PHIL 120: Jan 8 1.Basic notions of logic 2.More implications of the recursive definition of SL a....

PHIL 120: Jan 8

1.1. Basic notions of logicBasic notions of logic

2.2. More implications of the recursive definition More implications of the recursive definition of SLof SL

a.a. Kinds of sentencesKinds of sentences

b.b. Main connectives and sentential componentsMain connectives and sentential components

3.3. Symbolizing sentences with more than one Symbolizing sentences with more than one connectiveconnective

Argument

An argument is a set of 2 or more sentences, one of which is the conclusion and the others premises.

Standardized form Premise

Another premise (if there is one)

And another (if there is one) …

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Conclusion

Arguments

Premise indicator terms: ‘because’, ‘since’, ‘for the reason that’…

Conclusion indicator terms: ‘therefore’, ‘thus’, ‘it follows that’, ‘so’ …

One compound sentence can contain an argument:

‘Sarah will do well in logic because she studies and anyone who studies does well in logic’

Arguments

‘‘Sarah will do well in logicSarah will do well in logic becausebecause [she] [she] Sarah Sarah studiesstudies and and anyone who studies does well in anyone who studies does well in logiclogic’’

Sarah studies.Sarah studies.

Anyone who studies does well in logic.Anyone who studies does well in logic.

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Sarah will do well in logic.Sarah will do well in logic.

Arguments

Famous arguments:Famous arguments: Descartes’ “cogito”Descartes’ “cogito”

““I think, therefore I am”I think, therefore I am”““Cogito, ergo sum”Cogito, ergo sum”

John Locke (and Samuel Johnson)John Locke (and Samuel Johnson)Men (humans) are free by nature.Men (humans) are free by nature.As they are born free, no other man has the right to As they are born free, no other man has the right to claim complete and total power over another man.claim complete and total power over another man.

------------------------------------------------------------------------------------------------------------------------------Governments can only legitimately rule just as long as Governments can only legitimately rule just as long as men, who are free by nature, consent to it.men, who are free by nature, consent to it.

Arguments

Arguments and philosophy 2008 UW Philosophy Club tee shirt: On the front:

Sketches of Plato, Socrates, Aristotle, Newton, Descartes, etc.

“Great minds don’t just think”

On the back: “They argue”

Basic notions of logic: validity

An argument is deductively valid if and only if it is not possible for the premises to be true and the conclusion false.

An argument is deductively invalid if and only if it is possible for the premises to be true and the conclusion false.

Every argument is either deductively valid or deductively invalid.

Only arguments are valid or invalid.

Validity: A matter of form rather than content

All men are mortal.All men are mortal.Socrates is a man.Socrates is a man.--------------------------------------------------Socrates is mortal.Socrates is mortal.

IfIf it is true that all men it is true that all men are mortal and that are mortal and that Socrates is a man, Socrates is a man, thenthen it must be true it must be true that Socrates is a that Socrates is a mortal.mortal.

All men are green.All men are green.Socrates is a man.Socrates is a man.--------------------------------------------------Socrates is green.Socrates is green.

IfIf it true that all men are it true that all men are green…green…

MENMORTALS

Venn Diagram of the two categories and of their potential overlap

MEN MORTALS

The first premiseThere are no men who aren’t mortal, hence the part of

‘men’ that doesn’t overlap with ‘mortals’ is empty

XMEN MORTALS

The second premise using X for SocratesAs he is a man, he must be in that portion of the category “men” that is not empty – and that overlaps with “mortal”. So, once we diagram the two premises, the conclusion is already diagrammed.

Basic notions of logic that apply to sentences Logical truth: a sentence is logically true if

and only if it is not possible for the sentence to be false.

2 + 2 = 4 A rose is a rose If it rains, then it rains

Logical falsity: a sentence is logically false if and only if it is not possible for the sentence to be true.

If it rains, then it doesn’t rain. A rose is not a rose.

Basic notions of logic that apply to sentences Logical indeterminacy: a sentence is logically

indeterminate if and only if it is neither logically true nor logically false.

Most sentences are logically indeterminate though all sentences are either true or false.

The truth or falsity of logically indeterminate sentences is determined by the way the world is.

