Logical Connectives

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.

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.

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

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.

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.

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

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: LM• Lenny left for Bermuda and either Manny left for Bermuda

or Nancy stayed. (N=Nancy stayed.) Translation: L&(MN)• 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

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

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: (LM) [Alternate Translation: L&M]

• It’s not true that both Manny and Lenny left. Translation: (M&L) [or ML]

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.

Truth Table for Conditional

• We’ll use the arrow ⟶ (or the horseshoe ) to represent the truth functional ‘if...then…’; and we’ll call statements formed with the arrow (or the horseshoe) ‘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

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.

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

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 double arrow ⟷ (or the triple bar ) as an abbreviation for that. Also, we’ll call sentences formed by using the double arrow ‘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’.

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 complicated 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

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