Tweedledum: “I know what you’re thinking, but it isn’t so. No how.” Tweedledee: “Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn’t, it ain’t. That’s logic.”

PHIL 120: Introduction to Logic Professor: Lynn Hankinson-Nelson Instructors: Lars Enden Cheryl Fitzgerald Mitch Kaufman Joe Ricci

PHIL 120: Introduction to Logic Course website: Syllabus and course requirements Power point lectures Sample tests Office hours and locations; addresses. Announcements Course Text: The Logic Book, 5 th edition. McGraw Hill. A solutions manual is available online.

PHIL 120 Requirements I. I. Attendance and participation in lectures and discussion sections, including assigned homework and pop quizzes (20%) II. II. 5 tests (16% each) III. III. You may take 1 test over to raise the grade IV. IV. Practice, practice, practice… logic is not a spectator sport! V. V. Ask questions!

Logic   The study of reasoning   The drawing of inferences: what follows from what and what doesn’t follow   We can talk in terms of ‘ideas’, ‘beliefs’, and the like, but it’s more concrete to talk about ‘sentences’ – those entities that we use to express ideas, beliefs, and claims.   One focus: arguments   A set of at least two sentences, one of which is the conclusion and the other or others is/are reasons (premises) that support it.

Good arguments vs. bad arguments All men are mortal Socrates is a man. __________________ Socrates is mortal. This argument is truth- preserving. It is deductively valid. If it is true that all men are mortal, and true that Socrates is a man, then it must be true that Socrates is mortal All men are mortal Socrates is mortal. ____________________ Socrates is a man. This argument is not truth preserving. It is deductively invalid. Even if the premises are true, they do not guarantee the truth of the conclusion. Socrates could be the name of any living thing.

Logic and humor

Good arguments vs. bad arguments If you studied a lot, you did well in the logic course You studied a lot. _________________________ You did well in the logic course. This argument is truth- preserving. It is deductively valid. It is not possible for the premises to be true and the conclusion false If you studied a lot, you did well in the logic course You did well in the logic course. ____________________ You studied a lot. This argument is not truth preserving. It is deductively invalid. It is possible for the premises to be true and the conclusion false.

The language SL   A symbolic language used to illustrate the logical structure of sentences, of sets of sentences, of arguments, and of other relationships between sentences   In sentential logic, the most basic unit is the simple declarative sentence.   Simple: no logical connectives   Declarative: either true or false   We assume bivalence

The language SL The vocabulary of SL: 1. 1.Roman capital letters, A through Z, with or without subscripts (e.g., S and S 3 ) used to symbolize simple, declarative sentences (sentential) connectives: ~ (tilde) & (ampersand) v (wedge)  (horseshoe)  (triple bar) The first is a unary connective. The rest are binary connectives Punctuation: ( ) and [ ]

The language SL   Every sentence of SL is either simple/atomic or compound/molecular.   Simple/atomic sentences have no connectives.   Compound/molecular sentences have at least one connective. Meta variables Object language and meta language P, Q, R, and S are meta variables used to talk about sentences of SL.

The recursive definition of SL Every sentence letter is a sentence If P is a sentence of SL, ~P is a sentence of SL If P and Q are sentences, then (P & Q) is a sentence If P and Q are sentences, then (P v Q) is a sentence If P and Q are sentences, then (P  Q) is a sentence If P and Q are sentences, then (P  Q) is a sentence Nothing else is a sentence.

What the recursive definition of SL does   It tells us what will count as a sentence of SL   It also tells us what will not.   For example, these are not sentences of SL: A & clause 3: & is a binary connective B & C ? SL does not include ‘?’ ~ & ~ must be used before a sentence (clause 2); and & must connect 2 sentences (clause 3)

What the recursive definition of SL does   It tells us what will count as a sentence of SL   These are sentences of SL: A & B (clause 3) (B & B) & B (clause 3) ~~B (clause 2) A v B (clause 4) A  B (clause 5) A  B (clause 6)

Using SL to symbolize sentences Roman capital letters A through Z (with or without subscripts) to symbolize simple declarative sentences: ‘Mary went to the store’ (we could symbolize as M). ‘John went to the store’ (we could symbolize as J). These are NOT simple declarative sentences: Either Mary went to the store or John did. Mary did not (or didn’t) go to the store.

