Language, Proof and
Logic
The Logic of Atomic Sentences
Chapter 2
2.1a
Valid and sound arguments
• 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
2.1b
Valid and sound arguments
The Fitch format:
All humans have two legs
Teddy is a human
Teddy has two legs
2.2a
Methods of proof
• 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. c=b
3. Cube(b)
= Elim: 1,2
2.2b
Methods of proof
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?
99%
about 98%
2?
about 90%
10?
100? about 35%
1000000? practically 0%
2.2c
Methods of proof: Principles for identity
1. = Elim (indiscernibility of identicals):
If b=c, then whatever holds of b also holds of c.
2. = Intro (reflexivity of identity):
Sentences of the form b=b are always true.
3. Symmetry of identity: If b=c, then c=b.
4. Transitivity of identity: If a=b and b=c, then a=c.
1. x2 > x2-1
2. x2-1 = (x-1)(x+1)
3. x2 > (x-1)(x+1)
= Elim: 2,1
2.2d
Methods of proof: Principles for identity
1. = Elim (indiscernibility of identicals):
If b=c, then whatever holds of b also holds of c.
2. = Intro (reflexivity of identity):
Sentences of the form b=b are always true.
3. Symmetry of identity: If b=c, then c=b.
4. Transitivity of identity: If a=b and b=c, then a=c.
Principles (3) and (4), in fact, follow from (1) and (2):
1. a=b
2. b=c
1. b=c
2. b=b
3. c=b
= Intro
= Elim: 1,2
3. a=c
= Elim: 1,2
2.3a
Formal proofs
In the formal deductive system F, every proof of S from premises
P1,…,Pn looks like the following:
Premises
P1
.
.
.
Pn
Intermediate
Conclusions
I1
.
.
.
Ik
S
Conclusion
Justification1
.
.
.
Justificationk
Justificationk+1
2.3b
Formal proofs: =Intro, =Elim, Reit
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:
1. SameRow(a,a)
2. b=a
t=t
3. b=b
4. a=b
5. SameRow(b,a)
= Elim:
P(t)
…
Reit:
t=h
P
…
…
P(h)
P
= Intro
= Elim: 3,2
= Elim: 1,4
2.4
Constructing proofs in Fitch
Do “You try it”, pages 58 and 60
Genuine rules of Fitch vs. Con “rules”.
2.5
Demonstrating nonconsequence
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.