Define logic [fuc, ] Natural Deduction. Natural Deduction. Preamble. Akim D le June 14, 2016

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1 Define logic [fuc, ] Natural Deduction kim D le akim@lrde.epita.fr EPIT École Pour l Informatique et les Techniques vancées June 14, D le Natural Deduction 2 / 49 Natural Deduction Preamble The following slides are implicitly dedicated to classical logic. 3 dditional Material. D le Natural Deduction 3 / 49. D le Natural Deduction 4 / 49

2 Logical Formalisms Proof Types Proof Systems Proof Types Proof Systems 3 dditional Material 3 dditional Material. D le Natural Deduction 5 / 49. D le Natural Deduction 6 / 49 Terminal Symbols Terminal Symbols Propositional alculus onstants a, b, c,... Propositional Variables,,,... Unary onnective inary onnectives,, Punctuation (, ), [, ]. Predicate calculus Individual Variables x, y, z,... Functions f, g, h,..., with a fixed arity Predicates P, Q, R,..., with a fixed arity Quantifiers, Punctuation.. D le Natural Deduction 7 / 49. D le Natural Deduction 8 / 49

3 Propositional Formulas Terms formula ::= propositional variable formula formula formula formula formula formula formula With the proper arity. term ::= constant function ( term,...). D le Natural Deduction 9 / 49. D le Natural Deduction 10 / 49 First Order Formulas Syntactic onventions formula ::= propositional variable formula formula formula formula formula formula formula predicate ( term,...) individual variable formula individual variable formula ssociativity Precedence (increasing) 1, , are left-associative (unimportant) is right-associative (very important) With the proper arity.. D le Natural Deduction 11 / 49. D le Natural Deduction 12 / 49

4 Free Variables Proof Types FV(X ) = FV(P(x 1, x 2,, x n )) = {x 1, x 2,, x n } FV( ) = FV() FV( ) = FV() FV() FV( ) = FV() FV() FV( ) = FV() FV() FV( x ) = FV() {x} FV( x ) = FV() {x} Proof Types Proof Systems 3 dditional Material. D le Natural Deduction 13 / 49. D le Natural Deduction 14 / 49 Different Proof Types Different Proof Types. D le Natural Deduction 15 / 49

5 Proof Systems Proof Systems Proof Types Proof Systems Hilbertian Systems Natural Deduction Sequent alculus Natural Deduction in Sequent alculus 3 dditional Material. D le Natural Deduction 17 / 49. D le Natural Deduction 18 / 49 xioms Inference Rules xioms are formulas that are considered true a priori x x + 0 = x xiom schemes use meta-variables (that range over a specific domain) X + Y = Y + X xiom schemes are used when quantifiers are not welcome SXYZ XZ(YZ) KXY X H 1 H 2 H n Rule name xiom name xiom schemes are used when quantifiers do not apply. D le Natural Deduction 19 / 49. D le Natural Deduction 20 / 49

6 Logical Formalisms Hilbertian System single inference rule: the modus ponens modus ponens Many axioms to define the connectives ( ) ( ) ( ( )) ( ) David Hilbert ( ). D le Natural Deduction 21 / 49. D le Natural Deduction 22 / 49 Hilbertian System: Prove Natural Deduction ( (( ) )) ( ) ( ) ( ) Normalization 3 dditional Material. D le Natural Deduction 23 / 49. D le Natural Deduction 24 / 49

7 Deduction Normalization 3 dditional Material Deduction deduction is a tree whose root () is the conclusion and whose active leafs (Γ) is the set of hypotheses. ny formula is a valid hypothesis. Proof (Demonstration) Γ proof is a deduction without hypotheses.. D le Natural Deduction 25 / 49. D le Natural Deduction 26 / 49 Deductions Implication What s this? deduction of under the hypothesis. [] I E Deduction theorem, and Modus Ponens. Note the connection with (left) contraction: any number of (including 0) is discharged.. D le Natural Deduction 27 / 49. D le Natural Deduction 28 / 49

8 Proving in Natural Deduction onjunction [] I I le re. D le Natural Deduction 29 / 49. D le Natural Deduction 30 / 49 Universal Quantification bsurd [y/x] I x [] I x I x y FV(hyp()) x E [t/x] [] I I x ( ) E. D le Natural Deduction 31 / 49. D le Natural Deduction 32 / 49

9 Disjunction Existential Quantification li ri [] [] E [[y/x]] [t/x] I x x E y FV(, hyp()) For elimination, y hyp(), i.e., not in the hypotheses other than the discharged.. D le Natural Deduction 33 / 49. D le Natural Deduction 34 / 49 Negation Normalization [] I E Plus one of these equivalent formulations of the fact that classical negation is involutive. XM [ ] [ ] ontradiction Normalization 3 dditional Material. D le Natural Deduction 35 / 49. D le Natural Deduction 36 / 49

10 ut Normalization ut: Introduction of a connective followed by its elimination. I le The normalization process eliminates the cuts. I le. D le Natural Deduction 37 / 49. D le Natural Deduction 38 / 49 Normalizing onjunctions I le I re Normalizing Implications [] I E. D le Natural Deduction 39 / 49. D le Natural Deduction 40 / 49

11 Normalizing Universal Quantifiers Normalizing Disjunction I x E [t/x] [t/x] x must not be free in the hypotheses, otherwise the reduction would change them. li ri [] [] [] E [] E. D le Natural Deduction 41 / 49. D le Natural Deduction 42 / 49 dditional Material Logicians in a ar 3 dditional Material Three logicians walk into a bar, and the bartender asks Would you all like a drink? The first one says, Maybe. The second one says, Maybe. The third one says, Yes.. D le Natural Deduction 43 / 49. D le Natural Deduction 44 / 49

12 Edukera Edukera. D le Natural Deduction 45 / 49. D le Natural Deduction 46 / 49 Edukera ibliography Notes EE [Girard et al., 1989] short (160p.) book addressing all the concerns of this course, and more (especially Linear Logic). Easy and pleasant to read. Now available for free. [Girard, 2004] much more comprehensive book focusing on logic and its connections with computer science. In French.. D le Natural Deduction 47 / 49. D le Natural Deduction 48 / 49

13 ibliography I Fuck theory Experiments in visceral philosophy. dictionary-of-philosophy-logic. Girard, J.-Y. (2004). ours de Logique, Rome, utomne Girard, J.-Y., Lafont, Y., and Taylor, P. (1989). Proofs and Types. ambridge University Press. D le Natural Deduction 49 / 49