A better distinction

• If a deductive argument is valid, then its conclusion follows with equal necessity from its premises no matter what else may be the case.

P: All humans are mortalQ: Socrates is humanTherefore S: Socrates is mortal

P: All humans are mortalQ: Socrates is humanR: Socrates is uglyTherefore S: Socrates is mortal

• The new argument with an enlarged set of premises is valid

• The validity of the original argument It is a contradiction that all humans are mortal, Socrates is human, and Socrates is not mortal

• The situation where all humans are mortal, Socrates is human, Socrates is ugly but, at the same time, Socrates is not mortal?

• Still contradiction!!

• The conclusion follows strictly from the enlarged set of premises because it follows strictly from the two original premises initially given

P: All humans are mortalQ: Socrates is humanR: Socrates is not uglyTherefore S: Socrates is mortal

P: All humans are mortalQ: Socrates is humanR: Socrates is not humanTherefore S: Socrates is mortal

Corollary

• Any sentence is deducible from a contradiction

• The following argument is valid whatever Q may be:

P¬PTherefore, Q

• In general, suppose that Q legitimately follows from P1, P2, P3, . . . Then Q legitimately follows from R, P1, P2, P3, . ., whatever R may be

Reverse step?

• It is impossible to make a deductively valid argument it invalid by adding new premises

• It is possible to make a deductively invalid argument valid by adding new premises

Socrates is human Socrates is mortal

Socrates is human All humans are mortal Socrates is mortal

Inductive argument• Adding new premises to the original argument can serve either to weaken or to strengthen the result argument

Augustine is a philosopher and lived a long lifeAquinas is a philosopher and lived a long lifeBertrand Russell is a philosopher and lived a long lifeSungho Choi is a philosopherTherefore, Sungho Choi will live a long life

• Weakening the support

Augustine is a philosopher and lived a long lifeAquinas is a philosopher and lived a long lifeBertrand Russell is a philosopher and lived a long lifeSungho Choi is a philosopherSungho Choi has a terminal cancerTherefore, Sungho Choi will live a long life

• Strengthening the support

Augustine is a philosopher and lived a long lifeAquinas is a philosopher and lived a long lifeBertrand Russell is a philosopher and lived a long lifeSungho Choi is a philosopherSungho Choi doesn’t smoke and exercises on a regular basisTherefore, Sungho Choi will live a long life

• Deductive argument is a type of argument whose conclusion is claimed to follow from its premises with absolute necessity, this necessity not being a matter of degree

Truth and validity

• The properties of truth and falsehood are predicated of sentences

• Arguments are either valid or invaild

• Meaningless phrases: ‘valid sentences’, ‘true arguments’

• To say that an argument is valid is to say that the truth of its premises is inconsistent with the falsehood of its conclusion

All humans are mortalSocrates is humanTherefore, Socrates is mortal

All non-humans are immortalSocrates is non-humanTherefore, Socrates is immortal

All humans are mortalSocrates is humanTherefore, Socrates is a philosopher

All humans are mortalSocrates is mortalTherefore, Socrates is female

• The truth or falsehood of the conclusion of an argument doesn’t determine its validity or invalidity

Truth and validity cont’d

• The only constraint on the truth values of the sentences imposed by the validity of an argument is that we cannot have true premises and false conclusion at the same time

• The truth or falsehood of the conclusion of an argument doesn’t determine its validity or invalidity univocally

• The validity of an argument does not guarantee the truth of its conclusion

Soundness

• An argument is sound if and only if (1) it is valid; and (2) all of its premises are true

• The conclusion of a sound argument must be true

• The difference between soundness and validity: the first guarantees but the second does not guarantee the truth of the conclusion

• The falsehood of the conclusion unsound argument either it is invalid or some of its premises are false

Symbolic language

• The more symbols a symbolic language contains, the more representational power it has, and therefore, the more accurate account of deductive arguments it gives

• “it is not the case that” negation

• “if, . . Then” conditional

• Use the capital letters ‘P’ through ‘Z’ to symbolize English sentences

Negation

• Symbolic languages consist of symbolic sentences

•‘It is not the case that Socrates is bald’ is the negation of ‘Socrates is bald’

