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Symbolic Logic : Symbolic Logic

Study concepts, example questions & explanations for Symbolic Logic

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Example Questions

Example Question #1 : Sentential Logic

Which of the following statements is NOT a definition of sentential logic?

Possible Answers:

If and are formulas then $A\vee B$ is a formula as well.

If and are formulas then $A\rightarrow B$ is a formula as well.

If and are formulas then $A\Leftrightarrow B$ is a formula as well.

If is a formula then $\sim A$ is a formula as well.

Only , , , and are formulas.

Correct answer:

Only , , , and are formulas.

Explanation:

stuff

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Example Question #2 : Sentential Logic

Which of the following statements is NOT a definition of sentential logic?

Possible Answers:

If is a formula then $\sim A$ is a formula as well.

If and are formulas then $A\rightarrow B$ is a formula as well.

Only , , , and are formulas.

If and are formulas then $A\Leftrightarrow B$ is a formula as well.

If and are formulas then $A\vee B$ is a formula as well.

Correct answer:

Only , , , and are formulas.

Explanation:

It is important to recall that sentential logic has a very specific definition that outlines and describes different formulas.

There are seven different statement criteria when discussing sentential logic and they are as follows.

I. If is a formula then $\sim A$ is a formula as well.

II. If and are formulas then $A\vee B$ is a formula as well.

III. If and are formulas then $A\rightarrow B$ is a formula as well.

IV. If and are formulas then $A\Leftrightarrow B$ is a formula as well.

V. If and are formulas then $A\wedge B$ is a formula as well.

VI. All upper case letters are formulas

VII. Nothing else is a formula.

Looking at the possible answer selections, I, II, III, and IV are part of the sentential logic definition thus, "Only , , , and are formulas." is NOT in the definition. This can be verified by part VI in the definition which states that all upper case letters are formulas.

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Example Question #3 : Sentential Logic

Which of the following statements is part of the definition for sentential logic?

Possible Answers:

Anything and everything can be considered a formula.

If is a formula then so is $\sim a$ .

If and are formulas then $A\Leftrightarrow B$ is a formula as well.

If is a formula then so is $B\times A$ .

Every lower case letter is a formula.

Correct answer:

If and are formulas then $A\Leftrightarrow B$ is a formula as well.

Explanation:

It is important to recall that sentential logic has a very specific definition that outlines and describes different formulas.

There are seven different statement criteria when discussing sentential logic and they are as follows.

I. If is a formula then $\sim A$ is a formula as well.

II. If and are formulas then $A\vee B$ is a formula as well.

III. If and are formulas then $A\rightarrow B$ is a formula as well.

IV. If and are formulas then $A\Leftrightarrow B$ is a formula as well.

V. If and are formulas then $A\wedge B$ is a formula as well.

VI. All upper case letters are formulas

VII. Nothing else is a formula.

Looking at the possible answer selections only IV is part of the sentential logic definition thus, "If and are formulas then $A\Leftrightarrow B$ is a formula as well." is in the definition.

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Example Question #1 : Truth Tables

Looking at the following truth table, find the missing operator if

$\text{result}=p\bigstar q$ .

$\begin{tabular}{|c|c|c|} \hline p& q&result\\ \hline T&F&F\\ T&T&T\\ F&T&F\\ F&F&T\\ \hline \end{tabular}$

Possible Answers:

$\lnot$

$\rightarrow$

$\vee$

$\wedge$

Correct answer:

$\rightarrow$

Explanation:

To help solve for the missing operator in this truth table, first recall the different operators and there meanings.

$\\\vee:\text{Or} \\\wedge:\text{And} \\\lnot:\text{Not} \\\rightarrow:\text{Implies} \\\Leftrightarrow:\text{Equivalence}$

In truth tables when the "or" operator is used translates to, either and (the constants) being true. When the "and" operator is used that means that for the result to hold true both the constants must be true. The "not" operator negates the answer. The "implies" that the first constant results in the second constant . Lastly, the "equivalency" operator signifies that both constants are the same.

