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Homework 4- Propositional Logic: Truth Tables
Here we will see how much easier it is to determine the semantic properties of sentences and sets
of sentences if we use the truth table method.
Truth tables capture all of the same information as our semantic proofs did, i.e.:
¬ is a true if and only if  is false.
(˄  is a true if and only if is true and is true.
(˅  is a true if and only if is true or is true.
(→  is a true if and only if is false or is true.
(1)
(2)
(3)
(4)
With our syntax and semantics for L we have a procedure for determining the truth value of any
sentence of L from the truth value of the propositional variables in the sentence. The easiest way
to utilize this is via the method of truth tables. A truth table simply encodes the information
given in the semantics of L. Here are the truth tables for the logical connectives.
¬
T || F
F || T
|
T |
T |
F |
F |
(˄ 
T ||
T
F ||
F
T ||
F
F ||
F
|
T |
T |
F |
F |
(˅ 
T ||
T
F ||
T
T ||
T
F ||
F
|
T |
T |
F |
F |
(→ 
T ||
T
F ||
F
T ||
T
F ||
T
The Greek letters in the truth tables are schematic, standing for any wff of L. Thus, we use the
schematic letters in the above truth tables to construct bigger truth tables for any wff of L. For
example, take the sentence ‘¬(A ˄ B)’ of L. I can construct a truth table for this sentence in
three steps.
First I make the table itself.
A
T
T
F
|
|
|
|
B
T
F
T
|| ¬(A ˄ B)
||
||
||
2
F | F ||
Each row on the table corresponds to a certain assignment of truth-values to the propositional
variables in our sentence. So I next carry the truth values of the variables over to the right hand
side.
A
T
T
F
F
|
|
|
|
|
B
T
F
T
F
|| ¬(A ˄ B)
|| T
T
|| T
F
|| F
T
|| F
F
Then I utilize the above schematic truth tables to discern the truth value on each row for the
logical operator that binds A and B, in this case ‘˄’. Thus I have:
A
T
T
F
F
|
|
|
|
|
B
T
F
T
F
|| ¬(A ˄ B)
|| T T T
|| T F F
|| F F T
|| F F F
Note that we fill these in in exactly the same order as we do semantic proofs.
Now I’ve determined for each assignment of truth-values to propositional variables what the
truth-value of ‘(A ˄ B)’ is (the truth-value underneath the ‘˄’). Now I need to use the schematic
truth table for ‘¬’ to determine the truth-value of the whole sentence on each line.
A
T
T
F
F
|
|
|
|
|
B
T
F
T
F
||
||
||
||
||
¬(A ˄ B)
F
T
T
T
T
T
F
F
T
F
F
F
T
F
T
F
The truth-value under the negation sign corresponds to the truth-value of the whole sentence,
given the truth-values of propositional variables on the left (again, negation would be the last
operator added by a syntax proof and the last rule used in a semantic proof). For example, I’ve
now proven that when ‘A’ is true and ‘B’ is true, that ‘¬(A ˄ B)’ is false.
Here’s an example of a completed truth table for the sentence ‘(¬P → (Q ˄ R))’.
P
T
T
T
T
F
F
|
|
|
|
|
|
|
Q
T
T
F
F
T
T
| R ||
| T ||
| F ||
| T ||
| F ||
| T ||
| F ||
(¬P → (Q ˄ R))
FT
FT
FT
FT
TF
TF
T
T
T
T
T
F
T
T
F
F
T
T
T
F
F
F
T
F
T
F
T
F
T
F
3
F | F | T || T F F F F T
F | F | F || T F F F F F
Thus, we know that if ‘P’, ‘Q’, and ‘R’ are all true then we also know that ‘(¬P → (Q ˄ R))’ will
also be true.
A Note on Translation:
The intended interpretations of our logical constants are: ‘˅’ means ‘or’, ‘˄’ means ‘and’, ‘¬’
means ‘it is not the case that’, and ‘→’ means ‘implies’. Given this, a surprising large number of
natural language sentences can be translated into L. In this manner, we can determine the validity
of a surprisingly large number of natural language arguments by translating the sentences of the
argument into L and then using the truth-table method to determine if the conclusion of the
argument is entailed by the premises.
For example, consider the following argument.
If Jones goes to the bank, then he takes out money. Jones does not take out money. Therefore
Jones does not go to the bank.
Translation Manual:
Let:
P = Jones goes to the bank.
Q = Jones takes out money.
Premisses:
Conclusion:
(P → Q), ¬Q
¬P
Then we can use the truth-table method to verify that {(P → Q), ¬Q)} entails ¬P, which we will
do in the next homework assignment.
Some English words translate into L in ways that seem surprising at first. For example ‘P unless
Q’ translates into ‘(¬Q → P)’ and ‘P only if Q’ translates ‘P → Q’, and ‘P if Q,’ translates as ‘Q
→ P.’ One completely essential bit of advice- if you get negated sentences in the argument, only
translate the non-negated part of the sentence in your translation manual. When you use the
negated sentence in an argument, just put a ‘¬’ in front of it, as in the argument above.
