All men are mortal. ����� #�|Q=�H��l��h��B:��������̗�"-��Xk�-m�r�ot��]��. Statements in Predicate Logic P(x,y) ! Example: Party Planning We want to plan a party. �H�o\Ϣ�d�Z}� �R A5�u"�� s�M�e�?���L�f-�G��-����Y 13 0 obj <> stream %���� ]"��N!��^V'�r+�LN�]���i,�ʯg�_�``�d��c�6��%vܡ��~���Y�X��͂�Q��y+(g�j .��yںr�Ε�1[���RA�U�O�V�y�@�c� b��k��zj�ʒv��X�9�iy�����ws�7�)�,����|���]�2��ގNpN� �[U�����s܂ᴅ��R���_`���n���LWK���~�q�j��s�u��q����VFӜ�t� Truth Values An interpretation I is a function which assigns to any atomic formula p i a truth value I(p i) 2f0;1g: If I(p i) = 1, then p i is called true under the interpretation I. Translating into propositional logic a: you are a computer science major b: you are a freshman C: you can access the Internet from campus you can access the Internet from campus only if you are a computer science major or you are not a freshman, c →a V ¬b . 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A set of sentences defining a set of vectors is called the axiomatization of the set of vectors. %�쏢 x��\[��4��S��S���qt�`!,,�����d&$�$�K��~-YV˖s2���$. ��n}\G2��y�K]����ҚȲ�u�#U�$ R�u�TzŴ7��e|e����OXY�����Չ�&x�'����{�ܚ���-� ��M9ɇ�K�T�Ӂspi�R2#������H#�!p;��Y�揚܍��J,ͻ����t���$s��6��3�u�o�&m���.���kl�=��L�HcӉW>sG"�G'�!��i���t;�v�s�M�ID�ϐ�3���^f��"9�ܨɊ�]U/��-��z�_K�|�y�l�4i�4ק��s��B�2�H7�G5��D��ST��v�s��� ����J)��s)$9�����¾;v�|�-V,�m_LG��xM����}�z9YT�h+�fb��T�}4�l9��e�A�=S� & '�� jt9�%�V� ��C����� [��������O�v��^�MR�Q� �쥊va�E� %PDF-1.4 K�$�`9���PB�����{.q9"����m���7�ۉ]o����r�Ė4?&� ��X��k0��"{�0�%� Propositional Logic In this chapter, we introduce propositional logic, an algebra whose original purpose, dating back to Aristotle, was to model reasoning. ! Logic in Computer Science 4. Math 127: Propositional Logic Mary Radcli e 1 What is a proposition? 1�X��*oS�t~�f ��� md��TC(���T��g MpÓaԍ��������W&D�M�/�s��g2�,�l��z�'=�情q��,r��t�$d努~��jJHN"������/���=a,)���!`}V���u�DfU�ZN]5 ! Axiomatizability. propositional logic if and only if there is a signature of size n and a set of sentences from the corresponding language such that the vectors in the set correspond to the set of interpretations satisfying the sentences. ��eT��f�ZH- �F�Lr#�� p�EZm��� m�$7Ҟ 7�MT��f�ZH- �F0����b 8�� PROPOSITIONAL LOGIC: THE FULL LANGUAGE VL Logic, Lecture 2, WS 20/21 Armin Biere, Martina Seidl Institute for Formal Models and Verification Johannes Kepler University Linz. note formulas of propositional logic. '��f��t�7�\�ֿ�3Y���e�l�� ߭�ԫW� �Ro�|�'����P+��]o� YT�ָAJ��]Ѻ��;!n} 5 0 obj Alessandro Artale Logic: Propositional Logic (Part I) Propositional inference: Truth Table method Let α= A∨B and KB = {(A∨C),(B ∨¬C)} Is it the case that KB |= α? 3 0 obj <> endobj A predicate P describes a relation or property. If I(p i) = 0, then p i is called false under the interpretation I. G��jOuO~9(�È�=i�i� ����=G���V�M�Y~��+�K�&a�*>�G���B�� �(1�ﴸH������edf�0��Oc���'47�m� I ��4B���d+{14[t�JK_�.