Inference Rules 3. Domain: all dogs Determine the argument using P(x) x P(x). Addition p ∴ p ∨ q p → (p ∨ q) 2. Therefore, (p 2)2 = 2 > (3 2) 2 = 9 4. Modus ponens p p → q ∴ q p ∧ (p → q) → q 5. • Using the inference rules, construct a valid argument for the conclusion: “We will be home by sunset.” Solution: 1. Propositional Logic 2. Like most proofs, logic proofs usually begin with premises — statements that you’re allowed to assume. This is called universal instantiation. 1. << /Length 5 0 R /Filter /FlateDecode >> 1 Table of Inference Rules Number and Name of the Rule Rule of Inference Corresponding Tautology 1. � *ΗaڶL{ؿ���pĚ�c���L�ż��Hc�'L���,�oq[JP��9Gᄳ5�B���G��g��Q�掠ZsI�>����4����s���Fҭ-c�aw�VIW]7{BD&3��\Y�75�"x1�[('FNL1K�Y�4s���P�A��E�I���+Ba��MĬJM�xMŇ�GH_NF��j�`N� stream All dogs are cute. %PDF-1.3 x�I�%G���+j�z�r���BL���ba=�Ѝq����;�C�7��� ��U����ɈȬ��������Ͻ����w�������{�?�ߝ�{��Q�>eƌ�G�!i��~�S�hѾ���I�뵌?|ͩ �Q\�7+n������ M�勒Jr�����O�H-�o���? Choose propositional variables: p: “It is sunny this afternoon.” q: “It is colder than yesterday.” r: “We will go swimming.” s : “We will take a canoe trip.” t : “We will be home by sunset.” 2. ∧ A;B ∴ A∧B ∧ A∧B ∴ A,B ∨ A ∴ A∨B,B∨A ∨ A∨B;¬A ∴ B A B ∴ A → B A;A → B ∴ B 13 %��������� Inference rules for propositional logic plus additional inference rules to handle variables and quantifiers. (q _r ) … This slide discusses a set of four basic rules of inference involving the quantifiers. (p _q ) addition) p _q p _q [(p _q )^(:p _r )] ! Inference rules for propositional logic Two rules per binary connective: to introduce and eliminate it. x P(x) P(c) a rule of inference. Rules of Inference for Propositional Logic Determine whether the argument is valid and whether the conclusion must be true If p 2 > 3 2 then (p 2)2 > (3 2) 2. Rules of Inference 1. Modus Tollens (method of denying) premises: q, p q conclusion: p 6 What is wrong? If a statement is true about every single object, then it … Inference rules 1 The following rules make it possible to derive next steps of a proof based on the previous steps or premises and axioms: Rule of inference autologyT Name p ^q (p ^q ) !p simpli cation) p p [(p )^(q )] ! 4 0 obj Since a tautology is a statement which is “always true”, it makes sense to use them in drawing conclusions. predicate logic. (p ^q ) conjunction q) p ^q p p ! Solution: Determine individual propositional function P(x): x is cute. Most of the rules of inference will come from tautologies. Modus Ponens (method of affirming) premises: p, p q conclusion: q 2. }��������˾�/�?�w�~� y��x��^m��{��o�ۅ���?��wz������;��}�v7���2�?O���~����{��e�Fo��ǖF����|�Q�oz���W{;�)v(�������s�2�� �����xb��}3�7�,����ow-�ٔ�;}a+�}�o�3j*�����A��I�G��遹,�2�P��OQ�l�Z3�-�7���S�t�tś�,rA>�F�x~) i��p��6~c�����Kb'�_��X��z^[��� Is the argument valid? Domain: all dogs Therefore, P(his dog). Simplification p ∧ q ∴ p (p ∧ q) → p 3. Conjunction p q ∴ p ∧ q 4. 1. Intro Elim Intro Elim Direct Proof Rule Modus Ponens Direct Proof Rule is special: not like the other rules. If a statement is true about all objects, then it is true about any specific, given object. Predicate Logic 4. Rules of inference for quantified statement (example) State which rule of inference is applied in the following argument. The argument is valid: modus ponens inference rule. 6��5\�k�B��zw+Q�hՕ�o���8��x5�l�ȗH��H. Does the conclusion must be true? ∀ ∴ foranyarbitraryc 2. The rules of inference are the essential building block in the construction of valid arguments. Modus tollens ¬ q p → q … Therefore, his dog is cute. n��EL�'Y6~cn�Û!�����:$�m|?�.q��L߃���j/2�|����C ��2'�g�#mK��]�&�PF-+l��ƙ�^�Kۿʄ�����{���/m���Ը{wxwF��;��kN���B�P#�?r;��36C�Q���t-�����t���Qk��*"�Q�HJ�߹�$��5�Yg+;�*�I��� ��b�?Ru�)���:z�Ý�+���R�5 We know that p 2 > 3 2. Ò1PÀ±îrrÍ&{ÍÙ�(RÁö/£.�KD ÓƒÀWŞ)Å$6a(6ùD½¼fıGĞɸ‘�C}šü3(Ò!Ï.QId)/Xn£¸ÁR¡ä”�åñÛÚ|”À cn4ì˜lû:pëزÀ‡è¡ÓOÇÍòàT ×Å!Ì÷h¡G‘ÆĞ�,Q\ö¶Ğ%ÈC"”ª‚ºC_óÜŠ¡Ôú22œØ>Bæ�¨qB1RE¯â7°ÿ¦n‚sk!ÚĞ*Ñ#_PÍ¢ÿ7„Õß»’•_DdCåLö7+ø[ÇÉš°ûPU«‘¾Ì½EürñVsö‰0Üöaõ}qãu#ê¿Ï�~+X/Fí¼fää~`üÒÛZ+ÀÒ2ZpYEx_a4€ô P`½•�ÓVÛZï/ ®cI’ï)²Š\Ï£tEÜK¾hÀ•)´4�‘ë]5#‹Š¡YÇ�3;+Šªƒ„û«“Œ"d¸ğ×YÁâ–fd¹ÎaïŠ]‹Ëºàî–ö b¸Jg]YÃîäîÊ/KówuÁ¦;YWô2ù#ß4 6V!�‹wˆäO©ÔÁŠ(ÛSzgƒ³òê8j[%%Ó»©ûòJ½*­`èˆO`ŒïLÒly†gš?ñéşÆ}(9zX›ë2oçJ#È£ª81q�fğ[t0?Ë.

rules of inference pdf

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