2 Chapter 2 – Language And Its Common Usage

2.1 – The Nuts and Bolts of Language – Objects, Quantifiers, Negations, Conjunctions, Disjunctions, and Conditional Statements

When we communicate using any language, we often have five things that we use in a sentence: Objects, Quantifiers, Negations, Conjunctions, and Disjunctions.

Objects are things we can touch or events that we can talk about. Examples are things like “door”, “laptop”, “carrot”, “wedding”, “soccer match”, or “movie”.

Quantifiers are words that tells us how much of something we’re considering. Examples are “none”, “some”, or “all”. If we want to say how often something happens we can use similar quantifiers such as “never”, “sometimes”, or “always”.

Negations are the opposite of something. The negation of a sentence such as “I am going to my place of work” is “I am not going to my place of work”. If we negate a negation, we get the original statement. So, the negation of “I am not going to my place of work” is “I am going to my place of work”.

Conjunction – The word “and” is a conjunction. It connect two events or two things. Examples are “I am going to my place of work and I will bring my laptop”, “A rectangle is a parallelogram and a four-sided object”.

Disjunction – The word “or” is a disjunction. It’s used to indicate that more than one thing is possible. Examples are “I’m going to the museum or I’m going to get ice cream”, “Flowers are red, or yellow, or purple, or orange, or violet, or some other color”. When we use the word “or”, sometimes we mean to say that only one of the listed things are possible. This is said to be an exclusive use of “or”. An example of this is “The light is on or the light is off”. Sometimes, we use “or” to mean that any number of things are possible. An example is “Libraries let us borrow books or videos”. This last sentence says that we can borrow both books or videos, or books and videos.

Conditional Statement – A statement which uses “If” and “Then”. An example is “If it snows, then the campus is closed”. Be careful not to confuse a conditional statement with a bi-conditional statement. A bi-conditional statement says that “If and only if it snows, then the campus is closed”.

2.2 – Organization of Language – Syntax

Syntax is the order in which certain types of words appear in a sentence. Different languages have different ways of saying the same thing, although the syntax might be different. For example, if we translate the sentence “I walked home” into Hindi (मैं घर चला गया), the words translate to “I home walked”. Both languages say the same thing but with a different order of words.

2.3 – Logic and The Validity of A Statement  – Truth Tables

Logic is a process by which we can determine the validity of a statement, i.e., whether the statement is true or false . Children first learn the meaning of individual words, and then learn to use the words in a sentence by observing how others use those words. At some point, children learn to determine whether a statement is true or false by recalling what was said or what they saw, and then assigning truth values to the words or phrases in the statement.

Example: The validity of the statement “The dog barked” can be checked by recalling if you heard the dog bark. If you heard it bark, then the statement is true. Otherwise, the statement is false.

Example: The validity of the statement “The dog barked and the cat meowed” can be checked by checking whether we heard the dog bark, and checking whether we heard the cat meow. Then the statement “The dog barked and the cat meowed” is true if both the dog barked and the cat meowed. Otherwise, the statement is false.

Example: The validity of the statement “If it snows, then the campus is closed” can be checked by determining whether it is or has been snowing, and also whether the campus is closed. If both parts of the statement are true, the the whole statement is true. If the first part is true, but the second part is false, the the whole statement is false. It turns out that, if the first part is false, and the second part is true, then the whole statement is true. Also, if both parts are false, then the whole statement is true. Perhaps this seems strange, but we will use truth-tables to sort this out in just a bit.

Example: The statement “The light is on or the light is off” is always a true statement. If the light is on, then the statement is true. If the light is off, the statement is also true.

Example: The statement “The light is on and the light is off” is always a false statement. The light can’t be both on and off simultaneously (at least in the usual sense).

Implications and Truth Tables – We can formalize the results from the examples discussed above.

Logical conjunction truth table

If we have statement p and statement q, then the conjunction, p and q (written as p ∧ q), is true or false depending on the truth value of p and of q. As an example, let p be the statement, “The dog barked”, and q be the statement “The cat meowed”. Then the logical conjunction of p and q is the statement “The dog barked and the cat meowed”. The truth table for this or any conjunction looks like this:

In ordinary language terms, if both p and q are true, then the conjunction p ∧ q is true. For all other assignments of logical values (true or false values) to p and to q the conjunction p ∧ q is false.

Logical disjunction truth table

Using the same statements p and q, we get the following truth table for p or q (written as p ∨ q):

Stated in English, if p is true, or if q is true, then p ∨ q is true, otherwise p ∨ q is false.

Logical conditional truth table

The truth table for a conditional statement “If p then q” (written p→q) might be a little surprising. As an example let’s let p be the statement “It snows”, and q be the statement “The campus is closed”. Then the conditional statement “If it snows, then the campus is closed” has the following truth-table:

The first line of the this truth-table is easy to understand because, if it’s true that it snows and if it’s also true that the campus is closed, then the conditional statement “If it snows, then the campus is closed” makes sense and is true. The second line is also easy to understand because, if it true that it snows but it’s false that the campus is closed, then the conditional statement “If it snows, then the campus is closed” can reasonably be considered to be false.

Now, the third line says that, if it’s false that it snows (it doesn’t snow) but it’s true that the campus is closed, then the conditional statement “If it snows, then the campus is closed” is true. Why is it true? We might be tempted to think that it should be false. However, there are other things that might cause the campus to be closed, perhaps a power outage or a holiday or an earthquake or other possibilities. So, considering all the possible things that might cause the campus to be closed, one of them would be that it snows. We would like the conditional statement to take into account all things that might cause the campus to be closed. This is why the conditional statement is true in this case. Had we stated that the campus is closed only when it snows, then we have a bi-conditional statement which is true only when the “If” part and the “Then” part are both true or both false.

The fourth line says that if it’s false that it snows (it doesn’t snow) and it’s also false that the campus is closed (the campus is open), then the conditional statement “If it snows, then the campus is closed” is true. This also might seem confusing until we realize that a false statement implying a false statement makes the implication true because the implication doesn’t really say anything when both the “If” part and the “Then” part are both false.

If fact, when the “If” part is false, regardless of whether the “Then” part is true or false, we say that the implication is vacuously true. With an “If, Then” statement, we just don’t want to have a true statement implying a false statement.

Here’s a video that might be helpful.

Now we will practice determining whether a statement is true or false.