LECTURE 8

Question 1

Every statement has a truth value; what is it?

Answer 1

it is either true or false

Question 2

Whats a The truth value of a true statement

Answer 2

“true

Question 3

Whats a The truth value of a false statement

Answer 3

false

Question 4

do questions and exclamations have truth value?

Answer 4

They dont because they are neither true nor false.

Question 5

Whats a conjunction

Answer 5

Two simple statements joined by a connective to form a compound
statement

Question 6

whats a conjunct and give an example

Answer 6

Each of the component statements in a conjunction
Ex: Example: Julio is here (p), and Juan is here (q):
p & q

Question 7

What are terms/symbol that can express grammatical conjunction

Answer 7

“and”( &) an an ampersand is the main term but other examples are also: But, yet, nevertheless, while, also,
and moreover.
– In propositional logic, all these are logically
equivalent—they are therefore properly
symbolized by the ampersand (&).

Question 8

What does the The truth value of a conjunction depend on

Answer 8

The truth value of a conjunction depends on the truth values of its
conjuncts—that is, by the truth value of its conjuncts.
* Example: Last night I had a Coke, and I also had an order of
poutine.
– The conjunction is true if, and only if, it’s true both that you had
a Coke last night and that you had a poutine last night.

Question 9

How can we identify and keep track of all the possible truth values
of a conjunction?

Answer 9

we can create a truth table, which is a
graphic way of displaying all the possibilities:

Question 10

How does the truth table look for conjunctions

Answer 10

• Row 1: When p is true and q is true, p & q is true.
  – Row 2: When p is true and q is false, p & q is false.
  – Row 3: When p is false and q is true, p & q is false.
  – Row 4: When p is false and q is false, p & q is false.
• If one statement in the conjunction is false, the whole
  conjunction is false; only if both conjuncts are true is the
  whole conjunction true.

Question 11

When is a conjunction false

Answer 11

If one statement in the conjunction is false, the whole

Question 12

When is a conjunction true

Answer 12

only if both conjuncts are true is the
whole conjunction true.

Question 13

See The following example:
If it’s true that I had Coke last night and also true that I had
poutine last night, then the statement “I had Coke and also had
poutine last night”
a) Is it a conjunction?
b) Identify the truth value

Answer 13

a) Yes
b) Its True

Question 14

See the following example:
But if it’s true that I had Coke last night but false that I had
poutine last night, then “I had Coke and also had poutine last
night”
a) Is it a conjunction?
b) Identify the truth value

Answer 14

a) yes
b) as a whole,it is false.( becuase one of the staemnets is false making all false)

Question 15

Whats a disjunction

Answer 15

In a disjunction, we assert that either p or q is true (though both
might be), and that even if one of the statements is false, the whole
disjunction is still true

Question 16

Whats a disjunct

Answer 16

Each of the component statements in a disjunction is called a disjunct.
Example: Either Joan is angry (p) or Ann is serene (q): p v q

Question 17

The symbol for a disjunction is called a

Answer 17

The symbol for disjunction is called a ( v ) wedge; it is roughly
equivalent to the word “or.”
– The word “unless” is also sometimes used in place of “or.”

Question 18

What worlds usually signal the beginning of
a disjunction.

Answer 18

The words “either” and “neither” usually signal the beginning of
a disjunction.

Question 19

The truth table for a disjunction looks like this:

Answer 19

• Row 1: When p is true and q is true, p v q is true.
  – Row 2: When p is true and q is false, p v q is true.
  – Row 3: When p is false and q is true, p v q is True.
  – Row 4: When p is false and q is false, p v q is false.

Question 20

In a disjunction, p ∨ q is true in every possible combination of Ts and Fs except one,
where

Answer 20

both p and q are false (in the last row).

Question 21

For a disjunction to be true….

Answer 21

only one of the disjuncts must be
true.
* In English, the word “or” has two meanings:
– “One or the other, or both”: inclusive sense
– “Either but not both”: exclusive sense

Question 22

Whats a negation and what symbol is used

Answer 22

the denial of a statement, which we indicate with the
word “not” (or some equivalent term) and the “tilde” symbol (~p).