Looking at the truth table,

$\begin{tabular}{|c|c|c|} \hline p& q&result\\ \hline T&F&F\\ T&T&T\\ F&T&F\\ F&F&T\\ \hline \end{tabular}$

and result in a true statement whenever the first constant is the same as the second constant. Therefore, the missing operator is "implies".

In mathematical terms the missing operator is $\rightarrow$ .

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Example Question #2 : Truth Tables

Looking at the following truth table, find the missing operator if

$\text{result}=p\bigstar q$ .

$\begin{tabular}{|c|c|c|} \hline p& q&result\\ \hline T&F&T\\ T&T&T\\ F&T&T\\ F&F&F\\ \hline \end{tabular}$

Possible Answers:

$\vee$

$\wedge$

$\lnot$

Correct answer:

$\vee$

Explanation:

To help solve for the missing operator in this truth table, first recall the different operators and there meanings.

$\\\vee:\text{Or} \\\wedge:\text{And} \\\lnot:\text{Not} \\\rightarrow:\text{Implies} \\\Leftrightarrow:\text{Equivalence}$

Looking at the truth table,

$\begin{tabular}{|c|c|c|} \hline p& q&result\\ \hline T&F&T\\ T&T&T\\ F&T&T\\ F&F&F\\ \hline \end{tabular}$

and result in a true statement whenever one of the constants is true. Therefore, the missing operator is "or".

In mathematical terms the missing operator is $\vee$ .

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Example Question #3 : Truth Tables

Looking at the following truth table, find the missing operator if

$\text{result}=\bigstar p$ .

$\begin{tabular}{|c|c|} \hline p&result\\ \hline T&F\\ F&T\\ \hline \end{tabular}$

Possible Answers:

$\wedge$

$\lnot$

$\rightarrow$

$\Leftrightarrow$

$\vee$

Correct answer:

$\lnot$

Explanation:

To help solve for the missing operator in this truth table, first recall the different operators and there meanings.

$\\\vee:\text{Or} \\\wedge:\text{And} \\\lnot:\text{Not} \\\rightarrow:\text{Implies} \\\Leftrightarrow:\text{Equivalence}$

Looking at the truth table,

$\begin{tabular}{|c|c|} \hline p&result\\ \hline T&F\\ F&T\\ \hline \end{tabular}$

The result is always opposite of the value of . Therefore, the missing operator is "not".

In mathematical terms the missing operator is $\lnot$ .

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Example Question #1 : Predictions & Quantifiers

Which of the following symbols is a "quantifier"?

Possible Answers:

$\forall$

$\Rightarrow$

$\lnot$

$\therefore$

Correct answer:

$\forall$

Explanation:

Symbolic logic describes English statements using mathematical symbols. These mathematical symbols can be categorized into five areas.

I. Predicates: , $\perp$

II. Terms: Terms are the variables that represents the objects and constants of a statement.

III. Quantifiers: $\forall$ , $\exists$

IV. Punctuation: (,)

V. Connectives : $\&$ , $\vee$ , $\wedge$ , $\Rightarrow$ , $\Leftrightarrow$ , $\lnot$ , $\rightarrow$

This particular question asks to identify the "quantifier".

Since there are only two symbols that are categorized as "quantifiers", $\forall$ and $\exists$ ,and the "for all" symbol $(\forall)$ is the only one present in the answer choices, that is the correct answer.

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Example Question #2 : Predictions & Quantifiers

Which of the following symbols is a "quantifier"?

Possible Answers:

$\lnot$

$\therefore$

$\Rightarrow$

$\exists$

Correct answer:

$\exists$

Explanation:

Symbolic logic describes English statements using mathematical symbols. These mathematical symbols can be categorized into five areas.

