A better distinction

1
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 mortal
• Q Socrates is human
• Therefore S Socrates is mortal
• P All humans are mortal
• Q Socrates is human
• R Socrates is ugly
• Therefore S Socrates is mortal

2

• The new argument with an enlarged set of
  premises is valid
• The situation where all humans are mortal,
  Socrates is human, Socrates is ugly but, at the
  same time, Socrates is not mortal
• 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 mortal
• Q Socrates is human
• R Socrates is not human
• Therefore S Socrates is mortal

3
Corollary

• Any sentence is deducible from a contradiction
• The following argument is valid whatever Q may
  be
• P
• P
• Therefore, Q

4
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 life
• Aquinas is a philosopher and lived a long life
• Bertrand Russell is a philosopher and lived a
  long life
• Sungho Choi is a philosopher
• Therefore, Sungho Choi will live a long life
• Weakening the support
• Augustine is a philosopher and lived a long life
• Aquinas is a philosopher and lived a long life
• Bertrand Russell is a philosopher and lived a
  long life
• Sungho Choi is a philosopher
• Sungho Choi has a terminal cancer
• Therefore, Sungho Choi will live a long life

5

• Strengthening the support
• Augustine is a philosopher and lived a long life
• Aquinas is a philosopher and lived a long life
• Bertrand Russell is a philosopher and lived a
  long life
• Sungho Choi is a philosopher
• Sungho Choi doesnt smoke and exercises on a
  regular basis
• Therefore, 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

6
Truth and validity

• The properties of truth and falsehood are
  predicated of sentences
• The properties of validity and invalidity are
  predicated of 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 mortal
• Socrates is human
• Therefore Socrates is mortal

7

• All non-humans are immortal
• Socrates is non-human
• Therefore Socrates is immortal
• All humans are mortal
• Socrates is human
• Therefore, Socrates is a philosopher
• All humans are mortal
• Socrates is mortal
• Therefore, Socrates is female
• The truth or falsehood of the conclusion of an
  argument doesnt determine its validity or
  invalidity

8
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
• An unsound argument is such that either it is
  invalid or some of its premises are false

9
The task of logic

• The logician is not so much interested in the
  truth or falsehood of sentences as in the logical
  relations between them
• Determining the validity or invalidity of
  arguments falls squarely within the province of
  logic

10
The value of special symbols

• Arguments presented in English or any other
  natural languages are often difficult to appraise
• Yesterday there were a number of crickets singing
  in our garden
• Cricket consists of bats and balls
• Therefore, yesterday there were a number of bats
  and balls singing in our garden
• The ambiguity of the term cricket
• Any of various insects that produce a shrill
  chirping sound by rubbing the front wings
  together
• An outdoor game played with bats, a ball, and
  wickets
•  

11

• He took off his shoes
• He took off his socks
• Therefore, he took off his socks and (then) shoes
• And in the conclusion has the meaning of
  temporal succession.
• He took off his shoes and (then) socks
• He took off his socks and (then) shoes
• It would be convenient to set up an artificial
  symbolic language

12
The use of a special logical notation

• Aristotle
• The special symbols of modern logic helps us to
  exhibit with greater clarity the logical
  structures of sentences and arguments

13
One further value of logical notation

• The aid they give in the actual use and
  manipulation of sentences and arguments
• The real superiority of Arabic numerals is
  revealed only in computation
• Multiplying 113 by 9
• Multiplying two numerals in Chinese character
• The drawing of inferences and the appraisal of
  arguments is greatly facilitated by the adoption
  of a special logical notation