Example:
* The price of eggs in China is very high (p).
* Negation: The price of eggs in China is not very high: ~p

Question 23

A double negation is the same thing as ….

Answer 23

No negation

Question 24

The truth table for a negation looks like ..

Answer 24

• Row 1: When p is true, ~ p is false
  – Row 2: When p is false, ~p is true.

Question 25

The basic form of a conditional

Answer 25

“if . . . then . . .”
– Example: If the cat is on the mat (p: antecedent), then
the rat will stay home (q: consequent): p → q

Question 26

The truth table for conditionals looks like

Answer 26

• Row 1: When p is true and q is true, p → q is true.
  – Row 2: When p is true and q is false, p → q is false.
  – Row 3: When p is false and q is true, p → q is true.
  – Row 4: When p is false and q is false, p → q is true.

Question 27

A conditional is false if and only if

Answer 27

its antecedent is true and its
consequent is false. In all other possible combinations of truth values, a conditional is
true.

Question 28

Conditional statements can be expressed in ways other than the if–
then configuration (the standard form). Provide some examples

Answer 28

Examples of conditionals in various patterns:

– Non-Standard: You will fall off that ladder if you’re not careful.
– Standard: If you’re not careful, you will fall off that ladder.

– Non-Standard: I’ll ride the bus only if I’m late.
– Standard: If I ride the bus, then I’m late.

– Non-Standard: Whenever I think, I get a headache.
– Standard: If I think, I get a headache.

• Because of such variations in conditional statements, it’s
  important to translate conditionals into standard form before you
  try to assess their validity.

Question 29

If Adam is guilty (P), then either Bob is an accomplice (Q) or Carol is an accomplice (R).
Symbolization : P →(Q v R)

Answer 29

This is a conditional, whose consequent is a disjunction.
( refer to lecture 8 slide 9 )

Question 30

The truth table test for validity is based on…

Answer 30

an elementary fact
about validity: It’s impossible for a valid argument to have
true premises and a false conclusion.

Question 31

Devising truth tables for arguments can reveal the

Answer 31

underlying structure of the arguments, even those that
are fairly complex.

Question 32

CHECKING FOR VALIDITY IN
SIMPLE ARGUMENTS example

Answer 32

If the icebergs melt, the lowlands will flood.
The icebergs will not melt.
Therefore, the lowlands will not flood.
p → q
~p
~q

Question 33

CHECKING FOR VALIDITY IN
SIMPLE ARGUMENTS step by step
( lecture 8 slides 21 - 27)

Answer 33

1)
For a two-variable argument, we need four rows to account for the
four possible combinations of T and F: T T, T F, F T, and F F. We
place those values in the first two columns immediately, since they
are the same for every two-variable truth table:

2)
To figure out the content for the third column in the table, we need
to use our knowledge of how conditional claims work.
* To fill in the first blank in the third column, we look to the left.
There, we see that p is True and q is also True. When p is True and q
is also True, the conditional p → q is also True. So we know to write
“T” in that first position of the third column.
* We can now reason our way through the rest of the column.

3)
* Now we use the same method to fill in the fourth and fifth columns.
* In the truth table, the truth value of ~p is the contradictory of p,
and the truth value of ~q is the contradictory of q. Therefore,
whatever the truth value of a statement, the tilde (~) reverses it,
from true to false or false to true.

Question 34

You can begin checking the argument’s validity in two
different ways:

Answer 34

1. Inspect all rows that have a false conclusion and then see if the
   premises in that row are true, indicating an invalid argument.
2. Zero in on rows showing all true premises and then check to
   see if any of them have false conclusions.
   * In this case, the third row is the key: in that row, both
   premises are true, and the conclusion is false. So the
   argument is invalid.

Question 35

example of simple argument
( see slide 28 lecture 8 )

Answer 35

1. If you pass PHL214, I’ll buy you a donut. P → D
2. You don’t pass PHL214. ~ P
   .:3. I won’t buy you a donut. (from 1,2) .: ~D

Again, our question is this:
can we find a row where all
the premises are true, and
the conclusion is false? If
so, then we know it is an
invalid argument.

1. Either you will get an A or a B in SSH105. A v B
2. You don’t get an B in SSH105. ~B .
   .:3. You will get an A in SSH105. (from 1,2) .: A