DeMorgan s Laws and the Biconditional. Philosophy and Logic Sections 2.3, 2.4 ( Some difficult combinations )
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1 DeMorgan s aws and the Biconditional Philosophy and ogic Sections 2.3, 2.4 ( Some difficult combinations )
2 Some difficult combinations Not both p and q = ~(p & q) We won t both sing and dance. A negation of a conjunction. Both not p and not q = (~p & ~q) We won t sing and we won t dance. A conjunction of negations.
3 ~(p & q) (~p & ~q) hese are not equivalent to one another. he first says: it is not the case that both will happen. he second says: it is the case that both will not happen.
4 Difficulties, continued Neither p nor q = Not either p or q = It is not the case that either p or q = ~(p v q) We will neither sing nor dance. A negation of a disjunction. Either not p or not q = (~p v ~q) Either we won t sing or we won t dance. A disjunction of negations.
5 ~(p v q) (~p v ~q) hese too are not equivalent to one another. he first says: neither will happen. he second says: one or the other of them won t happen.
6 We need a fourth connective he preceding formulations use = incorrectly. Clearly the sentences are different sentences. What we want to say is that the two sentences have equivalent truth conditions No matter what pattern of on/off switches occurs in the world, these two sentences will light up at exactly the same times. One is on (true) if and only if the other is on (true). or this we need...
7 Biconditional symbol: (triple bar) translation: if and only if P Q P Q
8 Simplest rule: rue only if the truthvalues match P Q P Q
9 hat is: rue only if both are rue OR both are alse (it's the second clause that gets forgotten) P Q P Q
10 DeMorgan s aws A conjunction (p & q) is true if and only if both conjuncts are true. So it is false ( ~(p & q)) if one or the other of the conjuncts is false. hat is: ~(p & q) (~p v ~q) Not both p and q Either not p or not q.
11 DeMorgan s aws (2) A disjunction is false ( ~(p v q)) if and only if both disjuncts are false. So: ~(p v q) (~p & ~ q) Neither p nor q Both not p and not q.
12 Cross correlations ~(p & q) (~p v ~q) (~p & ~ q) ~(p v q) Not both p and q Either not p or not q Both not p and not q Neither p nor q
13 In English It is not the case that we will both sing and dance. Either we will not sing or we will not dance. ~(s & d) (~s v ~d) Both we will not sing and we will not dance. We will neither sing nor dance. (~s & ~d) ~(s v d)
14 ~(p & q) (~p v ~q) P Q
15 ~(p & q) (~p v ~q) P Q p & q
16 ~(p & q) (~p v ~q) P Q p & q ~(p & q)
17 ~(p & q) (~p v ~q) P Q p & q ~(p & q) ~p ~q
18 ~(p & q) (~p v ~q) P Q p & q ~(p & q) ~p ~q ~p v ~q
19 ~(p & q) (~p v ~q) P Q p & q ~(p & q) ~p ~q ~p v ~q ~(p& q) (~p v ~q)
20 ~(p & q) (~p v ~q) P Q p & q ~(p & q) ~p ~q ~p v ~q ~(p& q) (~p v ~q)
21 ~(p & q) (~p v ~q) P Q p & q ~(p & q) ~p ~q ~p v ~q ~(p& q) (~p v ~q)
22 ~(p & q) (~p v ~q) P Q p & q ~(p & q) ~p ~q ~p v ~q ~(p& q) (~p v ~q)
23 ~(p & q) (~p v ~q) P Q p & q ~(p & q) ~p ~q ~p v ~q ~(p& q) (~p v ~q)
24 ~(p & q) (~p v ~q) P Q p & q ~(p & q) ~p ~q ~p v ~q ~(p& q) (~p v ~q)
25 ~(p & q) (~p v ~q) P Q p & q ~(p & q) ~p ~q ~p v ~q ~(p& q) (~p v ~q)
26 ~(p & q) (~p v ~q) P Q p & q ~(p & q) ~p ~q ~p v ~q ~(p& q) (~p v ~q)
27 ~(p& q) (~p v ~q) ~p v ~q ~q ~p ~(p & q) p & q Q P ~(p & q) (~p v ~q)
28 Symbolize & test for validity Valerie is either a doctor or a lawyer. Valerie is neither a doctor nor a stockbroker. Hence Valerie is a lawyer. D: Valerie is a doctor. : Valerie is a lawyer. S: Valerie is a stockbroker.
29 Valerie is either a doctor or a lawyer. D v Valerie is neither a doctor nor a stockbroker. Hence Valerie is a lawyer. D: Valerie is a doctor. : Valerie is a lawyer. S: Valerie is a stockbroker.
30 Valerie is either a doctor or a lawyer. D v Valerie is neither a doctor nor a stockbroker. Hence Valerie is a lawyer. D: Valerie is a doctor. : Valerie is a lawyer. S: Valerie is a stockbroker.
31 Valerie is either a doctor or a lawyer. D v Valerie is neither a doctor nor a stockbroker. Hence Valerie is a lawyer. D: Valerie is a doctor. : Valerie is a lawyer. S: Valerie is a stockbroker.
32 he test Construct a truth table. Use a separate column for each premise. Put the conclusion on the rightmost end. Compute the values. Ask: Is there any row where all the premises are true and the conclusion is false? If yes: the argument is invalid. If no: the argument is valid.
33 D v herefore,
34 D v herefore, D S
35 D v herefore, D S D v
36 D v herefore, D S D v (D v S)
37 D v herefore, D S D v (D v S)
38 D v herefore, D S D v (D v S)
39 D v herefore, D S D v (D v S)
40 D v herefore, D S D v (D v S)
41 D v herefore, D S D v (D v S)
42 D v herefore, D S D v (D v S)
43 D v herefore, (D v S) D v S D
44 D v herefore, (D v S) D v S D
45 D v herefore, (D v S) D v S D
46 D v herefore, (D v S) D v S D
47 D v herefore, (D v S) D v S D
48 he test Is there any row where all the premises are true and the conclusion is false? If yes: the argument is invalid. If no: the argument is valid.
49 he test Is there any row where all the premises are true and the conclusion is false? Note that on any row we need to look only at the columns for the premises and for the conclusion. he other columns can be ignored! Mark the premise columns somehow, and put the conclusion in your last column.
50 D v herefore, (D v S) D v S D
51 D v herefore, (D v S) D v S D No such row. VAID!
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