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# Class Note for PHIL 382 at UMass

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This 4 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 13 views.

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Date Created: 02/06/15

Phil 382 Logic Supplement Negation not A IA T F F T other symbols 7 Conjunction and TT T TF F FT F FF F other symbols amp Disjunction or A B A v B T T T T F T F T T F F F Arguments Material Conditional if then A B A B T T T T F F F T T F F T other symbols 3 Note A 9 B is equivalent to A IB and is equivalent to A v B Material Biconditional if and only itquot TT T TF F FT F FF T other symbols E Note A lt gt B is equivalent to A BB A An argument is a set of formulas with one designated as the conclusion The formulas that are not the conclusion are the premises These premises are meant to provide support for the conclusion For example the following is an argument P L C Here C is the conclusion of the argument This is indicated by placing it beneath a horizontal line P and Q are the premises and are meant to support the conclusion C Validity We say that an argument is valid if and only if Necessarily if the premises are true then the conclusion is true Another way to put this an argument is valid if and only if Assuming the premises are true the conclusion is true too Entailment Often in this course we ll talk about entailment rather than validity The two concepts are very similar however We say that a set of formulas l entail a formula 3 if and only if Necessarily if the members of 4 are true then 3 is true Another way to put this a set of formulas l entail a formula 3 if and only if Assuming that the members of 4 are true then 3 is true too In other words a set of formulas l entail a formula 3 if and only if the argument with 3 as a conclusion and the set of formulas 4 as premises is valid Example A A B entails A There is one way for A A B to be true When we assign truth values in this way A is true So assuming that A A B is true A is true So A A B entails A Example A 9 B and A together entail B There are three ways to assign truth values so that A 9 B is true A B A 9 B To see if the entailment holds we also need to assume that A is true This can happen in only one way A B A 9 B A When we assign truth values in this way B is true So assuming that A 9 B is true and A is true B is true So A 9 B and A entail B Example A v B and IB together entail A There are three ways that A v B can be true A B A v B F F F There is one way that B can be true IB F W Putting these together we see that there is only one way that both A v B can be true and B can be true AB When we assign truth values in this way we see that A must be true So assuming A v B is true and IB is true A is true So A v B and IB entail A Quantifiers Occasionally we ll need to talk about quantified formulas in this course Here are some examples of quantified sentences All humans are mortal Some students study philosophy No man is an island Some philosophers are not male These are quantified sentences because at the beginning of them they talk about some quantity of things eg all things or some things or no things In logic we have two quantifiers to represent these sentences Vx 7 this is the universal quantifier it means something like for all x Elx 7 this is the existential quantifier it means something like for some x We can then represent quantified sentences like this Vx Ax Bx For all X if X is A then X is B All As are Bs ExAx A Bx For some X X is A and X is B 2 Some As are Bs Final Note This supplement is meant to help you not to confuse you or scare you away from this course We will make use of logic quite a bit in this course especially at the beginning so if you re confused about any of this you need to be sure to see me However I am confident that anyone in this course can learn the necessary logic without too much trouble

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