Propositional Logic: Truth Table and Validity of Arguments
In these notes, I will discuss the topic truth table and validity of arguments, that is, I will
discuss how to determine the validity of an argument in propositional logic using the
truth table method.
However, it must be noted that there are two basic methods in determining the validity
of an argument in symbolic logic, namely, truth table and partial truth table method.
Again, in this post, I will only discuss the truth table method, thus the topic “truth table
and validity of arguments”. I will discuss the partial truth table method in my next post.
Validity and Invalidity of Arguments
How do we know whether an argument is valid or invalid?
On the one hand, a truth-functionally valid argument form is an argument that is
composed of propositions that have truth-functional forms such that it is impossible for
its premises to be all true and its conclusion false. In other words, an argument is valid
if it does not contain the form “all true premises and false conclusion”.
On the other hand, a truth-functionally invalid argument form is an argument that is
composed of propositions that have truth-functional forms such that it is possible for its
premises to be all true and its conclusion false. In other words, an argument is invalid if
all of its premises are true and its conclusion false.
Let’s consider the example below.
1. If the squatters settle here, then the cattlemen will be angry and that there will be a
fight for water rights. The squatters are going to settle here. Therefore, there will be a
fight for water rights. (S, C, F)
So, how do we determine the validity of the argument above?
Before we can apply the truth table method in determining the validity of the argument
above, we need to symbolize the argument first. After symbolizing the argument, we
will construct a truth table for the argument, and then apply the rule in determining the
validity of arguments in symbolic logic.
But how do we symbolize the argument above?
, In case one does not know how to symbolize arguments in symbolic logic, please refer
to my previous post titled “Symbolizing Propositions in Symbolic Logic”,
http://philonotes.com/index.php/2018/02/14/symbolizing-propositions-in-symbolic-
logic/.
In symbolizing arguments in symbolic logic, we just need to apply the techniques that
we employed in symbolizing propositions. Hence, we symbolize arguments in symbolic
logic proposition by proposition or sentence by sentence.
Now, if we look at the argument above, the first proposition is “If the squatters settle
here, then the cattlemen will be angry and that there will be a fight for water rights.”
And then we see the constants “S, C, and F” at the end of the argument.
If we recall my discussion on symbolizing propositions, we learned that the variables or
constants provided after the proposition (argument in this case) represent the
propositions in the entire proposition (argument in this case) respectively. Hence, the
constant S stands for “The squatters settle here”, C for “The cattlemen will be angry”,
and F for “There will be a fight for water rights”. Thus, the first proposition “If the
squatters settle here, then the cattlemen will be angry and that there will be a fight for
water rights” is symbolized as follows:
S ⊃ (C • F)
The next proposition in the argument above says “The squatters are going to settle
here”. As we notice, this proposition is just a repeat of the proposition in the previous
statement, and this proposition is symbolized by the constant S. Hence, the second
proposition “The squatters are going to settle here” is symbolized as follows:
S
The third and last proposition is obviously the conclusion because of the signifier
“therefore”. This proposition is also a repeat of the proposition in the first sentence,
which is symbolized by the constant F. Hence, the conclusion “Therefore, there will be a
fight for water rights” is symbolized as follows:
F
At the end of it all, the argument “If the squatters settle here, then the cattlemen will
be angry and that there will be a fight for water rights. The squatters are going to
In these notes, I will discuss the topic truth table and validity of arguments, that is, I will
discuss how to determine the validity of an argument in propositional logic using the
truth table method.
However, it must be noted that there are two basic methods in determining the validity
of an argument in symbolic logic, namely, truth table and partial truth table method.
Again, in this post, I will only discuss the truth table method, thus the topic “truth table
and validity of arguments”. I will discuss the partial truth table method in my next post.
Validity and Invalidity of Arguments
How do we know whether an argument is valid or invalid?
On the one hand, a truth-functionally valid argument form is an argument that is
composed of propositions that have truth-functional forms such that it is impossible for
its premises to be all true and its conclusion false. In other words, an argument is valid
if it does not contain the form “all true premises and false conclusion”.
On the other hand, a truth-functionally invalid argument form is an argument that is
composed of propositions that have truth-functional forms such that it is possible for its
premises to be all true and its conclusion false. In other words, an argument is invalid if
all of its premises are true and its conclusion false.
Let’s consider the example below.
1. If the squatters settle here, then the cattlemen will be angry and that there will be a
fight for water rights. The squatters are going to settle here. Therefore, there will be a
fight for water rights. (S, C, F)
So, how do we determine the validity of the argument above?
Before we can apply the truth table method in determining the validity of the argument
above, we need to symbolize the argument first. After symbolizing the argument, we
will construct a truth table for the argument, and then apply the rule in determining the
validity of arguments in symbolic logic.
But how do we symbolize the argument above?
, In case one does not know how to symbolize arguments in symbolic logic, please refer
to my previous post titled “Symbolizing Propositions in Symbolic Logic”,
http://philonotes.com/index.php/2018/02/14/symbolizing-propositions-in-symbolic-
logic/.
In symbolizing arguments in symbolic logic, we just need to apply the techniques that
we employed in symbolizing propositions. Hence, we symbolize arguments in symbolic
logic proposition by proposition or sentence by sentence.
Now, if we look at the argument above, the first proposition is “If the squatters settle
here, then the cattlemen will be angry and that there will be a fight for water rights.”
And then we see the constants “S, C, and F” at the end of the argument.
If we recall my discussion on symbolizing propositions, we learned that the variables or
constants provided after the proposition (argument in this case) represent the
propositions in the entire proposition (argument in this case) respectively. Hence, the
constant S stands for “The squatters settle here”, C for “The cattlemen will be angry”,
and F for “There will be a fight for water rights”. Thus, the first proposition “If the
squatters settle here, then the cattlemen will be angry and that there will be a fight for
water rights” is symbolized as follows:
S ⊃ (C • F)
The next proposition in the argument above says “The squatters are going to settle
here”. As we notice, this proposition is just a repeat of the proposition in the previous
statement, and this proposition is symbolized by the constant S. Hence, the second
proposition “The squatters are going to settle here” is symbolized as follows:
S
The third and last proposition is obviously the conclusion because of the signifier
“therefore”. This proposition is also a repeat of the proposition in the first sentence,
which is symbolized by the constant F. Hence, the conclusion “Therefore, there will be a
fight for water rights” is symbolized as follows:
F
At the end of it all, the argument “If the squatters settle here, then the cattlemen will
be angry and that there will be a fight for water rights. The squatters are going to
