Practice Problems: Symbolization
The following are some practice problems on symbolization; i.e., they cover Part II of forall x: Calgary.
In the following exercises, you will have to enter formulas of TFL. You don't have to enter the actual symbols of TFL, but instead use symbols found on your keyboard. Here's a list:
|if then →||
|if and only if ↔||
These are practice exercises, which cannot be submitted. On an actual problem set, each exercise will have a "submit" button. Make sure you click that button once you have correctly solved an exercise. Carnap will remember which problems you have completed (i.e., submitted) but will not remember your solutions.
Identifying the main connective
Let's first look at some exercises to make sure you understand the syntax of TFL. Every sentence of TFL is built up from sentence letters using one of the connectives of TFL (¬, ∧, ∨, →, ↔). The last connective used in the construction of a sentence is its main logical operator. In the exercises below, type the main logical operator of the sentence, then hit return. If you pick the right connective, the system will show you the sub-sentence(s) from which the sentence is constructed using the main logical operator you just entered. It may be a sentence letter, or itself constructed from simpler sentences. Enter the main logical operator of the highlighted (in red) subsentence, until only sentence letters are left.
Symbolizing a sentence of English means finding a sentence of TFL which has the same truth conditions as the original English sentence. It requires a symbolization key which pairs basic sentences with sentence letters of TFL. Here's the symbolization key we'll use:
- C: Sanjiv is from Calgary
- E: Sanjiv is from Edmonton
- H: Mandy likes hiking
- S: Mandy likes skiing
To symbolize negation, you should paraphrase grammatical negation (e.g., "Mandy doesn't like hiking") using "not" or "it is not the case that" followed by the sentence being negated (e.g., "not [Mandy likes hiking]"). You can then symbolize the sentence by replacing "not" with the "¬" symbol and any basic sentences by the corresponding sentence letters in the symbolization key. Try it by putting "~H" in the text box below, and hit return to check your solution:
(Note that once you enter a correct solution, the system will replace the sentence with "Success!". If you want to see the original sentence again, you can reload the page. But be careful! Solutions which you have not yet submitted will be lost, even if you've entered a correct solution. So once you've solved a problem, always click "submit"!)
"And", "although", "but" are symbolized using ∧. Remember to start by dealing with pronouns and coordination by first paraphrasing to make all the basic sentences explicitly part of our sentence. E.g., "Mandy likes skiing but she doesn't like hiking" amounts to "Both [Mandy likes skiing] and not [Mandy likes hiking]".
Be careful with "if", "provided", and "only if". These are all symbolized with →, but the order is important.
It's important to be careful and symbolize the parts of your sentence step by step once they become complicated. In the following example, we provide a partial paraphrase to help you along:
Neither-nor and unless
"Neither A nor B" can be paraphrased as "Both not A and not B" (and alternatively as "Not either A or B"). Then symbolize this paraphrase as you ordinarily would. So for instance you can paraphrase
Mandy likes neither hiking nor skiing
alternatively as one of
Both [it is not the case that [Mandy likes hiking] and [it is not the case that [Mandy likes skiing]]
It is not the case that [either [Mandy likes hiking] or [Mandy likes skiing]]
Try both ways---the system will accept either as a correct solution.
"A unless B" can be paraphrased as "Either A or B" or as "A if not B". Once you have paraphrased it, symbolize it as you ordinarily would. Try both ways:
(If you've provided a solution that exactly matches the model solution, the system will say "perfect match!" when you hit enter. Other possible symbolization will earn you a "logically equivalent to the standard translation". That's ok: you'll still get full credit for it.)
Let's try a complex example that uses all of what we've done: