LECTURE 8 Flashcards
Every statement has a truth value; what is it?
it is either true or false
Whats a The truth value of a true statement
“true
Whats a The truth value of a false statement
false
do questions and exclamations have truth value?
They dont because they are neither true nor false.
Whats a conjunction
Two simple statements joined by a connective to form a compound
statement
whats a conjunct and give an example
Each of the component statements in a conjunction
Ex: Example: Julio is here (p), and Juan is here (q):
p & q
What are terms/symbol that can express grammatical conjunction
“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 (&).
What does the The truth value of a conjunction depend on
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.
How can we identify and keep track of all the possible truth values
of a conjunction?
we can create a truth table, which is a
graphic way of displaying all the possibilities:
How does the truth table look for conjunctions
- 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.
When is a conjunction false
If one statement in the conjunction is false, the whole
When is a conjunction true
only if both conjuncts are true is the
whole conjunction true.
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
a) Yes
b) Its True
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
a) yes
b) as a whole,it is false.( becuase one of the staemnets is false making all false)
Whats a disjunction
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
Whats a disjunct
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
The symbol for a disjunction is called a
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.”
What worlds usually signal the beginning of
a disjunction.
The words “either” and “neither” usually signal the beginning of
a disjunction.
The truth table for a disjunction looks like this:
- 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.
In a disjunction, p ∨ q is true in every possible combination of Ts and Fs except one,
where
both p and q are false (in the last row).
For a disjunction to be true….
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
Whats a negation and what symbol is used
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
A double negation is the same thing as ….
No negation
The truth table for a negation looks like ..
- Row 1: When p is true, ~ p is false
– Row 2: When p is false, ~p is true.
The basic form of a conditional
“if . . . then . . .”
– Example: If the cat is on the mat (p: antecedent), then
the rat will stay home (q: consequent): p → q
The truth table for conditionals looks like
- 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.
A conditional is false if and only if
its antecedent is true and its
consequent is false. In all other possible combinations of truth values, a conditional is
true.
Conditional statements can be expressed in ways other than the if–
then configuration (the standard form). Provide some examples
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.
If Adam is guilty (P), then either Bob is an accomplice (Q) or Carol is an accomplice (R).
Symbolization : P →(Q v R)
This is a conditional, whose consequent is a disjunction.
( refer to lecture 8 slide 9 )
The truth table test for validity is based on…
an elementary fact
about validity: It’s impossible for a valid argument to have
true premises and a false conclusion.
Devising truth tables for arguments can reveal the
underlying structure of the arguments, even those that
are fairly complex.
CHECKING FOR VALIDITY IN
SIMPLE ARGUMENTS example
If the icebergs melt, the lowlands will flood.
The icebergs will not melt.
Therefore, the lowlands will not flood.
p → q
~p
~q
CHECKING FOR VALIDITY IN
SIMPLE ARGUMENTS step by step
( lecture 8 slides 21 - 27)
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.
You can begin checking the argument’s validity in two
different ways:
- Inspect all rows that have a false conclusion and then see if the
premises in that row are true, indicating an invalid argument. - 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.
example of simple argument
( see slide 28 lecture 8 )
- If you pass PHL214, I’ll buy you a donut. P → D
- 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.
- Either you will get an A or a B in SSH105. A v B
- You don’t get an B in SSH105. ~B .
.:3. You will get an A in SSH105. (from 1,2) .: A
