The Logic of Atomic Sentences

1
The Logic of Atomic Sentences
Language, Proof and Logic
Chapter 2
2
Valid and sound arguments
2.1a

• An argument is a series of statements in which
  one, called the
• conclusion, is meant to be a consequence of the
  others, called the
• premises.
• An argument is valid if the conclusion must be
  true in any circumstance
• in which the premises are true. We say that the
  conclusion of a valid
• argument is a logical consequence of its
  premises.
• An argument is sound if it is valid and the
  premises are true.
• Is the following argument valid? Is it also sound?

Premises All humans have two legs
Teddy is a
human Conclusion Teddy has two legs


3
Valid and sound arguments
2.1b
The Fitch format
All humans have two legs
Teddy is a human Teddy has two legs


4
Methods of proof
2.2a

• To show that an argument is invalid, you need to
  find a counterexample,
• i.e. a circumstance that makes all premises
  true but the conclusion false.
• To show that an argument is valid, you need to
  find a proof, i.e. a
• step-by-step demonstration that the conclusion
  follows from the premises.
• Usually involves many intermediate steps.
• Informal proof Usually uses English perhaps
  intermixed with symbolic
• expressions. Lazily combines series of obvious
  steps. All proofs that
• you have seen in geometry (or elsewhere) are
  informal.
• Formal proof Follows a strictly predefined set
  of rules, such as, say,
• 1. Cube(c)
• 2. cb
• 3. Cube(b)
  Elim 1,2

5
Methods of proof
2.2b
Rigor At every step in the proof, the conclusion
should follow from the premises with absolute
certainty. All formal proofs are, of course,
rigorous. But so can and should be
(good) informal proofs. Why rigor is
necessary Imagine the steps/transitions relied
upon only provide 99 certainty. What would be
the probability that the conclusion indeed
follows from the premises if the number of steps
in the proof is 1? 2? 10? 100? 1000000?
99
about 98
about 90
about 35
practically 0


6
Methods of proof Principles for identity
2.2c

• Elim (indiscernibility of identicals)
• If bc, then whatever holds of b also
  holds of c.
• Intro (reflexivity of identity)
• Sentences of the form bb are always true.
• 3. Symmetry of identity If bc, then cb.
• 4. Transitivity of identity If ab and bc,
  then ac.

1. x2 gt x2-1 2. x2-1 (x-1)(x1) 3. x2 gt
(x-1)(x1) Elim 2,1


7
Methods of proof Principles for identity
2.2d

• Elim (indiscernibility of identicals)
• If bc, then whatever holds of b also
  holds of c.
• Intro (reflexivity of identity)
• Sentences of the form bb are always true.
• 3. Symmetry of identity If bc, then cb.
• 4. Transitivity of identity If ab and bc,
  then ac.

Principles (3) and (4), in fact, follow from (1)
and (2)
1. ab 2. bc 3. ac Elim 1,2
1. bc 2. bb Intro 3. cb
Elim 1,2


8
Formal proofs
2.3a
In the formal deductive system F, every proof of
S from premises P1,,Pn looks like the
following P1
.
.
. Pn
I1
Justification1
.
. .
.
.
. Ik
Justificationk
S
Justificationk1
Premises Intermediate Conclusions Conclusion


9
Formal proofs Intro, Elim, Reit
2.3b
Below t and h are any constant terms, i.e. terms
with no variables, i.e. constants or
combinations of constants through function
symbols.
Intro tt Elim P(t) th
P(h)

• 1. SameRow(a,a)
• 2. ba
• 5. SameRow(b,a)

Intro
3. bb
Elim 3,2
4. ab
Elim 1,4
Reit P P


10
Constructing proofs in Fitch
2.4
Do You try it, pages 58 and 60 Genuine rules
of Fitch vs. Con rules.
11
Demonstrating nonconsequence
2.5
Through finding a counterexample (defense
attorney vs. prosecutor) In the blocks language,
a counterexample would be a world where all
premises are true but the conclusion is not.