In these notes, I will discuss how to symbolize arguments in propositional or symbolic logic, which uses all the basic symbols, especially the use of parentheses. As I have mentioned in my other notes, symbolizing arguments in logic is important because before we can determine the validity of an argument in symbolic logic, we need to symbolize the argument first.
In symbolizing arguments in symbolic logic, we need to do the following:
First, we need to symbolize the argument sentence by sentence.
Second, we have to identify the major connectives in each sentence of the argument. This is important because once we have identified the major connective we will be able to punctuate the sentence or proposition properly.
Third, we need to remember that the variables or constants, such p and q or Y and Z, stand for the entire sentence or proposition, and not for the words within the sentence or proposition itself.
Lastly, we need to put proper punctuations and negation signs if necessary.
Let us consider the example below.
If the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth, then the captain either had good motives for controlling the world or his motives were evil. The airship Albatros had a powerful weapon if and only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground. It is either the case that if the Albatros could destroy objects on the ground its captain could enforce his will all over the earth, or it is the case that if he attempted to blow up the British vessel then his passengers would recognize the hoax. It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax. Furthermore, the captain’s motives for controlling the world were not evil. Therefore, his motives were good. (A, D, W, G, E, S, B, P)
As we can see, the argument above is quite long and indeed complicated. But again, we can easily symbolize this argument because, as I already mentioned, we will symbolize this argument sentence by sentence (or proposition by proposition). So, let’s start with the first sentence.
Sentence 1
If the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth, then the captain either had good motives for controlling the world or his motives were evil.
If we analyze this sentence, it is clear that the major connective is “if…then” or just “then”. Hence, it is a conditional proposition. Now, in symbolizing this sentence, we need to punctuate the antecedent and the consequent.
If we look at the antecedent, we notice that it is a compound proposition whose conjuncts are both conditional propositions. Because there are several connectives in the sentence, then we also need to punctuate the antecedent. Hence, the antecedent (which reads: the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth) is symbolized as follows:
(A ⊃ D) • (D ⊃ W)
As we can see, the consequent of the proposition above is an exclusive disjunction. Thus, we need to underscore the wedge to differentiate it from an inclusive disjunction. The consequent (which reads: the captain either had good motives for controlling the world or his motives were evil) is symbolized as follows: G v E.
Please note that the constants provided at the end of the argument above represent the propositions in the entire argument respectively. Thus, in the first proposition, the constant A stands for “the airship Albatros had powerful weapon”, D stands for “it could destroy objects on the ground”, W stands for “the captain could enforce his will all over the earth”, G stands for “the captain either had good motives for controlling the world”, and E stands for “his motives were evil”.
Now, when we symbolize the entire proposition, we need to punctuate both the antecedent and the consequent because, as the rule says, there should only be one major connective in each proposition. Thus, the proposition above is symbolized as follows:
[(A ⊃ D) • (D ⊃ W)] ⊃ (G v E)
Note: Please apply the principles discussed above in symbolizing the rest of the sentences below. If you have questions or clarifications, please leave a comment below. The PHILO-notes team is happy to respond to them.
Sentence 2
The airship Albatros had powerful weapon if and only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground.
(A ≡ S) • (S ⊃ D)
Sentence 3
It is either the case that if the Albatros could destroy objects on the ground its captain could enforce his will all over the earth, or it is the case that if he attempted to blow up the British vessel then his passengers would recognize the hoax.
(D ⊃ W) v (B ⊃ P)
Sentence 4
It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax.
~ (B ⊃ P)
Sentence 5
Furthermore, the captain’s motives for controlling the world were not evil.
~ E
Sentence 6 (which is the conclusion)
Therefore, his motives were good.
G
In the end, the argument above is symbolized as follows:
or
