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2.02 Language of logic

Lesson

Let's have a look at some of the ways to represent statements with symbols.  This short-hand will allow us to study the validity of our logical arguments in the future.

 

Negation

Recall that for every proposition $p$p there exists another proposition $\sim p$~p meaning the negation of $p$p.

As an example we might write:

$p$p $:$: The plate is dirty
$\sim p$~p $:$: The plate is not dirty
     

 

Note that the negation of a negation becomes the original proposition. In symbols we could write that $\sim\left(\sim p\right)$~(~p) is the same as (or more formally, is equivalent to) $p$p.

The symbol $\sim$~ is an example of an operator and there are many operators defined in the study of logic. 

 

Compound propositions

Two or more propositions can be joined with a connective (a connective is another type of operator) to become a compound proposition.

 

The conjunction connective

An example of a connective is what logicians call a conjunction, often symbolized as $\wedge$. Its meaning is the same as the English word "and" when used to indicate the conjoining of two ideas, such as "It's raining and I'm cold".

We have to be a little bit careful with the word "and" when used in other circumstances. For example the word "and" used in the phrase "I tripped and fell" is not a conjunction because we need $p\wedge q$pq to have the same sense as $q\wedge p$qp and the phrase "I fell and tripped" doesn't really make any sense. 

In addition, a phrase like "Paul left the party and Ed arrived at the party", if taken as a compound proposition, can imply no causality, so that we cannot assume that the reason Ed arrived is because Paul left.

 

The disjunction connective

Another important connective is the disjunction connective, symbolized by $\vee$. It is equivalent to the English word "or" in the inclusive sense. By inclusive we mean that the compound proposition $p\vee q$pq will be true if either $p$p or $q$q or both are true. In fact, the only way $p\vee q$pq will not be true if $p$p and $q$q are both false. 

For example, the compound proposition "the number is prime or the number is odd" means that either the number is prime, or else the number is odd, or else the number is both prime and odd.  

Careful!

The English language use of the word or can be ambiguous. For example, the word "or" in the sentence "I'll go to school or I'll get a low paying job" is used in an exclusive sense. What is really implied is contained in the sentence "I'll go to school or else I'll get a low paying job" - it's either one or the other but not both. In this case, the disjunction connective $\vee$ doesn't apply (Incidentally there is another special connective for the "exclusive or" ).

 

Changing order and grouping

In mathematics speak, both the conjunction and disjunction connectives exhibit the commutative property - namely that $p\wedge q$pq is equivalent to $q\wedge p$qp and $p\vee q$pq is equivalent to $q\vee p$qp.  Swapping the order of the individual propositions should make no difference to the sense of the compound proposition. 

In addition, we can create compound propositions that use both connectives. As an example, consider the following propositions:

$p$p $:$: John is having roast beef for dinner
$q$q $:$: John is having Yorkshire pudding for dinner
$r$r $:$: John is having dessert for dinner
     

From these, we could form a number of compound propositions that involve all three simple propositions.

For example consider the compound proposition  $\left(p\vee q\right)\wedge r$(pq)r.

It would read:

 John is having roast beef or Yorkshire pudding, and having dessert for dinner.

Note that the main connective is a conjunction. The expression is a conjunction of two propositions, the first of which is a compound proposition (formed by a disjunction). Remember that the disjunction is inclusive in that John could end up eating the roast beef, the Yorkshire pudding and the dessert.  

We could also write a proposition with $p$p,$q$q and $r$r in the same order, the disjunction and conjunction connectives in the same order, but the bracketing grouped around the last two propositions, so that our compound proposition becomes  $p\vee\left(q\wedge r\right)$p(qr)

It now reads differently:

John is having roast beef, or he is having Yorkshire pudding and dessert.

What has happened is that the main connective is now a disjunction. The expression is a disjunction of two propositions with the second proposition a compound proposition formed by a conjunction.

This example illustrates how parenthesis may have a profound effect on the meaning of the statement. 

 

The conditional

The symbol used for the conditional connective is a right arrow $\Rightarrow$

As an example, suppose a doctor tells you that if you eat an apple you will feel better. We could think about this statement logically. We could define two propositions:

$p$p $:$: I will eat an apple
$q$q $:$: I will feel better
     

We could then define a compound proposition, based on the doctor's advice, that says: 

If $p$p then $q$q 

In symbolic logic we would write a compound proposition:

$p$p $\Rightarrow$ $q$q

The connective, as a right arrow, tells us that the statement can only be read one way. 