The earth is a sphere Copper conducts electricity All men are mortal

Basic notions of logic that apply to sentences Logical equivalence: the members of a pair

of sentences are logically equivalent iff it is not possible for one of the members of the pair to be true while the other is false.

“Neither Mary nor John went to the store” “Both Mary didn’t go to the store and John

didn’t go to the store” It follows that:

The sentences of any pair of logical truths (logically true sentences) are logically equivalent

The sentences of any pair of logical falsehoods (logically false sentences)

Basic notions of logic that apply to sentences

Strange sentences:“This sentence is false.”(The liar’s paradox)

Strange pairs of sentences:On the front of a tee shirt:The sentence on the back of this shirt is false.On its back:The sentence on the front of this shirt is true.

Basic notions of logic that apply to sets of sentences

Logical consistency: A set of sentences is logically consistent iff it is possible for all the members of that set to be true.

A set of sentences is logically inconsistent iff it is not possible for all the members of that set to be true.

Put another way: logically consistent sets of sentences do not contain any contradictions; logically inconsistent sets of sentences do.

Basic notions of logic that apply to sets of sentences

Questions: If a set of sentences includes only logically

indeterminate sentences could it be logically consistent?

If a set of sentences includes one logically false sentence could it be logically consistent?

Could a set of sentences each one of which is logically true be logically inconsistent?

Could a set of sentences, none of which includes the ~, be logically inconsistent?

Implications of the definition of validity

An argument is deductively valid iff it is not possible for the premises to be true and the conclusion false.

From which it follows that valid arguments can have:

True premises and a true conclusion False premises and a false conclusion False premises and a true conclusion

Only arguments with true premises and a false conclusion cannot be valid.


A valid argument with a false premise and false conclusion:

All men are green.Socrates is a man.-------------------------Socrates is green.

A valid argument with false premises and a true conclusion:

If the sun didn’t rise 01/06/2009, PHIL 120 met at 11:30 that day as scheduled.

The sun didn’t rise 01/06/2009.------------------------------------------------Phil 120 met at 11:30 that day as scheduled.


“Special” cases of validity:1. All arguments containing one logically-false

premise are deductively valid.Why?

2. All arguments whose premises form a logically inconsistent set are deductively valid.Why?

3. All arguments with a logically true conclusion are deductively valid.Why?

Logical soundess An argument is deductively sound if and only

if it is deductively valid and its premises are true.

So valid arguments with one or more false premises are not sound arguments.

And valid arguments with one or more logically false premises are not sound arguments.

And valid arguments whose premises form an inconsistent set are not sound arguments.

Part 2

Further implications of the recursive definition of SL

Implications of the recursive definition All sentences of SL are one of the following:

An atomic or simple sentence (A, B… Z) A negation (a sentence of the form ~P) A conjunction (a sentence of the form (P & Q) A disjunction (a sentence of the form (P v Q) A material conditional (a sentence of the form

(P Q) A biconditional (a sentence of the form (P Q)

This means that no matter how many connectives a sentence may contain, it has one (and only one) main connective.

Implications of the recursive definition

Note: in practice, we drop the outside premises of a whole sentence.

Take the sentence:(A v B) & ~C

Our question: which kind of sentence is it? Put another way, what is the main connective of this entire sentence?

One clue: the & is the only binary connective that is not inside parentheses

So the entire sentence is a conjunction whose conjuncts are: A v B and ~C


The sentence (A v B) & ~Cis a conjunction whose conjuncts are:

A v B and ~CTo determine whether the entire sentence is

true or false, we need to determine if the left conjunct (A v B) is true or false and whether the right conjunct (~C) is true or false.

The parentheses around ‘A v B’ indicate that it is a disjunction that forms one conjunct of the entire sentence.


Compare the sentence (A v B) & ~C

To the sentenceA v (B & ~C)

In the second sentence ‘v’ is the only binary connective outside of parentheses indicating that the entire sentence is a disjunction whose disjuncts are ‘A’ and ‘B & ~C’

The left disjunct is the atomic sentence A; the right disjunct is the conjunction B & ~C, the left conjunct is B and the right is ~C



To the sentence(A v B) ~C

Here the only binary connective outside of parentheses is the indicating that the sentence is a material conditional whose antecedent is ‘A v B’ and whose consequent is ~C.

A v B is a disjunction.And ~C is a negation.



