Rules(A)-(E)whys & hows

Rules(A)-(E)whys & hows - Handout I PHIL 100...

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1 Handout PHIL 100 §§207-209 TA: Wm Kallfelz September 23, 2005 I. LOGIC (Cont.) Recall the “challenge exercise” I suggested in the previous handout: using Rules A-E, how can we determine the total number of valid standard form categorical syllogisms (SFCS)? Recall the total number of possible combinations is: 4 4 = 256 cases Let’s ask a “simpler” question: What are the total number of invalid cases , independent of syllogistic figure ? We’ve gained enough familiarity with Rules A-E to list explicitly their invalid cases: Rule A: No SFCS is valid that has two negative premises. According to this rule, cases like EOX, OEX are therefore forbidden (where X means any syllogistic form.) Why? Recall that E and I are negative categorical forms (universal and particular, respectively.) Listing these cases explicitly: EOX = { EOA, EOE, EOI, EOO } OEX = { OEA,OEE, OEI, OEO } Therefore, in set notation (i.e., in notation using operations involving sets) the total number of forbidden cases according to Rule A = 8, listed explicitly as 1 : EOX OEX = { EOA, EOE, EOI, EOO, OEA,OEE, OEI, OEO } Rule B: No SFCS is valid that has either a negative premise but does not have a negative conclusions or vice versa. 2 1 Note: The symbol: “ ”, if you’re not familiar with it, means: “union.” That is to say, the “union” of two sets is defined (informally) as follows: A B = {all elements x such that x is in A or x is in B }. It’s extremely important to understand the sense of “or” used here: it’s inclusive , and therefore not exclusive ! What does this mean? Here’s an example: “Either I am alive or I am dead” is an example of “exclusive or ,” (i.e., I cannot obviously be both alive and dead.) On the other hand, “either she will name her newborn daughter ‘Sue’ or she will name her newborn daughter ‘Ann’” is an example of inclusive ‘or.’ Why? Because there’s no logical reason why the mother to her new daughter cannot name her child: “Sue Ann.” In other words, inclusive ‘or’ involves the maximum number of cases with respect to ‘or.’ That is to say, inclusive ‘or’ occupies both the regions of A alone and B alone and where they may happen to overlap. Unless otherwise (explicitly) noted, in logic we always use “or” in this inclusive sense. 2 Note that the sense of “or” used in Rule B is clearly exclusive. Why? Explain this to yourself. (Hint: Consider the introduction of “ either or …”
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2 Clearly this forbids the following cases: EX { A or I } or OX { A or I } or XE { A or I } or XO { A or I } Or: ( EXA EXI ) ( OXA OXI ) ( XEA XEI ) ( XOA XOI )= { EAA, EEA, EIA, EOA , EAI, EEI, EII, EOI } { OAA , OEA , OIA , OOA , OAI,OEI,OII,OOI } { AEA , EEA , IEA , OEA , AEI,EEI,IEI,OEI } { AOA , EOA , IOA , OOA , AOI,EOI,IOI,OOI } Whew! But are all these cases mutually exclusive? No! Performing the (pre-)union, I
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Rules(A)-(E)whys & hows - Handout I PHIL 100...

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