Sentential Logic

Sentential Logic

 Complete, then copy and paste the entire worksheet or the truth tables into the Discussion Forum as your post

Step 1.  Represent these statements with variables and symbols. (For example, E could stand for “Elmo is red,” and B for “Big Bird is green,” but feel free to change these variables per your own preference). Complete this table below:

Given Sentence	Your Formal Representation
Elmo is red.	E
Big Bird is green.	B
Either Elmo is red or Big Bird is green.	E v B
Big Bird is green and Elmo is red.	B & E
It is not the case that Big Bird is green	~ B

Step 2.  Complete the following simple table with four combinations of T and F, when T (stands for true) and F (for false):

E	B
T	T
T	F
F	T
F	F

Step 3. Now complete the Truth Table for the rest of the sentences (T stands for true, and F stands for false). The color-coding will help you determine where some of the same information can be transferred over from the previous tables.

Elmo is red	Big Bird is green.	Either Elmo is red or Big Bird is green.	Big Bird is green and Elmo is red.	It is not the case that Big Bird is green
E	B	E v B	B & E	~ B
T	T	T	T	F
T	F	T	F	T
F	T	T	F	F
F	F	F	F	T

Step 4.

Interpret what this Truth table tells you.  What’s your claim regarding Elmo and Big Bird’s colors using the truth table you constructed?  Does the information (T and F) from your Truth Table align with reality? How is this process and constructing a Truth Table useful?

From the table, it is clear that;

• E is True
• B is False
• E v B is True
• B & E is False

Accordingly, the truth table aligns with the reality of the colors of Elmo and Big Bird.

• E is True because Elmo is indeed red in color in reality.
• B is False because Big Bird is not green in color but is yellow in color, and therefore, the truth value of that sentence is False hence (~B)
• E v B is True, the truth value of E v B is T if at least one value of E or B is T

From the statement either Elmo is red or Big Bird is green (E v B), we know E=T while B= F, and therefore, the truth value is T.

• B & E is False; the Truth value of E & B is False if at least one value of E or B is F

From the statement, Big Bird is green and Elmo is red, we know B = F while E = T and therefore, the truth value of the statement is false (F)

The depiction of these conditions in a truth table simplifies the interpretation and understanding of the truth values and sentence logic of the sentences.

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Unit 10 Discussion Assignment

Take a Break!

This assignment first asks you to “Take a Break with Elmo and Big Bird.” If you’re not familiar with these Sesame Street characters, watch this video.

Now that we’ve established our context, let’s have some fun with Elmo, Big Bird and sentential logic (SL), also known as propositional logic. Let’s remind ourselves of several features of SL:

• SL is a truth-functional logic, in the sense that every statement has only one truth value — it must be either true or
• An atomic sentence is a sentence that contains no sentential
• A compound sentence is a sentence that contains one or more sentential
• The truth value of a compound sentence is a function of the sentential connectives it contains.
• Each sentential connective has a corresponding truth table which makes explicit the circumstances under which it is considered true (or false)

Here are the sentential connectives of SL and their corresponding symbols:

Conjunction (frequently rendered as “and”)	&
Disjunction (frequently rendered as “or”)	∨
Negation (frequently rendered as “not”)	~
Conditional (frequently rendered as “if … then”)	⊃
Biconditional (frequently rendered as “if and only if”)	≡

Let’s now consider the following set of propositions:

• Elmo is
• Big Bird is
• Either Elmo is red or Big Bird is green.
• Big Bird is green and Elmo is
• It is not the case that Big Bird is green

Once we know the truth-value of the first two sentences (whether they are true or false), we can establish the truth-values of the others by applying the rules for their respective connectives.

For this assignment, complete the worksheet below and add it to the Discussion Forum. Then read and comment on at least two students’ posts by reviewing the accuracy of their truth tables.

Worksheet

(Complete, then copy and paste the entire worksheet or the truth tables into the Discussion Forum as your post)

Step 1. Represent these statements with variables and symbols. (For example, E could stand for “Elmo is red,” and B for “Big Bird is green,” but feel free to change these variables per your own preference). Complete this table below:

 

Given Sentence	Your Formal Representation
Elmo is red.	E
Big Bird is green.	B
Either Elmo is red or Big Bird is green.
Big Bird is green and Elmo is red.
It is not the case that Big Bird is green

Step 2. Complete the following simple table with four combinations of T and F, where T stands for “true” and F stands for “false.” Use the standard array as explained in our text.

E	B

Step 3. Now complete the truth table for the rest of the sentences (again, T stands for “true,” and F stands for “false”). The color-coding will help you determine where some of the same information can be transferred over from the previous tables.

Elmo is red	Big Bird is green.	Either Elmo is red or Big Bird is green.	Big Bird is green and Elmo is red.	It is not the case that Big Bird is green
E	B

Step 4.

Interpret what this truth table tells you. What’s your claim regarding Elmo and Big Bird’s colors using the truth table you constructed? Does the information (T and F) from your truth table align with reality? How is the process of constructing a truth table useful?

End of the Worksheet