• P : ‘Socrates is bald’

• ~P : ‘It is not the case that Socrates is bald’

• The negation of ‘P’

• NB. No parenthesis is required

Conditional

• P: Diogenes is canine Q: Diogenes is carnivorous

• (P Q) : ‘If Diogenes is canine, then Diogenes is carnivorous’

• Conditional formed from ‘P’ and ‘Q’

• ‘P’ is the antecedent of the conditional and ‘Q’ is the consequent of the conditional

• NB. Conditional symbols are accompanied by parentheses

Sentential connectives

• Refer to the phrases like ‘it is not the case that’ and ‘if then’

• ‘or’ and ‘and’

• Their main function is to connect sentences to one another to form a compound sentence

• Logical connectives

Michellanies

• the relation between a capital letter and the sentence it abbreviates is subject to change

• Sentence letters – P, Q, R, . . Z, P1, Q2, R5, Z0,

The most elementary symbolic language

• The only logical symbol is the negation symbol

• P, Q, R, . . Z, ~P, ~Q, ~R, . . ~Z, ~~P, ~~Q, . . ~~Z

• How to exhaustively characterize the class of symbolic sentences

1. Sentence letters are symbolic sentences2. Negations formed from symbolic sentences are symbolic sentences 3. Nothing other than sentences letters and negations formed from symbolic sentences are symbolic sentences

Inductive definition

• Alternative characterization

1. Sentences letters are symbolic sentences2. If φ is a symbolic sentence, then so is ~φ

• The class of natural numbers

1. The number 1 is a natural number2. If n is a natural number, n+1 is also a natural number

• The class of my ancestors

• Examples of symbolic sentences

Grammatical Tree

• We can represent this generation of the sentence by means of a grammatical tree that displays its genealogy

• Each initial node is a sentence letter

• The top node is the symbolic sentence whose genealogy is being displayed

A new symbolic language

• The only logical symbol is the conditional symbol

• P, Q, R, . . Z, (P P), (PQ), (PR), . . (PZ), (QP), (QR), (P(PP)), ((PQ) Q), . .(P ((PZ) R) Z)…

• Sentence letters are symbolic sentences• Conditionals formed from symbolic sentences are symbolic sentences• Nothing other than sentences letters and negations formed from symbolic sentences are symbolic sentences

• Alternative characterization

1. Sentences letters are symbolic sentences2. If φ and ψ are symbolic sentences, then so is (φ ψ)

• Examples

• Grammatical tree

A more complex symbolic language

• A symbolic language that contains both the negation sign and conditional sign

• P, Q, ~R, . .~~ Z, (~P P), (~~PQ), (P~~~R), . . (PZ), (QP), (QR), (P(PP))

• Sentences letters are symbolic sentences• If φ is a symbolic sentence, then so is ~φ• If φ and ψ are symbolic sentences, then so is (φ ψ)

• Examples

Definitions

• Atomic sentences vs. compound sentences

• The main connective of a compound sentence is the connective that is used at the last step in building the sentence

• Examples

• Conditional sentence and negation sentence

• Examples

Grammatical tree

• Each nonbranching node is of the form ‘~φ’; and it has the symbolic sentence pi as its sole immediate ancestor

• Each branching node is of the form (φ ψ), having the symbolic sentence φ as its immediate left ancestor and the symbolic sentence ψ as its immediate right ancestor

• Any expression that can be generated as the top node of a grammatical tree is a symbolic sentence

Parenthesis

• The function of parentheses is just like that of punctuation in written language

• The teacher says John is a fool

• PQR

• (~P Q) vs. ~(P Q)

• “If it doesn’t rains, I go out without an umbrella” “It is not the case that if it rains, I go out without an umbrella”

Informal notation

• No confusion will arise if we omit the outermost parentheses of a sentence

• When parentheses lie within parentheses, some pair may be replaced by pairs of brackets for the sake of display and recognition

• In official notation, a symbolic sentence is enclosed by a single pair of outermost parentheses but in informal notation it is not

• Chapter 1, Section 1 of Terence Parsons’ article

Translation

• Translation and symbolization

• Translation into English

• A scheme of abbreviation correlates a sentence letter with an English sentence

• Two steps of translation: literal vs. free translation

• Free translation is a liberal version of literal translation

Literal translation

1. Restore any parentheses that may have disappeared as a result of informal conventions

2. Replace sentence letters by English sentences in accordance with the given scheme of abbreviation

3. Replace the negation sign with ‘it is not the case that’

4. Replace the conditional sign with ‘if then’

Free translation

• A free translation or translation simpliciter is a sentence we can get from a literal translation only by changing its style

• A free translation of φ into English is a stylistic variant of the literal translation of φ into English

• How to determine whether a sentence is a stylistic variant of the literal translation of φ?

