2. Truth Functions
• Declarative sentences (statements) are either true or
false but not both. They cannot be neither.
• Whether a sentence is true or false determines its truth
value.
• Functions take input values to unique output values.
That is, the input values determine the output values.
• There are some ways of combining sentences into
longer ones so that the truth value of the longer
sentence is determined by the truth value of the parts
that were combined to form it.
• So, we’ll call the connectives that combine sentences
this way truth functions.
3. And
• ‘And’ is often a truth functional connective.
Inserting ‘and’ between two statements gives a
longer sentence the truth of which is determined
by the truth of the parts.
• For example, consider: (A) Al admires aardvarks.
(B) Barb bakes bologna. The truth of ‘Al admires
aardvarks and Barb bakes bologna’ is determined
by the truth of A and of B.
• If A and B are both true then the longer sentence
is true. If either of A or B is false then so is the
longer sentence.
4. A Truth Table for ‘And’
• We can express all four
possibilities for A and B in
table form.
• Here’s how to read the
table. The second
(horizontal) row says that
when A is true and B is
true then A and B is true.
The bottom row says that
when A is false and B is
false then A and B is false.
A B A and B
T T T
T F F
F T F
F F F
5. Conjunction
• We’ll use a dot ‘∙’ (or an ampersand ‘&’) to represent the
truth functional connective that is captured by the truth
table on slide four.
• We’ll call sentences formed with the dot (or ampersand)
conjunctions. We’ll call the parts conjuncts. For example,
‘A∙B’ is a conjunction and ‘A’ is its left conjunct.
• Not every instance of ‘and’ in English can be symbolized
with the dot. Contrast ‘Al and Barb are chefs’ (It means: Al
is a chef ∙ Barb is a chef.) with ‘Al and Barb are enemies.’ (It
doesn’t mean: Al is an enemy ∙ Barb is an enemy.)
• Some words besides ‘and’ can be symbolized with the dot
or ampersand. For example, ‘but’ and ‘also’ form
conjunctions.
6. Or
• ‘Or’ is often used as a truth functional
connective.
• Using the sentences from slide three, the truth of
‘A or B’ can be determined if we know the truth
value of A and the truth value of B.
• If Al doesn’t admire aardvarks and Barb doesn’t
bake bologna then ‘A or B’ is false. Otherwise it’s
true.
• We’ll symbolize the truth functional (inclusive)
‘or’ with a wedge ‘ ’ and we’ll call the resulting
sentences disjunctions. The parts are disjuncts.
7. Truth Table for Disjunction
• The four possibilities for
sentences A and B are
represented by the left two
(vertical) columns.
• The top row of Ts and Fs is the
possibility where A is true and
B is true. On that possibility ‘A
B’ is true.
• The bottom row is the
possibility where A and B are
both false. In that case ‘A B’
is also false.
• Disjunction differs from
conjunction on the middle two
rows.
A B A B
T T T
T F T
F T T
F F F
8. Examples
• Lenny and Manny left for Bermuda. (L=Lenny left for
Bermuda. M=Manny left for Bermuda.) Translation:
L∙M
• Either Lenny or Manny left for Bermuda. Translation:
L M
• Lenny left for Bermuda and either Manny left for
Bermuda or Nancy stayed. (N=Nancy stayed.)
Translation: L∙(M N)
• Lenny left for Bermuda but Manny did too.
Translation: L∙M
• Either Manny left for Bermuda and Nancy stayed or
else Lenny left for Bermuda. Translation: (M∙N) L
9. Negation
• Prefixing a statement with ‘it is
not the case that’ flips the
truth value.
• The symbol to represent that
truth function is ‘ ’. The
symbol is called ‘tilde’ or just
‘squiggle’.
• Unlike the other connectives,
tilde doesn’t connect two
sentences. It just flips the
value of a single sentence.
• Also unlike the other
connectives, we don’t usually
put parenthesis around a
negation.
A A
T F
F T
10. Examples
• Manny left but Lenny did not leave.
