Proofs (prof. would rather call it “Derivations”)
(1) A is B, (2) no C is D, (3) B is D, no B is C.
(from 1 and 3) A is D. This becomes premise (4)
if 4 and 2, no C is A.
A derivation is a sequence of annotated inference steps.
The validity of each of the steps is absolutely clear.
All individual inference steps are necessarily truth-preserving.
As is known, an argument can be valid or sound.
A derivation or proof can also be valid or sound.
How to annotate a derivation
each step is annotated
which previous steps does the current step depend on?
which basic argument form is being invoked?
basic argument forms:
hypothetical syllogism (HS): if A then B, if B then C, so if A then C.
modus tollens (MT): if A then B, not B, so not A.
if Lorenz is a logician, then he’s smart; if Lorenz is smart, then he knows Greek; Lorenz doesn’t know Greek; therefore he’s not a logician
(1) if Lorenz is a logician, then he’s smart
(2) if Lorenz is smart, he knows Greek
(3) Lorenz doesn’t know Greek
(4) if Lorenz is a logician, he knows Greek (from 1,2 by HS)
(5) Lorenz is not a logician (from 3, 4 by MT)
valid argument form: if A then B, if B then C, not A, not C, so not A (*)
valid: consider the following derivation
1. if A then B
2. if B then C
3. not C
4. if A then C (from 1, 2 by HS)
5. not A (from 3, 4 by MT)
the validity of (*) follows from those of HS and MT
logic is a highly systematic enterprise
valid argument form ⇒ all concrete instances are valid
relations between valid argument forms
later in this course we will have the proof system of natural deduction