Formal Reasoning

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Formal Reasoning

• CS311
• Western Washington University

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Why Formal Reasoning?
We have repeatedly shown the short-comings of
truth tables. We need a system that more
accurately resembles how we communicate with
each other. We need a formal reasoning
system. What makes up a formal reasoning
system? 1) a set of wffs 2) a set of axioms 3)
a set of inference rules The wffs represent the
statements The axioms represent the fundamental
truths. An axiom is a wff that we know to be
true a tautology.
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Inference Rules
An inference rule maps one or more wffs, called
premises, hypotheses, or antecedents, to a single
wff, called the conclusion or consequent.
Modus Ponens(revisited) MP(A, A? B) B The
general form of an inference rule is R(P1, P2,
, Pk) C which is also written P1, P2, ,
Pk ? C An inference rule, R(P1, P2, , Pk) C,
if the following wff is a tautology P1 ? P2 ?
? Pk ? C
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Modus Ponens (MP) A? B, A ?B Modus Tolens (MT) A
?B, ?B ??A Conjunction (Conj) A, B ?A ?
B Simplification (Simp) A ? B ?A Addition
(Add) A ?A ? B Disjunctive Syllogism (DS) A ? B,
?A ?B Hypothetical Syllogism (HS) A ?B, B ?C ?A
?C
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Reasoning in a Formal System
Proofs! A proof is a finite sequence of wffs
with the property that each wff in the sequence
is either an axiom or can be inferred from
previous wffs in the sequence. To create a
consistent theory, choose only axioms that are
tautologies and inference rules that map
tautologies to tautologies. The other optionthe
bad option.is an inconsistent theory W, where
both W and ?W can be proved as theorems.
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Proofs
Suppose the following sequence of wffs is
a proof W1, W2, , Wn W1 must be an axiom
because there arent any previous wffs in the
sequence. For all other Wi, Wi can be an axiom
or the conclusion from applying an inference
rule. And Wn is a theorem. The format of a
proof is tabular. The first column will contain
a step number, the second column will contain a
Wi, and the final column will contain a reason.
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Conditional Proofs
Most of the statements that we want to prove are
in the conditional formor some permutation
thereof. Typically, we will want to reduce the
wordiness to A ? B ? C ? D where A, B, C, and D
represent statements or clauses. The key to
proving conditionals is the Conditional Proof
Rule(CP) Suppose we wish to prove a conditional
of the form A1 ? A2 ? An ? B Start the
proof by listing each of the premises, Ai, on a
separate line with the letter P(premise) as the
reason. Treat these premises as axioms, and
construct a proof of B.
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Heres the format Prove A ? B ? C ? D 1.
A P 2. B P 3. C P k. D k1. A ? B
? C ? D 1, 2, 3, k, CP The k1 step above may be
replaced with QEDmore Latin! Lets try it (A
? B) ? (A ? C) ? ?A ? B ? C
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Subproofs
Some proofs require sub-conditional proofs. use
indentation to denote a subproof. For example,
lets prove ((A ? B) ? (B ? C)) ? (B ? C) ?
D This wff requires a subproof because there is
a conditional in its conclusionNOT because
there is a conditional in the premise. The proof
would start like this 1. (A ? B) ? (B ? C)
P 2. B P start subproof of B ? C 3. A ? B
2, Add 4. B ? C 1, 3 MP 5. C 4 Simp 6. B ?
C 2, 5, CP finish subproof 7. (B ? C) ? D 6,
Add 8. QED 1, 7 CP
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One thing to note about subproofs Statements
that are made as part of the subproof may not be
used to infer any statement that occurs after the
subproof has finished. Unlessthe statement is
independent of any of the premises of the
subproof. Confused yet?
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Simplifying Conditional Proofs
At this point, our proofs have only used P, CP,
and inference rules from chapter 6 as reasons.
What about all those equivalences could we use
them here? Yep. They can be used as
theorems. We can either write out the theorem or
just denote it with a T in the reason column.
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How do conditional proofs relate to how we
communicate in our reasoning system? Consider
the following paragraph The team wins or I am
sad. If the team wins, then I go to a movie. If
I am sad, then my dog barks. My dog is
quiet. Therefore I go to a movie. How do we
evaluate this jumble?
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Indirect Proofs
Sometimes we may not be able to get going on a
proof using the conditional proof technique. For
instance, we may feel like there arent enough
premises to do anything with, I.e. A ? B You
have the option of proving a statement using an
indirect method. 1. Prove the contrapositive.
This is a valid proof technique because of the
equivalence A ? B ? ?B ? ?A But this may not
give us much more to work with. 2. Prove by
contradiction.
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Proof by Contradiction
A proof by contradiction is based on the
equivalence A ? B ? A ? ?B ? false Now we have
at least 2 premises to start with! What makes an
indirect proof work? Indirect Proof
Rule(IP) Suppose we wish to construct an indirect
proof of the conditional A1 ? A2 ? An ?
B Start the proof by writing each premise on a
separate line with the letter P in the reason
column. Then place the wff ?B on the next
line, and write P for IP as the reason. That
stands for Premise for Indirect Proof. Then
prove a false statement.
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Heres the format of an indirect proof of A ? B
? C ? D 1. A P 2. B P 3. C P 4. ?D P for
IP k. false k1. QED 1, 2, 3, 4, k,
IP Lets do an examplewe can prove
the paragraph of jumble using the IP rule. (W ?
S) ? (W ? M) ? (S ? B) ? ?B ? M
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General Proof Notes
1. Practice makes perfectso practice
practice practice! 2. Not all wffs can be
provenwhich means that not all wffs are
tautologies! 3. You cant add extra premises to
a proof to make it easier. If you do, you
are proving a different statement! 4. You
cant apply inference rules to
subexpressionsonly to premises or
inferences, I.e. A ? ((A ? B) ? C) ? B 1.
A P 2. (A ? B) ? C P 3. B 1,2 MP
ERROR! QED 1, 2, 3, CP