To the sentence~[(A v B) C]

Here the only connective outside of parentheses is the ~ indicating that the sentence is a negation, a sentence of the form ~P, where P is the conditional (A v B) C

The antecedent of this conditional is the disjunction ‘A v B’ and the consequent is the atomic sentence ‘C’

So every sentence of SL that is not atomic has a main connective (and this determines what kind of sentence it is).

For a sentence of the form ~P, P is its immediate component

For sentences of the forms P & Q, P v Q, P Q, or P Q, P and Q are the immediate components

If an immediate component itself has a connective, it too has a component or 2 components.

And the atomic components of a sentence are all the components that are atomic.

Finally, the sentential components of a sentence include the sentence itself.

Main connectives and sentential componentsMain connectives and sentential components

Take the sentence: (A v B) Take the sentence: (A v B) ~(~A & ~B) ~(~A & ~B)Its sentential components include itIts sentential components include itand its immediate components:and its immediate components:(A v B) and ~(~A & ~B)(A v B) and ~(~A & ~B)That in turn have components, respectively:That in turn have components, respectively:A, B and ~A & ~BA, B and ~A & ~BA and B are atomic (so atomic components of the A and B are atomic (so atomic components of the

whole sentence)whole sentence)~A & ~B has the components ~A and ~B~A & ~B has the components ~A and ~BThat, respectively, have the components A and BThat, respectively, have the components A and B

Part 3

Symbolizing sentenceswith more than 1 connective

SymbolizingSymbolizing

Sentences with more than one connectiveSentences with more than one connectiveSentences with one binary connective and a tildeSentences with one binary connective and a tildeSentences with more than one binary connective Sentences with more than one binary connective

(with or without one or more tildes)(with or without one or more tildes) Compare the sentences:Compare the sentences:

Bob jogs regularly but Carol doesn’t.Bob jogs regularly but Carol doesn’t.Although Carol doesn’t jog regularly, Bob does.Although Carol doesn’t jog regularly, Bob does.

B & ~CB & ~C~C & B~C & B

They are They are logically equivalentlogically equivalent..


UnderlineUnderline simple declarative sentences and choose simple declarative sentences and choose sentence letters for themsentence letters for them

Identify the main connective (if there is one), and any Identify the main connective (if there is one), and any other connectivesother connectives

Begin to build the sentenceBegin to build the sentence‘‘Bob jogs regularly if Carol doesn’tBob jogs regularly if Carol doesn’t

BB CC‘‘Bob jogs regularlyBob jogs regularly ifif it is not the case that it is not the case that Carol jogs Carol jogs

regularlyregularly~C ~C B B


Alice and Bob jog regularly if and only if Carol or Dan do Alice and Bob jog regularly if and only if Carol or Dan do not [jog regularly]not [jog regularly]

A: Alice jogs regularlyA: Alice jogs regularly B: Bob jogs regularlyB: Bob jogs regularlyC: Carol jogs regularlyC: Carol jogs regularly D: Dan jogs regularlyD: Dan jogs regularlyMain connective: Main connective: Immediate sentential components:Immediate sentential components:A & B and ~C v ~DA & B and ~C v ~D

(A & B) (A & B) (~C v ~D) (~C v ~D) OROR

(A & B) (A & B) ~(C & D) ~(C & D)

SymbolizingSymbolizingEither Alice, Bob or Carol jogs regularly.Either Alice, Bob or Carol jogs regularly.A: Alice jogs regularlyA: Alice jogs regularly B: Bob jogs regularly B: Bob jogs regularlyC: Carol jogs regularlyC: Carol jogs regularlyMain connective: vMain connective: vEither:Either:

(A v B) v C(A v B) v C ORORA v (B v C)A v (B v C)

NOT:NOT:A v B v CA v B v C


If Carol jogs regularly, then if Bob does too, so If Carol jogs regularly, then if Bob does too, so does Alice.does Alice.

C: Carol jogs regularlyC: Carol jogs regularlyB: Bob jogs regularlyB: Bob jogs regularlyA: Alice jogs regularlyA: Alice jogs regularlyMain connective: Main connective: But there are 2 sentences of the form But there are 2 sentences of the form P P Q Q..Which contains the main connective?Which contains the main connective?

C C (B (B A) A)

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