 

The bi-conditional

Recall that sometimes (but certainly not always) the conditional connective works both ways. That is, sometimes when we say "If $p$p then $q$q", then we can also say "If $q$q then $p$p".

As an example, the statement:

If the triangle is equilateral then the triangle has three equal angles.

is precisely the same as the statement :

If the triangle has three equal angles, then the triangle is equilateral. 

When this happens we can replace the two statements with the "if and only if" clause, so that we can say:

The triangle is equilateral if and only if it has three equal angles.

Suppose we call $p$p the proposition "The triangle is equilateral" and $q$q is the proposition "The triangle has three equal angles".

In symbolic language, we would say $p$p$\Leftrightarrow q$q to denote the biconditional.

 

Practice questions

Question 1

Answer the following.

  1. Which of the following is true for a negation?

    It is symbolized by $\sim$~ and is read "Not".

    A

    It is symbolized by $\Rightarrow$ and is read "If-then".

    B

    It is symbolized by $\wedge$ and is read "And".

    C

    It is symbolized by $\vee$ and is read "Or".

    D

    It is symbolized by $\Leftrightarrow$ and is read "If and only if".

    E

    It is symbolized by $\sim$~ and is read "Not".

    A

    It is symbolized by $\Rightarrow$ and is read "If-then".

    B

    It is symbolized by $\wedge$ and is read "And".

    C

    It is symbolized by $\vee$ and is read "Or".

    D

    It is symbolized by $\Leftrightarrow$ and is read "If and only if".

    E
  2. Which of the following is true for a conjunction?

    It is symbolized by $\Rightarrow$ and is read "If-then".

    A

    It is symbolized by $\sim$~ and is read "Not".

    B

    It is symbolized by $\Leftrightarrow$ and is read "If and only if".

    C

    It is symbolized by $\vee$ and is read "Or".

    D

    It is symbolized by $\wedge$ and is read "And".

    E

    It is symbolized by $\Rightarrow$ and is read "If-then".

    A

    It is symbolized by $\sim$~ and is read "Not".

    B

    It is symbolized by $\Leftrightarrow$ and is read "If and only if".

    C

    It is symbolized by $\vee$ and is read "Or".

    D

    It is symbolized by $\wedge$ and is read "And".

    E
  3. Which of the following is true for a disjunction?

    It is symbolized by $\vee$ and is read "Or".

    A

    It is symbolized by $\sim$~ and is read "Not".

    B

    It is symbolized by $\Leftrightarrow$ and is read "If and only if".

    C

    It is symbolized by $\Rightarrow$ and is read "If-then".

    D

    It is symbolized by $\wedge$ and is read "And".

    E

    It is symbolized by $\vee$ and is read "Or".

    A

    It is symbolized by $\sim$~ and is read "Not".

    B

    It is symbolized by $\Leftrightarrow$ and is read "If and only if".

    C

    It is symbolized by $\Rightarrow$ and is read "If-then".

    D

    It is symbolized by $\wedge$ and is read "And".

    E

Question 2

Let

  $p$p: The lemonade is sour.
  $q$q: The rice is hot.

What are the following statements in symbolic form?

  1. "The rice is not hot if and only if the lemonade is sour."

    $\sim$~$p$p$\wedge$$q$q

    A

    $\sim$~$p$p$\Rightarrow$$q$q

    B

    $p$p$\Rightarrow$$\sim$~$q$q

    C

    $\sim$~$q$q$\Leftrightarrow$$p$p

    D

    $\sim$~$p$p$\wedge$$q$q

    A

    $\sim$~$p$p$\Rightarrow$$q$q

    B

    $p$p$\Rightarrow$$\sim$~$q$q

    C

    $\sim$~$q$q$\Leftrightarrow$$p$p

    D
  2. "If the lemonade is sour, then the rice is not hot."

