Warm-Up Problem. Let be a Predicate logic formula and a term. Using the fact that. (which can be proven by structural induction) show that 1/26

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1 Warm-Up Problem Let be a Predicate logic formula and a term Using the fact that I I I (which can be proven by structural induction) show that 1/26

2 Predicate Logic: Natural Deduction Carmen Bruni Lecture 13 Based on slides by Jonathan Buss, Lila Kari, Anna Lubiw and Steve Wolfman with thanks to B Bonakdarpour, A Gao, D Maftuleac, C Roberts, R Trefler, and P Van Beek Predicate Logic Semantics 2/26

3 Last Time State and prove the Relevance Lemma Define what it means for a set of [well-formed] Predicate Logic formulas to semantically entail a [well-formed] formula Solve problems using this definition Predicate Logic Semantics 3/26

4 Learning Goals Use new rules for Natural Deduction over Predicate Logic Solve problems using these new rules Predicate Logic Semantics 4/26

5 Natural Deduction for Predicate Logic Natural Deduction for Predicate Logic extends Natural Deduction for propositional logic by including rules for introduction and elimination of quantifiers Other proof techniques and tricks remain the same as Natural Deduction for propositional logic In fact, all the rules from Natural Deduction extend to our Predicate Logic setting, however we need new rules to handle quantifiers Predicate Logic Semantics 5/26

6 e and i Elimination of and introduction of are fairly straightforward Name -notation inference notation -elimination ( e) -introduction ( i) If ND then ND If ND, then ND Given that a formula is true for every value of, conclude it is true for any particular value, such as that of Given that a formula is true for a particular value (of ), conclude it is true for some value Predicate Logic -Elimination and -Introduction 6/26

7 Analogy Notice the similarity between e and e, and between i and i: Name -notation inference notation -elimination ( e) -elimination ( e) -introduction ( i) -introduction ( i) If ND then ND If ND, then ND and ND If ND, then ND If ND, then ND and ND Predicate Logic -Elimination and -Introduction 7/26

8 Example Using e and e, prove that ND Predicate Logic -Elimination and -Introduction 8/26

9 Example Using e and e, prove that ND Proof: 1 Premise 2 e: 1 3 i: 2 Notice that the we use must be a symbol that hasn t appeared yet Predicate Logic -Elimination and -Introduction 8/26

10 Example: e Example Show ND Predicate Logic -Elimination and -Introduction 9/26

11 Example: e Example Show ND Proof: 1 Premise 2 Premise 3 e: 2 4 e: 1,3 Predicate Logic -Elimination and -Introduction 9/26

12 Example: i Example Show ND Predicate Logic -Elimination and -Introduction 10/26

13 Example: i Example Show ND Proof: 1 Premise 2 Assumption 3 e: 2, 1 4 e: 3 5 i: i: 5 Predicate Logic -Elimination and -Introduction 10/26

14 Note to the example The general form of rule i: Use in the previous example: We took for However, knowing what is, does not determine what is We could also take for ; thus the derivation step would be But the formula is not what we wanted to prove Predicate Logic -Elimination and -Introduction 11/26

15 Proving a Universal The -introduction rule follows ordinary mathematical usage To prove a property holds for all integers, one often starts with Let be an integer This means the same as Assume that the variable refers to an integer Then one proves that has the property Since we know nothing about the value, except that it is an integer, this justifies that every integer has the property One could also start the proof with Let be anything If is an integer, then The conclusion is essentially the same Predicate Logic -Introduction 12/26

16 Fresh Variables Definition: A variable is fresh in a subproof if it occurs nowhere outside the box of the subproof For e we do not need fresh variables! Fresh variables are available inside a nested subproof but not outside its subproof Do not reuse variable names for fresh variables in subproofs For i, you will need to start with a fresh variable For e, you will need to start with for a fresh variable (see next slides) Predicate Logic -Introduction 13/26

17 Not Free Variables Definition: A variable is not free in a set of well-formed Predicate logic formulas if and only if it is not free in any Predicate Logic -Introduction 14/26