Translation: M∙ L
• Either Nancy did not stay and Lenny left or
Manny didn’t leave. Translation: ( N∙L) M
• Neither Lenny nor Manny left. Translation:
(L M) [Alternate Translation: L∙ M]
• It’s not true that both Manny and Lenny left.
Translation: (M∙L) [or M L]
11. If..Then…
• Some uses of ‘if..then…’ are truth functional and
some are not. We’ll only care about the truth
functional ones.
• Consider: If you whistle loudly then the dog will
come. There’s really just one scenario that shows
the sentence to be false: whistle loudly and have
the dog not come. If you whistle and the dog
comes then the sentence was true. If you don’t
whistle then no matter whether the dog comes
or not, you didn’t show the sentence wrong.
12. Truth Table for Conditional
• We’ll use a horseshoe ‘ ’
(or an arrow ‘⟶’) to
represent the truth
functional ‘if...then…’;
and we’ll call statements
formed with the
horseshoe (or the arrow)
‘conditionals.’
• The top two rows of the
truth table for conditional
are uncontroversial. The
bottom two are less
obvious.
A B A B
T T T
T F F
F T T
F F T
13. Order Matters
• Notice on the table that the order of A and B matter. A
true and B false yields a different value than A false and
B true. So, unlike conjunction and disjunction where
order doesn’t matter, we have a different names for
the different parts of the conditional.
• For A B, A is called the ‘antecedent’ and B is called
the ‘consequent’.
• There are many English expressions that can be
translated as conditionals. The trick to symbolizing
them correctly is to identify the antecedent and the
consequent.
14. Examples of Conditionals
• Lenny left if Manny left. Translation: M L
• Lenny left only if Manny left. Translation: L M
[Think about this one.]
• Lenny left in case Manny left. Translation: M L
• Lenny left provided that Manny left. Translation:
M L
• Lenny left unless Manny left. Translation: M L
• Whenever Lenny leaves, Manny leaves.
Translation: L M
15. If and Only If
• ‘A only if B’ is translated ‘A B’ because when we say ‘A
only if B’ we are saying that A can’t be true without B; so, if
A is true then so is B. Hence, A B.
• ‘A if B’ is the same as ‘if B then A’ and so is translated ‘B A’.
• So, if we say ‘A if B and A only if B’ or in other words ‘A if
and only if B’ we could translate that as ‘(B A)&(A B)’.
We’ll use the triple bar ‘ ’ (or double arrow ‘⟷’ ) as an
abbreviation for that. Also, we’ll call sentences formed by
using the triple bar ‘biconditionals’.
• Sometimes you see ‘iff’ between two sentences. It’s not a
typo. It’s an abbreviation for ‘if and only if.’ So ‘A iff B’
would be translated as ‘A B’.
16. Truth Table for Biconditional
• Think of the biconditional as
saying that two sentences have
matching truth value. The longer
sentence is true when both parts
are true or when both parts are
false.
• Notice that order doesn’t matter;
so we don’t have special names
for the letter that comes first.
Unlike regular conditional, ‘A B’
is equivalent to ‘B A’.
• The sentence ‘A B’ is just an
abbreviation for the more
complex sentence ‘(B A)∙(A B)’
so next I’ll show you where the
values in the table to the right
came from.
A B A B
T T T
T F F
F T F
F F T
17. Justification for the Biconditional Table
• The table gives the value of
‘(B A)∙(A B)’ in each of the four
possibilities for A and B.
• The left two columns list the
possibilities for A and B.
• The sentence we care about is a
conjunction of two conditionals.
To figure out its truth value we
need to know the truth value of
the conditionals. Then we can
use it to figure out the truth value
of the conjunction.
• The 3rd column gives the values
for ‘B A’ and the 5th column
gives values for ‘A B’. The 4th
column was computed last using
the values from columns 3 and 5.
A B (B A) ∙ (A B)
T T T T T
T F T F F
F T F F T
F F T T T
