. the A and the B to the left of the double
lines). Then you use the truth value of the A and B to
determine the truth value of the sentence on that line. I put
a T under the & on row 1 because on that line both A is T
and B is T. Given the definition of the truth function of &
given earlier, an & sentence is true only when both of its
conjuncts are true. Then I put in the truth value for –A. On
row 1, A is T, so –A must be F. I then use those truth
values to determine the truth value of the whole sentence
(i.e. the wedge). Since a disjunction (an “or” sentence) is
A B (A & B) v –A
T T T T F
F T F T T
T F F F F
F F F T T
true as long as at least one of its disjuncts is true, I put a T
under the wedge on row 1.
I follow this same procedure to fill out the T’s and F’s for
each of the last three rows.
Argument Practice: (A ? – B) & – A + B
A B (A ? – B) & – A B
T T F F F F T
F T T F T T T
T F T T F F F
F F T T T T F
*

2 1 4 3
The asterisk marks the main connective of the formula that
serves as our only premise in this sequent. The numbers
(1-4) are meant to show you the order in which I filled in
the T’s and F’s in order to arrive at the truth value of the
main connective (the last column for that sentence).
After filling out the table for this argument (the premises,
in this case we only have one premise, are in their own
column, and the conclusion (B) is in its own column), we
can examine the table to see whether this argument is valid
or invalid. Here is how:
Valid vs. Invalid
? An argument is invalid iff there is one or more rows
on its truth table where all the premises are true and
the conclusion is false.
? Strategy:
1. Fill out the table.
2. Find all lines in which all premises are True.
3. Check to see if any of the lines with all True
premises has a False conclusion.
4. If you find a line with all True premises and a False
conclusion, the argument is INVALID.
5. If all lines with all True premises have True
conclusions, the argument is VALID.
6. If there is no line in which all of the premises are
true, then the argument is VALID.
So, what about the argument we did a truth table for?
A B (A ? – B) & – A B
T T F F F F T
F T T F T T T
T F T T F F F
F F T T T T F *
*

2 1 4 3
Look at each row of the premise. Find the ones where there
is a T, then look to see if B has an F in that row. On row 2,
the premise is true, so look at the conclusion (B). B is true
also, so that does not show the argument is invalid.
However, look at row 4. On row 4, the premise is true and
the conclusion is false. This row shows that the argument
is invalid. To indicate this table, and specifically this row,
is a proof of the argument’s invalidity, put an asterisk next
to that row and identify it as invalid.
The 4th row shows that it is possible for the premise to be
true and the conclusion is false. This means the argument is
invalid.
A v – A, A ? (C & B), -A ? – (C v B) + – (C & – B)
A B C Av-A A ? (C & B) -A ? – (C v B) – (C & – B)
T T T T T F T F T F
F T T T T T F F T F
T F T T F F T F F T
F F T T T T F F F T
T T F T F F T F T F
F T F T T T F F T F
T F F T F F T T T F
F F F T T T T T T F
* * * *
No rows in this table are such that all of the premises are
true and the conclusion is false. In each row where the
premises are all true, the conclusion is also true. This shows
that there is no possible truth assignment such that the
premises are true and the conclusion is false. That means
this argument is valid.
Brief Truth Tables
We will not be doing exercises with brief truth tables,
but just in case you wondered:
? Full Truth Tables can be way too super, ultra,
fantastically huge.
? Brief T.T.’s are a way to shorten the procedure.
? Set up a table just as you normally would, without
filling in any T’s and F’s yet.
? Place T’s in each column that contains a premise.
? Place an F in the conclusion column.
? Fill out the other columns according to the way that
each connective works (truth functionally).
? If the sequent you are evaluating allows the
remaining columns to be completed consistently,
the sequent is invalid.
? If upon assigning T’s to the premises and F to the
conclusion, your remaining assignments force a
contradiction, the sequent is valid.
? Since an argument that can have true premises and a
false conclusion is an invalid argument, the brief
Truth Table will allow us to examine whether our
argument can have this structure, i.e. whether it is
valid or invalid.
Practice:
A ? (B & C), C ? D + D ? A
– W ? (C & G), W ? (B & C) + C
ALPP Ch 10 ex 2-10
Sequents:
2. TvP, P + -T
3. in book
4. (C?-F)&-F + C
5. P?W, -(P&W) + -(PvW)
6. E?P, (E&P)?T, T + E
7. S?M, R?M, R + S
8. A?C, C?A, (A&C)?E + (AvC)?E
9. [Nv-(NvP)]?-R, P + R
10. Rv-R, R?(F&S), -R?-(FvS) + -(F&-S)
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