18 Not Free Variables Definition: A variable is not free in a set of well-formed Predicate logic formulas if and only if it is not free in any As an example, let L be a language consisting of variables and predicate symbols, Then if we see that each of and are not free in (only is free in a formula of ) Predicate Logic -Introduction 14/26

19 Rule -Introduction Name -notation inference notation -introduction ( i) If ND and not free in or, then ND fresh In other words, in order to prove, prove for arbitrary Notes: It s safest to always use variables that aren t in any formula in and not in (these are always not free variables) Your fresh variable must be used only in the subproof They cannot escape boxes Use different fresh variables in different subproofs to avoid confusion Predicate Logic -Introduction 15/26

20 Analogy For -Introduction The analogy to draw is with i: Name -notation inference notation -introduction ( i) -introduction ( i) If ND and not free in or, then ND If ND and ND, then ND fresh Instead of two formulas, you re combining all the possible formulas that could occur Predicate Logic -Introduction 16/26

21 Example: i Example Show ND Predicate Logic -Introduction 17/26

22 Example: i Example Show ND Proof: 1 Premise 2 fresh 3 e: 1 4 i: 2-3 Predicate Logic -Introduction 17/26

23 Example: i Example Show ND Predicate Logic -Introduction 18/26

24 Example: i Example Show ND Solution: 1 fresh 2 Assumption 3 Reflexive: 2 4 i: i: 1 4 Note above that is so that is Predicate Logic -Introduction 18/26

25 Example: i Example Show ND Predicate Logic -Introduction 19/26

26 Example: i Example Show ND Solution: 1 Premise 2 Premise 3 fresh 4 e: 1 5 e: 2 6 e: 4,5 7 i: 3 6 Predicate Logic -Introduction 19/26

27 Elimination of an Existential Quantifier Name -notation inference notation -elimination ( e) If ND, with fresh, then ND fresh In e, the variable should not occur free in,, or (Of course, will normally be free in ) Compare this to e: Name -notation inference notation elimination ( e) If, ND and, ND, then, ND Predicate Logic -Elimination 20/26

28 Example: e Example Show ND Predicate Logic -Elimination 21/26

29 Example: e Example Show ND Proof: 1 Premise 2 fresh Assumption 3 i: 2 4 e: 1,2-3 Predicate Logic -Elimination 21/26

30 Example: e Example Show ND Predicate Logic -Elimination 22/26

31 Example: e Example Show ND Proof: 1 Premise 2 fresh Assumption 3 fresh 4 e: 2 5 i: 4 6 i: e: 1,2-6 Predicate Logic -Elimination 22/26

32 Example: e Example Show ND Predicate Logic -Elimination 23/26

33 Example: e Example Show ND Proof: 1 Premise 2 Premise 3 fresh Assumption 4 e: 2 5 e: 3,4 6 i: 5 7 e: 1,3 6 Predicate Logic -Elimination 23/26

34 Example: e Example Show ND Predicate Logic -Elimination 24/26

35 Example: e Example Show ND Proof: 1 Premise 2 fresh Assumption 3 Assumption 4 i:3 5 i:4 6 Assumption 7 i:6 8 i:7 9 e:2,3 5, e:1,2 9 Predicate Logic -Elimination 24/26

36 Natural Deduction Proof Questions (Avoid using derived rules!) 1 ND 2 ND 3 ND 4 ND 5 ND 6 ND 7 ND 8 ND 9 ND 10 ND Predicate Logic -Elimination 25/26

37 Natural Deduction Proof Questions (Avoid using derived rules!) De Morgan s Laws in Predicate Logic: 1 ND 2 ND 3 ND 4 (Assignment problem!) ND Predicate Logic -Elimination 26/26

Warm-Up Problem. Write a Resolution Proof for. Res 1/32

Warm-Up Problem. Write a Resolution Proof for. Res 1/32 Warm-Up Problem Write a Resolution Proof for Res 1/32 A second Rule Sometimes throughout we need to also make simplifications: You can do this in line without explicitly mentioning it (just pretend you