Practice Problems: Symbolization

The following are some practice problems on symbolization. They cover Part II of forall x: Calgary.

In the following exercises, you will have to enter sentences of TFL into Carnap. You don't have to enter the actual symbols of TFL, but instead can use symbols found on your keyboard. Here's a list:

not ¬ -, ~, not
and /\, &, and
or \/, or
if then ->, >, only if
if and only if ↔︎ <->, <>, 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 show you the solutions you have submitted. If you want to save them, you should copy and paste them into a separate document, or take a screen shot.

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. When you have completely analyzed the sentence, the box will turn green and Carnap will put a checkmark next to it. Then you could, on an actual problem set, click the "Submit" button.

II.1
II.2

Negation

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 (or ~ on your keyboard) 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:

II.3
Mandy doesn't like hiking

Note that once you enter a correct solution, the system will show you a dialog that either says "Perfect match" or ("Logically equivalent to a standard translation" if your solution wasn't exactly identical to the solution Carnap knows about). The English sentence will then be highlighted in green, and Carnap puts a checkmark next to it.

Remember that Carnap forgets the solutions you entered but haven't submitted yet. So once you've solved a problem, always click the "Submit" button!)

Conjunction

"And", "although", "but" are symbolized using ∧. On the keyboard, you can use /\ or & instead. 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]".

II.4
Mandy likes skiing but she doesn't like hiking

Conditionals

Be careful with "if", "provided", and "only if". These are all symbolized with → (or -> on the keyboard), but the order is important.

II.5
Mandy likes skiing provided Sanjiv is from Calgary
II.6
Mandy likes hiking only if Sanjiv is not from Edmonton

Complex symbolization

It's important to be careful and to symbolize the parts of your sentence step by step once the sentence becomes complicated. In the following example, we provide a partial paraphrase to help you along.

You'll also need to symbolize "either or" (disjunctions) in this problem. Remember that "either or" is meant to be inclusive, unless we explicitly specify that it is exclusive (by also saying "but not both"). Disjunctions are symbolized using the symbol ∨, or \/ on your keyboard.

II.7
if [it is not the case that [Sanjiv is from Calgary or from Edmonton]] then [Mandy likes hiking and skiing]

Remember that in TFL only two sentences can be combined using ∧ and ∨. In the following example, the proposed symbolization does not work. try to fix it by putting in parentheses.

II.8
C /\ H /\ S

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.

II.9
Mandy likes neither hiking nor skiing

"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:

II.10
Sanjiv is from Calgary unless he's from Edmonton

(Again, 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. You will also need to know how to symbolize "if and only if".

II.11
Mandy likes hiking unless Sanjiv is from neither Calgary nor Edmonton, but she likes skiing if and only if Sanjiv is from Edmonton