One might reasonably ask right now if the semantic interpretation of our formulas really do
license the translations that we perform. In particular, does the ‘˅’ of L really mean the same
thing as the ‘or’ of English, and does the ‘→’ of L really mean the same thing as ‘implies’ or ‘if.
. . then’? The answer to both of these questions is ‘no’, but they do mean the same thing as their
English translates in a wide variety of cases. Moreover this wide variety of cases cover much of
the deductively relevant English usage.
First we’ll consider ‘˅’. Remember that the truth table for ‘˅’ is,
| (v 
T | T ||
T

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T | F ||
F | T ||
F | F ||
T
T
F
The ‘˅’ for our language is often called ‘inclusive disjunction’ because it is true if both of its
disjuncts are true. This strikes some people as weird because they reason that the value at the
first row of the table should be ‘F’. In fact one could define truth-functionally an ‘exclusive
disjunction’ with the following truth table:
|
T |
T |
F |
F |
(˅E 
T ||
F
F ||
T
T ||
T
F ||
F
There are four kinds of reasons that we don’t take the exclusive disjunct to be an operator in our
language. The first is that most English uses of ‘or’ are really inclusive disjunctions. For
example, if I say, ‘I’m going to the store or I’m going to a movie’, and I end up doing both,
translating the ‘or’ as exclusive disjunct would have made me say something false. For
embedded uses of ‘or’, such as in, ‘If I eat at the restaraunt, then I will fall asleep from the MSG
in the food, or I will get hungry thirty minutes later,’ it is even clearer that exclusive disjunct
doesn’t as an adequate translation.
Second, in cases where someone says a disjunction, both disjuncts are true, and we feel we’ve
been mislead, this is arguably (according to Grice) always only because the person knew both
disjuncts were true but did not assert a conjunction. It is a little bit like saying there is an animal
behind someone when you see that it is a bear. You said something true, but violated the
conversational maxim to say the logically strongest thing for which you have evidence.
Thirdy, we can express exclusive disjunct in L. The sentence ((P ˅ Q) ˄ ¬(P ˄ Q)), is logically
equivalent to exclusive dijunction. Check it by doing its truth-table.
The third reason concerns using inclusive disjunction as a basic operator when doing certain
mathematical proofs about logical languages. If you continue to study logic you will see that in
these contexts it is much better to deal with a language that takes the inclusive disjunction as
primitive.
Many people are perplexed by the truth table for ‘→’.
|
T |
T |
F |
F |
(→ 
T ||
T
F ||
F
T ||
T
F ||
T
Some of this perplexity can be cleared up by the realization that we are taking every sentence of
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our language to be true or false. Thus there has to be some value for the last two rows of the
truth table for ‘-->’. Consider the three other possibilities (Since everyone will agree that the
first two rows of the truth table are correct there are only three other possibilities).

| (→1  | (→2  | (→3 
T | T ||
T
T | T ||
T
T | T ||
T
T | F ||
F
T | F ||
F
T | F ||
F
F | T ||
T
F | T ||
F
F | T ||
F
F | F ||
F
F | F ||
T
F | F ||
F
None of these correspond to anything like the intuitive meaning of ‘if. . . then’. The first one
would have (A → B) mean the same thing as B. The second one is the same truth table as ‘if and
only if’ (which we can formulate in our language as ((A → B) ˄ (B → A)), and the third one is
the same truth table as ‘˄’. Since we know that none of these mean the same thing as ‘if. . .
then,’ and there are only four choices, we use the truth table we’ve picked. Later in the quarter,
after we have done work on deduction, we will be able to show why this choice does not get us
into any trouble.
This being said, the overwhelming majority of philosophers now accept that many natural
language uses of ‘if. . . then’ do not have the same meaning. Consider the sentence “If I’m going
to the store, then I’m going to buy some cigarettes.” Now assume I do not go to the store and I
do not buy cigarettes. Is the sentence true or false?
Many people have the intuition that the sentence is true if it were the case that had I gone to the
store then I would have bought cigarettes and false if it were the case that had I gone to the store
then I would not have bought cigarettes. But that means that the truth-value of the sentence is not
merely a function of the truth-value of its parts. As a result of this, the logical semantics of
natural language conditionals must involve more than just truth-values. It typically also involves
possible worlds, and is typically taught in modal logic classes (such as LSUs 4011).
--------------------------------------------------------------------------------------------------------------------Homework 4
Construct truth-tables for the following sentences.
1. ((¬P ˄ Q) ˅ P)
2. ((P → Q) → ¬P)
3. ¬(P ˅ Q) → (¬P ˄ Q)
4. ((P ˄ R) ˄ (P ˅ ¬Q))
5. ((P → ¬R) → ¬Q) → ¬P))