��+Ū��=��^H0��pB�&I�l��!�](��[��9[v����.0�ǽ ��|>�}^��� 3��%=_f���_���l[��OOZ��ő���~xR ��Qc��_K)�1c��&����pv��~E���=Ȓ+�\�xpM�� ����w��|q���la> ����2J���'����6��~Ͱ5tdi�K��_�!Y�r���Q�,�����Д��#�� ��K�t�}��$c�e}u� &��o�e~���?����k#���f"��7� �rRr��R!�Iw�"JWT*V��3�0Ҏx�f�� {��l�[���K�`�R���*R��`$��j�;.�&���q�����Q٘����*琂��ɐ_Qj�U9f����I�Zli}� Z��z�S�yO� �x"tb�v�v�X3���M"�B�_��,l��ts�Dž-ӠʍƂ�T���eIR���4\m����@�A#�Ԗ�k0A����HD$���9����!zg����-Q8�����}�4�����ە�ڰ{=f.�Oa{>3�sZ�l�7d���H���$�`V�����S�aKl��nW�:�K�,�l�w��-V��P�|�yI��u� �P�z�D(�W�-�HD�c#��,E�@�V�V�������nA:al�7�BP��b���0�/��՟���"�t�/ ��l��n����[�a�\�$ڮ�L�m�H���nL1і��q�Y�]^��$g�5�\+����6��T]f�-�Y����6ռ#�NT��uը���F�$��3��D����a k�� �z� ;��R�. <> PROPOSITIONAL FORMULAS: SYNTAX VL Logic Part I: Propositional Logic. For example, Chapter 13 shows how propositional logic can be used in computer circuit design. Variables (x,y) can take arbitrary values from some domain. {iFm�ڟ���/˽���1�]Tct������j�93|~8�R The fundamentals of proofs are based in an understanding of logic. 12 0 obj <> /XObject <> /ProcSet [/PDF /Text /ImageC ] >> /Group 14 0 R /Trans <> /Parent 22 0 R /MediaBox [0 0 453.543 255.118 ] /TrimBox [0 0 453.543 255.118 ] /Rotate 0 >> endobj In order to consider and prove mathematical statements, we rst turn our attention to understanding the structure of these statements, how to manipulate them, and how to know if they are true. �6��Bj�6���i��k��Th��R �̈́�H{: 6�B�B�� ��Z ��h&�G����z:Zm��� mD3ᅑ�� �. Predicate logic can express these statements and make inferences on them. M[�4J� �;Bi+|��7fV8�-C? stream ����vw�}�����ݝ�rw�s���7�{����������]���̴ �n{ �N{� �[�(������S��f�=?5D�RN7�6�o�-d��ϸ�ax�+�7�^}�� �~8��~�t�4�"�/ 엻��8|y�,AJlu��Q�Nk�pARG+���"��J���M�q�VFHkI"0ZK�������`-B��(��أrf�0�J���������:�CZ8T*)O*���!M�F���߃�#v��T��b�_d�RMC�h�8�j������p9O~=_]�ɡ�:zdb�#�2�?�q��I�M(�g�c�U�����RC��C[;���+��:���Ii�uΦ���@��S^�C>��x��C�ZL�ype�'�6�|"�M#��L-E�"�ߙ��(�L�כY]1uE'lї�m���$�l�%�^`S?,s&`��p��]���3��� ֣��hh��v�+��=�G`I�٤m��G��$��F㈴ZX~ݚȬ��c�����8$S��ι}��3��̢��)O/\q������q�6��>��T=��Pu��/Gh���!�h��y�I*���n���� [h�&R�K�f�=ۘj�I-��o�k�T�0:���i�#k�v�Փ�������ڱ��W{�r Propositional logic is the simplest logic illustrates basic ideas usingpropositions P 1, Snow is whyte P 2, oTday it is raining P 3, This automated reasoning course is boring P i is an atom or atomic formula Each P i can be either true or false but never both The values true or false assigned to each proposition is called truth value of the proposition. 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