Using SL to symbolize sentences   ‘Mary did not (or didn’t) go to the store’   The logic of this sentence is: ‘It is not the case that Mary went to the store’   ~ symbolizes ‘it is not the case that’   So, using M for ‘Mary went to the store’, we use ~M to symbolize ‘It is not the case that Mary went to the store’   Sentences whose main connective is the tilde are called negations.

The characteristic truth table for the ~ P ~P~P~P~P TF FT

Using SL to symbolize sentences   ‘Mary and John went to the store’   This is a compound/molecular sentence to be translated as:   ‘Mary went to the store and John went to the store’   We can use M to symbolize ‘Mary went to the store’, and J to symbolize ‘John went to the store’   We use the & for ‘and’   So we have: M & J

The characteristic truth table for the & PQ P & Q TTT TFF FTF FFF

Using SL to symbolize sentences   Sentences whose main connective is the & are called conjunctions and each of the sentences connected by the & is called a conjunct. We can refer to them as the right or the left conjunct.   We use & as a connective in all cases in which the compound sentence is only true if both of its component sentences are true.   This is the case for: ‘Mary went to the store but John did too’ the logical structure of which is: ‘Mary went to the store and John went to the store’

Using SL to symbolize sentences   ‘Mary went to the store but John did not’   Is paraphrased as:   ‘Mary went to the store and it is not the case that John went to the store’   M can symbolize ‘Mary went to the store’   And & is used for but. So far we have:   M &   ~J can symbolize ‘it is not the case that John went to the store’   So we have: M & ~J

Using SL to symbolize sentences   ‘Either Mary went to the store or John did’   Is translated as:   ‘Either Mary went to the store or John went to the store’   We use the v to symbolize ‘either/or’   If we use M to symbolize ‘Mary went to the store’ and J to symbolize ‘John went to the store’ we symbolize the whole sentence as: M v J

The characteristic truth table for v PQ P v Q TTT TFT FTT FFF

Using SL to symbolize sentences   Because ‘either/or’ and the v assume the inclusive sense of ‘or’ (at least one is true), we will need to do more if we believe a sentence makes use of the exclusive sense of ‘or’ (at most one of the two) and should be symbolized to reflect this.   On a restaurant menu, for example, the phrase ‘either soup or salad is included’ reflects the exclusive sense of ‘or’.   Context may tell us that a claim comes to ‘Either Mary went to the store or John did, but not both’

Using SL to symbolize sentences   ‘If Mary went to the store, John did’   ‘If Mary went to the store, then John went to the store’   M: Mary went to the store   J: John went to the store   We use  to symbolize ‘if, then’   So we have: M  J

The characteristic truth table for the  PQ P  Q TTT TFF FTT FFT

Using SL to symbolize sentences   The reasoning behind the    Consider the following claim:   If the operation is a success, the patient survives.   The condition, ‘the operation is a success’, is a sufficient but not a necessary condition for the patient’s survival.   The docs decided not to operate and the patient survived… they discovered they were wrong about the need for an operation…   If the claim was ‘Only if the operation is a success, the patient survives’, then the operation’s success would require the patient’s survival for the claim to be true … but that is not what ‘If’ by itself entails.

Using SL to symbolize sentences   ‘Mary went to the store if and only if John did’   ‘Mary went to the store if and only if John went to the store’   We use  for ‘if and only if’   So we have: M  J   ‘P if and only if Q’ is equivalent to:   (If P then Q) and (If Q then P)

The characteristic truth table for the  PQ P  Q TTT TFF FTF FFT