Deduction

1
Deduction

• In addition to being able to represent facts, or
  real-world statements, as formulas, we want to be
  able to manipulate facts, e.g., derive new facts
  from a set of statements. For example, if we know
  that
• If it is raining, my dog is wet.
• If my dog is wet, there will be water on the
  floor.
• There is no water on the floor.
• We should be able to conclude that it is not
  raining.

2
Deduction (cont'd)

• In propositional form, this argument would look
  like
• P -gt Q
• Q -gt R
• R
• P

3
Natural Deduction

• Is there a formal (a systematic, unambiguous way)
  of deriving new facts? We could have a set of
  rules that are apply to formulas that can be used
  to create (derive) new formulas which follow the
  original ones. One such rule is Modus Ponens
• P -gt Q
• P
• Q

4
Another Rule

• Another such rule is called Modus Tollens
• P -gt Q
• Q
• P
• We could use two applications of Modus Tollens to
  show that it is not raining.

5
Conjunction Rules

• In fact, there are other rules, some very
  obvious, so not so obvious, that we use to derive
  new facts from old ones. Three obvious ones deal
  with conjunction
• P P ? Q P ? Q
• Q P Q
• P ? Q

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7
Boxes

• In the previous slide, the blue box introduces an
  additional assumption which is at the top of the
  box. The scope of the assumption is within the
  box. After the box, the formula is no longer
  assumed to be true and may not be used. There may
  be additional formulas inside the box between the
  first and last lines. Also, boxes may be nested
  to introduce several assumptions.

8
Implication Rules

• Two rules involve implication. The second is MP
• P P
• Q P -gt Q
• P -gt Q Q

9
Other Rules

• P P
• false P false P
• P false P P

10
Rule Names

• In general, for each operator there are two (sets
  of) rules an introduction rule which introduces
  the operator, that is, the operator appears in
  the conclusion, and an elimination rule which
  removes an operator, that is, the operator
  appears in one of the hypotheses. The is one
  conjunction introduction rule, and two
  conjunction elimination rules, one disjunction
  elimination rule (proof by cases) and two
  disjunction introduction rules.

11
Rules Names (cont'd)

• Implication elimination is the Modus Ponens rule
  and there is a corresponding implication
  introduction (assume that P is true, prove Q, and
  you have the implication P-gtQ. The other four
  rules are negation introduction, negation
  elimination, contradiction elimination, and
  double negation elimination.

12
Natural Deduction

• In all, there are twelve rules (Modens Tollens
  can be derived from the other rules). These rules
  codify one approach that people take toward
  deduction (hence the name Natural Deduction). A
  natural deduction proof is a sequence of steps
  where each step is either a hypothesis (formula
  assumed to be true) or is derived from previous
  formulas by one of the rules.

13
Simple Proof

• Here is a proof that if we have hypothesis P-gtQ,
  and Q-gtR, we can derive the formula P-gtR
• 1. P -gtQ Hyp.
• 2. Q-gt R Hyp.
• 3. P Assume
• 4. Q MP
• 5. R MP
• 6. P -gt R Implication intro

14
Modus Tollens

• Modus Tollens is a derived rule, that is, any
  proof which uses Modus Tollens could be done by
  using the other rules instead. To see this,
  consider
• 1. P -gt Q
• 2. Q
• 3. P Assume
• 4. Q -gt Elim 3,1
• 5. false Elim 4, 2
• 6. P Intro 3-5

15
Derivability

• Since we have a system of natural deduction, we
  have a way to derive new facts from a set of
  given facts (hypotheses). We construct a proof
  which each line is either a hypothesis or derived
  from previous lines by a proof rule. The last
  line of the proof is the conclusion. If such a
  proof exists, we say that the conclusion C can be
  derived from the hypotheses H, H ? C (read H
  derives C.)

16
Semantic Entailment

• We also have a way of relating a formula to a set
  of formulas which we assume to be true (the
  hypotheses). If every interpretation which makes
  the hypotheses true (satisfies the hypotheses),
  also makes the conclusion true, i.e., whenever
  the hypotheses are true, the conclusion is also
  true, we say the the hypotheses semantically
  entails the conclusion, H ? C.

17
Tautologies and Contradictions

• A formula that is true under every interpretation
  is said to be valid and is also called a
  tautology.
• A formula that is false under every
  interpretation is call a contradiction and said
  to be unsatisfiable.
• A formula that is true under some interpretation
  is said to be satisfiable.

18
Proof vs. Truth

• In short, we have two notions Proof and Truth.
  We have facts that we can prove from the set of
  hypotheses (using the rules of natural deduction)
  and we have facts that we know are true given
  that the hypotheses are true (which can check by
  constructing the truth table). It turns out that
  for the natural deduction method for proposition
  calculus, these two notions coincide.

19
Soundness and Completeness

• If we can prove formula C from the set of
  hypotheses H, then H must also semantically
  entail C. In other words, if we can prove it, it
  must be true. This is known as soundness.
• If C is semantically entailed from H, then there
  is a proof of C from hypotheses H. In other
  words, if it is true, we can prove it. This is
  known as completeness.

20
Are All Systems Sound and/or Complete?

• No. Consider a system with one rule
• P
• that is, you can prove anything. Such a system is
  complete (if it's true you can prove it because
  you can prove anything), but it is not sound (you
  can prove things that aren't true).
• To construct a system that is not sound, delete
  one of the twelve rules of natural deduction.