I. Predicates: , $\perp$

II. Terms: Terms are the variables that represents the objects and constants of a statement.

III. Quantifiers: $\forall$ , $\exists$

IV. Punctuation: (,)

V. Connectives : $\&$ , $\vee$ , $\wedge$ , $\Rightarrow$ , $\Leftrightarrow$ , $\lnot$ , $\rightarrow$

This particular question asks to identify the "quantifier".

Since there are only two symbols that are categorized as "quantifiers", $\forall$ and $\exists$ ,and the "exists" symbol $(\exists)$ is the only one present in the answer choices, that is the correct answer.

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Example Question #3 : Predictions & Quantifiers

Which of the following symbols "predicates"?

Possible Answers:

$\forall$

$\wedge$

$\exists$

$\vee$

Correct answer:

Explanation:

Symbolic logic describes English statements using mathematical symbols. These mathematical symbols can be categorized into five areas.

I. Predicates: , $\perp$

II. Terms: Terms are the variables that represents the objects and constants of a statement.

III. Quantifiers: $\forall$ , $\exists$

IV. Punctuation: (,)

V. Connectives : $\&$ , $\vee$ , $\wedge$ , $\Rightarrow$ , $\Leftrightarrow$ , $\lnot$ , $\rightarrow$

This particular question asks to identify the "quantifier".

Since there are only two symbols that are categorized as "predicates", and $\perp$ ,and the "equal" symbol (=) is the only one present in the answer choices, that is the correct answer.

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Example Question #1 : First Order Logic

Identify the complex sentence of the following statement:

Sally has a basketball and she sells it to her friend Bob.

Possible Answers:

$\text{Sold(Sally, Basketball, Bob) }\Rightarrow \text{Owns(Sally, Basketball)}$

$\lnot\text{Sold(Sally, Basketball, Bob) }\Rightarrow \text{Owns(Sally, Basketball)}$

$\text{Own(Sally, Basketball, Bob) }\Rightarrow \text{Sold(Sally, Basketball)}$

$\text{Own(Sally, Basketball, Bob) }\Rightarrow \lnot\text{Sold(Sally, Basketball)}$

$\text{Sold(Sally, Basketball, Bob) }\Rightarrow \lnot\text{Owns(Sally, Basketball)}$

Correct answer:

$\text{Sold(Sally, Basketball, Bob) }\Rightarrow \lnot\text{Owns(Sally, Basketball)}$

Explanation:

First-order logic statements can be described in complex sentences by using logic symbols.

Recall the following logic symbols.

$\lnot$ means "not"

$\Rightarrow$ means "implies"

$\vee$ means "or"

$\wedge$ means "and"

$\Leftrightarrow$ means "equivalent"

For this particular problem the starting sentence is,

"Sally has a basketball and she sells it to her friend Bob."

First, identify the first-order statements and write them in symbolic form. This particular sentence has two first-order statements.

Statement 1: Sally has a basketball

$\text{Owns(Sally, basketball)}$

Statement 2: Sally sells her basketball to her friend Bob.

$\text{Sold(Sally, basketball, Bob)}$

To combine these statements into one complex sentence, it needs to be understood that once Sally sells her basketball she no longer has it therefore, the statement becomes:

$\text{Sold(Sally, Basketball, Bob) }\Rightarrow \lnot\text{Owns(Sally, Basketball)}$

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Symbolic Logic : Symbolic Logic

Study concepts, example questions & explanations for Symbolic Logic

All Symbolic Logic Resources

Example Questions

Example Question #1 : Sentential Logic

Example Question #2 : Sentential Logic

Example Question #3 : Sentential Logic

Example Question #1 : Truth Tables

Example Question #2 : Truth Tables

Example Question #3 : Truth Tables

Example Question #1 : Predictions & Quantifiers

Example Question #2 : Predictions & Quantifiers

Example Question #3 : Predictions & Quantifiers

Example Question #1 : First Order Logic

All Symbolic Logic Resources

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