Guideline

• Negation

• “It is not the case that John has 4 limbs”

• “John does not have 4 limbs”

• “John fails to have 4 limbs”

• Conditional

• “If John has 4 limbs then John has 2 siblings”

• “Provided that John has 4 limbs then John has 2 siblings”

• “On the condition that John has 4 limbs then John has 2 siblings”

• “John has 4 limbs only if John has 2 siblings”

• To assert that A only if B is to deny that A is true but B is false. This is to assert that if A then B

• Chapter 1 Section 2 of Terence Parson’s article

Cautionary note

• John owns a car Stylistic variants of one another?• John owns an automobile

• John is an unmarried man Stylistic variants of one another? • John is a bachelor

• John doesn’t own a car Stylistic variants• It is not the case that John owns a car

• If John is old, he can own a car Stylistic variants• In case John is old, he can own a car

• What is the difference?

• In the second case, the expressions at issue are phrases of connection but this is not true in the first case

• The expressions, “car” and “automobile”, are not phrases of connection

• Two synonymous sentences are stylistic variants of each other only if their difference concerns phrases of connection

Symbolization

• A symbolic sentence φ is a symbolization of an English sentence ψ iff ψ is a free English translation of φ.

• φ is a symbolization of an English sentence ψ iff ψ is a stylistic variant of the literal English translation of φ

Procedure

• Introduce ‘it is not the case that’ and ‘if . . Then’ in place of their stylistic variants.

• Replace ‘if . . .then’ with the conditional sign

• Replace ‘it is not the case that’ with the negation sign

• Replace English sentences by sentence letters in accordance with the given scheme of abbreviation

• Omit outermost parentheses according to the informal convention

Grouping together

• ‘If he does not greet, she will be distraught’

• ‘If’

‘She will be distraught if he greets’

• ‘only if’

‘She will be distraught only if he greets’

Ambiguous sentences

• ‘It is not the case that she will be distraught if he does not greet’

~(P Q)

P ~Q

• ‘if Wilma leaves Xavier stays if Yolando sings’

(Yolando sings) ((Wilma leaves) (Xavier stays))

(Wilma leaves) ((Yolando sings) (Xavier stays))

Commas

• A comma indicates that the symbolizations of sentences to its left or the symbolization of sentences to its right should be combined into a single sentence

• ‘If Wilma leaves, Xavier stays if Yolando sings’

Requiring that ‘Xavier stays’ and ‘Yolando sings’ are grouped together

• ‘If Wilma leaves Xavier stays, if Yolando sings’

‘Wilma leaves’ and ‘Xavier stays’ are required to be grouped together

Logical derivation

• A criterion for validity for those arguments that are formulated in the symbolic language under discussion

A symbolic argument is an argument whose premises and conclusion are symbolic sentences

• A derivation consists of a sequence of steps from the premises of a given argument to its conclusion

• Each step constitutes an intuitively valid argument

Mathematical derviation

X = 7+8+9+10+11+12+13+14Therefore, x = 84

X = 7+8+9+10+11+12+13+14 X = 15+9+10+11+12+13+14X = 24+10+11+12+13+14…..Therefore, X = 84

• By going through all of these steps, we can get from the premise of the original argument to its conclusion

Four inference rules

• Modus Ponens (MP): φ ψφ,Therefore, ψ

• Modus Tollens (MT)

φ ψ~ψ,Therefore, ~φ

• Double Negation (DN)

φTherefore, ~~φ

~~φTherefore, φ

• Repetition

ΨTherefore, ψ

Three types of derivation

• Direct derivation

• Conditional derivation

• Indirect derivation