    $p$p$\Rightarrow$$\sim$~$q$q

    A

    $p$p$\wedge$$q$q

    B

    $\sim$~$p$p$\Rightarrow$$\sim$~$q$q

    C

    $\sim$~$p$p$\Leftrightarrow$$\sim$~$q$q

    D

    $p$p$\Rightarrow$$\sim$~$q$q

    A

    $p$p$\wedge$$q$q

    B

    $\sim$~$p$p$\Rightarrow$$\sim$~$q$q

    C

    $\sim$~$p$p$\Leftrightarrow$$\sim$~$q$q

    D
  3. "The lemonade is not sour, but the rice is hot."

    $p$p$\Leftrightarrow$$\sim$~$q$q

    A

    $\sim$~$p$p$\Rightarrow$$q$q

    B

    $\sim$~$p$p$\wedge$$\sim$~$q$q

    C

    $\sim$~$p$p$\wedge$$q$q

    D

    $p$p$\Leftrightarrow$$\sim$~$q$q

    A

    $\sim$~$p$p$\Rightarrow$$q$q

    B

    $\sim$~$p$p$\wedge$$\sim$~$q$q

    C

    $\sim$~$p$p$\wedge$$q$q

    D
  4. "Neither is the lemonade sour nor is the rice hot."

    $\sim$~$p$p$\wedge$$\sim$~$q$q

    A

    $p$p$\wedge$$q$q

    B

    $\sim$~$q$q$\Leftrightarrow$$\sim$~$p$p

    C

    $p$p$\Leftrightarrow$$q$q

    D

    $\sim$~$p$p$\wedge$$\sim$~$q$q

    A

    $p$p$\wedge$$q$q

    B

    $\sim$~$q$q$\Leftrightarrow$$\sim$~$p$p

    C

    $p$p$\Leftrightarrow$$q$q

    D
  5. "It is false that the lemonade is sour or the rice is hot."

    $\sim$~$($($p$p$\vee$$q$q$)$)

    A

    $p$p$\vee$$\sim$~$q$q

    B

    $\sim$~$p$p$\wedge$$q$q

    C

    $\sim$~$p$p$\Rightarrow$$q$q

    D

    $\sim$~$($($p$p$\vee$$q$q$)$)

    A

    $p$p$\vee$$\sim$~$q$q

    B

    $\sim$~$p$p$\wedge$$q$q

    C

    $\sim$~$p$p$\Rightarrow$$q$q

    D
  6. "It is false that if the rice is not hot, then the lemonade is sour."

    $\sim$~$($($\sim$~$q$q$\Rightarrow$$p$p$)$)

    A

    $\sim$~$($($p$p$\vee$$q$q$)$)

    B

    $\sim$~$p$p$\Leftrightarrow$$q$q

    C

    $\sim$~$($($p$p$\Rightarrow$$q$q$)$)

    D

    $\sim$~$($($\sim$~$q$q$\Rightarrow$$p$p$)$)

    A

    $\sim$~$($($p$p$\vee$$q$q$)$)

    B

    $\sim$~$p$p$\Leftrightarrow$$q$q

    C

    $\sim$~$($($p$p$\Rightarrow$$q$q$)$)

    D

Question 3

"It is false that if you eat healthy foods, then you will not gain fat."

  1. What is the statement in symbolic form?

    Let

      $p$p: You eat healthy foods.
      $q$q: You gain fat.

    $\sim$~$p$p$\Rightarrow$$\sim$~$q$q

    A

    $\sim$~$($($\sim$~$p$p$\Rightarrow$$q$q$)$)

    B

    $\sim$~$($($p$p$\Rightarrow$$\sim$~$q$q$)$)

    C

    $\sim$~$p$p$\Leftrightarrow$$q$q

    D

    $\sim$~$p$p$\Rightarrow$$\sim$~$q$q

    A

    $\sim$~$($($\sim$~$p$p$\Rightarrow$$q$q$)$)

    B

    $\sim$~$($($p$p$\Rightarrow$$\sim$~$q$q$)$)

    C

    $\sim$~$p$p$\Leftrightarrow$$q$q

    D
  2. What type of statement is this?

    conditional

    A

    negation

    B

    disjunction

    C

    conjunction

    D

    biconditional

    E

    conditional

    A

    negation

    B

    disjunction

    C

    conjunction

    D

    biconditional

    E

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