Truth Functions• Declarative sentences (statements) are either true orfalse but not both. They cannot be neither.• Whether a sentence is true or false determines its truthvalue.• Functions take input values to unique output values.That is, the input values determine the output values.• There are some ways of combining sentences intolonger ones so that the truth value of the longersentence is determined by the truth value of the partsthat were combined to form it.• So, we’ll call the connectives that combine sentencesthis way truth functions.
And• ‘And’ is often a truth functional connective.Inserting ‘and’ between two statements gives alonger sentence the truth of which is determinedby the truth of the parts.• For example, consider: (A) Al admires aardvarks.(B) Barb bakes bologna. The truth of ‘Al admiresaardvarks and Barb bakes bologna’ is determinedby the truth of A and of B.• If A and B are both true then the longer sentenceis true. If either of A or B is false then so is thelonger sentence.
A Truth Table for ‘And’• We can express all fourpossibilities for A and B intable form.• Here’s how to read thetable. The second(horizontal) row says thatwhen A is true and B istrue then A and B is true.The bottom row says thatwhen A is false and B isfalse then A and B is false.A B A and BT T TT F FF T FF F F
Conjunction• We’ll use a dot ‘∙’ (or an ampersand ‘&’) to represent thetruth functional connective that is captured by the truthtable 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 symbolizedwith the dot. Contrast ‘Al and Barb are chefs’ (It means: Alis a chef ∙ Barb is a chef.) with ‘Al and Barb are enemies.’ (Itdoesn’t mean: Al is an enemy ∙ Barb is an enemy.)• Some words besides ‘and’ can be symbolized with the dotor ampersand. For example, ‘but’ and ‘also’ formconjunctions.
Or• ‘Or’ is often used as a truth functionalconnective.• Using the sentences from slide three, the truth of‘A or B’ can be determined if we know the truthvalue of A and the truth value of B.• If Al doesn’t admire aardvarks and Barb doesn’tbake bologna then ‘A or B’ is false. Otherwise it’strue.• We’ll symbolize the truth functional (inclusive)‘or’ with a wedge ‘ ’ and we’ll call the resultingsentences disjunctions. The parts are disjuncts.
Truth Table for Disjunction• The four possibilities forsentences A and B arerepresented by the left two(vertical) columns.• The top row of Ts and Fs is thepossibility where A is true andB is true. On that possibility ‘AB’ is true.• The bottom row is thepossibility where A and B areboth false. In that case ‘A B’is also false.• Disjunction differs fromconjunction on the middle tworows.A B A BT T TT F TF T TF F F
Examples• Lenny and Manny left for Bermuda. (L=Lenny left forBermuda. 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 forBermuda 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 orelse Lenny left for Bermuda. Translation: (M∙N) L
Negation• Prefixing a statement with ‘it isnot the case that’ flips thetruth value.• The symbol to represent thattruth function is ‘ ’. Thesymbol is called ‘tilde’ or just‘squiggle’.• Unlike the other connectives,tilde doesn’t connect twosentences. It just flips thevalue of a single sentence.• Also unlike the otherconnectives, we don’t usuallyput parenthesis around anegation.A AT FF T
Examples• Manny left but Lenny did not leave.Translation: M∙ L• Either Nancy did not stay and Lenny left orManny 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]
If..Then…• Some uses of ‘if..then…’ are truth functional andsome are not. We’ll only care about the truthfunctional ones.• Consider: If you whistle loudly then the dog willcome. There’s really just one scenario that showsthe sentence to be false: whistle loudly and havethe dog not come. If you whistle and the dogcomes then the sentence was true. If you don’twhistle then no matter whether the dog comesor not, you didn’t show the sentence wrong.
Truth Table for Conditional• We’ll use a horseshoe ‘ ’(or an arrow ‘⟶’) torepresent the truthfunctional ‘if...then…’;and we’ll call statementsformed with thehorseshoe (or the arrow)‘conditionals.’• The top two rows of thetruth table for conditionalare uncontroversial. Thebottom two are lessobvious.A B A BT T TT F FF T TF F T
Order Matters• Notice on the table that the order of A and B matter. Atrue and B false yields a different value than A false andB true. So, unlike conjunction and disjunction whereorder doesn’t matter, we have a different names forthe different parts of the conditional.• For A B, A is called the ‘antecedent’ and B is calledthe ‘consequent’.• There are many English expressions that can betranslated as conditionals. The trick to symbolizingthem correctly is to identify the antecedent and theconsequent.
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 ‘Aonly if B’ we are saying that A can’t be true without B; so, ifA 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 ifand only if B’ we could translate that as ‘(B A)&(A B)’.We’ll use the triple bar ‘ ’ (or double arrow ‘⟷’ ) as anabbreviation for that. Also, we’ll call sentences formed byusing the triple bar ‘biconditionals’.• Sometimes you see ‘iff’ between two sentences. It’s not atypo. 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 assaying that two sentences havematching truth value. The longersentence is true when both partsare true or when both parts arefalse.• Notice that order doesn’t matter;so we don’t have special namesfor the letter that comes first.Unlike regular conditional, ‘A B’is equivalent to ‘B A’.• The sentence ‘A B’ is just anabbreviation for the morecomplex sentence ‘(B A)∙(A B)’so next I’ll show you where thevalues in the table to the rightcame from.A B A BT T TT F FF T FF F T
Justification for the Biconditional Table• The table gives the value of‘(B A)∙(A B)’ in each of the fourpossibilities for A and B.• The left two columns list thepossibilities for A and B.• The sentence we care about is aconjunction of two conditionals.To figure out its truth value weneed to know the truth value ofthe conditionals. Then we canuse it to figure out the truth valueof the conjunction.• The 3rd column gives the valuesfor ‘B A’ and the 5th columngives values for ‘A B’. The 4thcolumn was computed last usingthe values from columns 3 and 5.A B (B A) ∙ (A B)T T T T TT F T F FF T F F TF F T T T