In these notes, I will be discussing the topic “symbolizing statements in propositional (or symbolic) logic.” This is very important because, as I have already said in my earlier post before we can determine the validity of an argument in symbolic logic by applying a specific rule, we need to symbolize the argument first. So, how do we symbolize propositions in symbolic logic?
First, we need to identify the major connective. This is because once we have identified the major connective, we will be able to punctuate the proposition properly.
Second, we have to keep in mind that the variables or constants, such p and q or Y and Z, stand for the entire proposition, and not for the words within the proposition itself.
Third and last, we need to put proper punctuation and negation if necessary.
Let us consider the examples below.
- If the squatters settle here, then the cattlemen will be angry and there will be a fight for water rights. (p, q, r)
As we can see, this example is a combination of a conditional proposition and a conjunctive proposition. However, if we analyze the proposition, it becomes clear to us that it is a conditional proposition whose consequent is a conjunctive proposition. Thus, the major connective in this proposition is “then.” Hence, when we symbolize the proposition, we need to punctuate the consequent. So, if we let p stand for “The squatters settle here,” q for “The cattlemen will be angry,” and r for “There will be a fight for water rights,” then the proposition is symbolized as follows:
p ⊃ (q • r)
- If either the butler or the maid is telling the truth, then the job was an inside one; however, if the lie detector is accurate, then both the butler and the maid are telling the truth. (p, q, r, s)
This example is indeed a complicated one. But it can be easily symbolized.
If we analyze the proposition, it becomes clear that it is a conjunctive proposition whose conjuncts are both conditional propositions with a component inclusive disjunction and conjunction respectively.
Now, if we let
p stands for “The butler is telling the truth”
q for “The maid is telling the truth”
r for “The job was an inside one” and
s for “The lie detector is accurate”
then we initially come up with the following symbol: p v q ⊃ r • s ⊃ p • q
The symbol above, however, is not yet complete. In fact, it remains very complicated. So, we have to punctuate it.
Since the major connective of the proposition is “however,” then we have to punctuate the component conjuncts. Thus, we initially come up with the following symbol:
[p v q ⊃ r] • [s ⊃ p • q]
However, the symbolized form of the proposition remains complicated because the component conjuncts have not been properly punctuated. As already said, there should only be one major connective in a proposition. So, let us punctuate the first conjunct.
Since it is stated in the first conjunct that the proposition is a conditional proposition whose antecedent is an inclusive disjunction, then we have to punctuate p v q. Thus, we initially come up with the following symbol:
[(p v q) ⊃ r] • [s ⊃ p • q]
And then let us punctuate the second conjunct. Since it is stated in the second conjunct that the proposition is a conditional proposition whose consequent is a conjunctive proposition, then we have to punctuate p • q. Thus, we come up with the following symbol:
[(p v q) ⊃ r] • [s ⊃ (p • q)]
Now, the symbol appears to be complete. Thus, the final symbol of the proposition “If either the butler or the maid is telling the truth, then the job was an inside one; however, if the lie detector is accurate, then both the butler and the maid are telling the truth” is as follows:
[(p v q) ⊃ r] • [s ⊃ (p • q)]
- Neither Lucas is hard-working nor is he intelligent. (p, q)
This example is obviously an inclusive disjunction; hence, we may initially symbolize the proposition as p v q. However, the words “Neither…nor” is a signifier of a negation, and these words suggest that the entire proposition is negated. Thus, we finally symbolize the proposition “Neither Lucas is hard-working nor is he intelligent” as follows:
~ (p v q)
Please note that ~ (p v q) is not the same with ~ p v ~ q. And ~ p v ~ q is not the proper symbol of example #3 because the words “Neither…nor” suggest that the proposition has to be completely negated. As we learned in my previous post titled “Punctuating Propositions in Symbolic Logic” (see http://philonotes.com/index.php/2018/02/11/punctuating-propositions-in-symbolic-logic/), when the proposition is completely negated, then the entire proposition has to be punctuated.
But let me explain why ~ (p v q) is not the same with ~ p v ~ q. If we recall, the rules in inclusive disjunction say “The inclusive disjunction is true if at least one of the disjuncts is true.” With this, let us determine the truth value of ~ (p v q) and ~ p v ~ q in order to prove that they are not the same.
Let us assign the truth value “true” for p and “false” for q.
- It is not the case that the manager will resign if she does not receive a salary increase. (p, q)
Please note that since the negation sign “It is not the case” precedes the entire proposition, then the entire proposition has to be negated. Thus, we need to punctuate the entire proposition and put the negation sign outside of it.
As I discussed in one of my previous posts, we learned that 1) the variables provided after the proposition represent the propositions in the entire proposition respectively, and 2) since in the example above the antecedent is written after the consequent, then q must be our antecedent and p our consequent. Hence, we initially come up with the following symbol: ~q ⊃ p. Please note that q is negated because it is clearly specified in the proposition. In other words, the proposition contains a negation sign “not.”
Now, since the negation sign “It is not the case” precedes the entire proposition, then, again, the entire proposition must be negated. Thus, we finally symbolize the proposition “It is not the case that the manager will resign if she does not receive a salary increase” as follows:
~ (~q ⊃ p)
- If it is not the case that the professor will take a leave of absence if and only if the administration allows him to, then there must be another good reason why the professor will take a leave of absence. (p, q, r)
In this example, since the negation sign “It is not the case” does not precede the entire proposition, then we do not negate the entire proposition. We only negate the proposition where the negation sign immediately precedes. Thus, the negation sign in the example above only negates the proposition “The professor will take a leave of absence if and only if the administration allows him to.” It does not clearly negate the proposition “There must be another good reason why the professor will take a leave of absence.”
Now, if we analyze the proposition, we notice that:
p stand for “The professor will take a leave of absence”
q for “The administration allows him to” and
r for “There must be another good reason why the professor will take a leave of absence.”
Please note that we do not repeat the variable “p” for the proposition “There must be another good reason why the professor will take a leave of absence” because the thought of the proposition is completely changed. This is because of the addition of the idea “There must be another good reason.” Thus, instead of repeating the variable “p,” we use the variable “r” to represent the proposition “There must be another good reason why the professor will take a leave of absence.”
So, we symbolize the proposition “If it is not the case that the professor will take a leave of absence if and only if the administration allows him to, then there must be another good reason why the professor will take a leave of absence” as follows:
~ (p ≡ q) ⊃ r