Reason and ArgumentChapter 6 (1/4)

Common misperceptions about logic:The science of Deduction and Analysis is one which can only be acquired by long and patient study, nor is life long enough to allow any mortal to attain the highest possible perfection in it. From A Study in Scarlet

A joke:Sherlock Holmes and Dr. Watson go on a camping trip. After a good dinner and a bottle of wine, they retire for the night, and go to sleep. Some hours later, Holmes wakes up and nudges his faithful friend. "Watson, look up at the sky and tell me what you see." "I see millions and millions of stars, Holmes" replies Watson. "And what do you deduce from that?" Watson ponders for a minute. "Well, astronomically, it tells me that there are millions of galaxies and potentially billions of planets. Astrologically, I observe that Saturn is in Leo. Horologically, I deduce that the time is approximately a quarter past three. Meteorologically, I suspect that we will have a beautiful day tomorrow. Theologically, I can see that God is all powerful, and that we are a small and insignificant part of the universe. What does it tell you, Holmes?" Holmes is silent for a moment.

The punchline:"Watson, you idiot!" he says. "Someone has stolen our tent!"

Common misperceptions about logic:Typically, Spock said nothing about logic, per se. Whenever Spock would claim that something was or was not logical, he generally meant rational.

A note about the rules of logic:The rules of logic are not made up or stipulated, or even proved (though they can be derived from one another like the axioms of Euclidian geometry, and as with any axiomatic system, at least one axiom will always remain unproven)The rules of logic are discovered, and are lent force by the very fact that theyre obvious, otherwise we wouldnt be able to call them rules of logic. Well start with the simplest rules: the rules for and

But first, Propositions:The textbook authors are (understandably) imprecise about talk of propositions. To wit: John is Tall is not a proposition, just like 3 is not a number.

PropositionsJohn is tall is a sentence that expresses the proposition that John is tall.3 is a numeral that expresses what we mean by the number 3. This is important because one should not get the idea that any sentence can express a proposition. To express a proposition, a sentence must be an example of a linguistic act.

Propositional Form and Substitution Instancesp & q represents any two joined propositions. Though the rule in the book (p. 144) allows that different variables be replaced by the same proposition, for practical purposes we will never do this. Also contrary to the text, Roses are red and violets are blue does not represent a single proposition, but instead a conjunction of two of them.The 8th edition of the book inserts a justification for this based on an analogy to mathematics. My reply: logic, whatever it looks like, is not precisely math. We have good reasons to limit one proposition to one propositional variable and vice versa.

Something else about propositions:Propositions are bearers of truth-values.That means that any given proposition can have the property of being true or the property of being false, and all propositions have one or the other (which is why we insist on the linguistic act constraint)Does that mean that we must view truth as a black-and-white kind of thing? Well, yes, it does.

Got a problem with black and white? Why dont you tell me that to my face! Thought so

Bearers of truth-valueSince propositions are expressed by sentences that are meaningful, they reflect states of affairs. In other words, they reflect the way things are or are not.Take the proposition expressed by the sentence John is tall.The proposition is true if it is considered in a state of affairs in which John is tall and it is false if it is considered in a state of affairs in which John is not tall. Notice that whether we agree about the state of affairs is a different question.

How truth tables work:The leftmost columns are called reference columns and contain each individual propositional variable (or sentence), usually in alphabetical order.There is one remaining column for each connective (&, v, ~, ) used.Each row of a truth table corresponds to one possible state of affairs. Every possible state of affairs is represented on a truth table. The number of rows is 2n where n is the number of reference columns.

The truth table for &

pqp & qTTTTFFFTFFFF

Propositional versus nonpropositional conjunction:Exercise IV:1. nonpropositional2. nonpropositional3. nonpropositional4. propositional5. propositional6. nonpropositional 7. nonpropositional8. either (ambiguous)9. nonpropositional

Validity for &

conclusionpremisepqp & qTTTTFFFTFFFF

How about this one?

conclusionpremisepqp & qTTTTFFFTFFFF

What about this?

premiseconclusionpqp & qTTTTFFFTFFFF

Notice:

premiseconclusionpqp & qTTTTFFFTFFFF

How about:

premisepremiseconclusionpqp & qTTTTFFFTFFFF

So any substitution instances of the following will ALWAYS be valid.p & qp

p & qq

pq___p & q

Exercise V 1. valid2. not valid3. valid (trivially)4. valid5. valid (though conversationally, a different meaning is implied)6. nonpropositional, but valid (would require predicate logic to demonstrate)

Exercise VI1. True2. True3. True

Exclusive vs. Inclusive orSometimes when a person says something of the form p or q they mean p or q or both and sometimes they mean p or q and not both. The former is an inclusive or and the latter is exclusive.Most logicians default to the inclusive or. Some even claim that all uses of or are inclusive, and it is conversational implication that makes some of them exclusive. In any case, it is important to examine cases where or is used to determine which is which, because it will affect the validity of any argument that or is used in.

Disjunction

pqp v qTTTTFTFTTFFF

NegationIt is tempting to say that Smurfs are blue and Smurfs are not blue are sentences that express two propositions. That is not the case. What is going on is that the same proposition is involved, and in one case the proposition is negated. If s stands for Smurfs are blue and ~ is our symbol for negation, then Smurfs are not blue is formalized as ~s.

Be careful with NegationSometimes not is syntactically ambiguous. Translating ~ as it is not the case that can help to disentangle ambiguity.Be careful with opposites. nobody owns Mars is the negation of somebody owns Mars because it is not the case that somebody owns Mars means the same thing as nobody owns MarsHowever, some opposites are not binary. Consider Cheering for the Yankees is moral. The negation of this should just be It is not the case that cheering for the Yankees is moral. Resist the temptation to translate the negation as Cheering for the Yankees is immoral. This is because actions that are not moral could be either amoral or immoral (but not both).The point is, just be strict in translating ~ as it is not the case that

Further ambiguity in negation:Consider ~(Everyone loves running)Not everyone loves runningEveryone does not love runningEveryone loves not runningNo one loves runningEveryone hates runningEveryone loves walkingFor the sake of Pete, just say It is not the case that everyone loves running

Disjunctive SyllogismConsider the argument:

p v q ~pq

CP2P1pq~pp v qTTTFFTFF

Disjunctive SyllogismConsider the argument:

p v q ~pq

VALID

CP2P1pq~pp v qTTFTTFFTFTTTFFTF

Consider the Argument:p v qp___~q

P2CP1pq~qp v qTTTFFTFF

Consider the Argument:p v qp___~q

INVALID

P2CP1pq~qp v qTTFTTFTTFTFTFFTF

Pay attention to parenthesesNotice that ~a & g means something different than ~(a & g).Substitute Annie is rich for a and Gina is happy for g.The first phrase translates to It is not the case that Annie is rich and it is the case that Gina is happy.The second phrase translates to It is not the case that both Annie is rich and Gina is happy.How about ~a & ~g?

Logic and MathI know that logic LOOKS for all the world like math, but resist the temptation to treat mathematical symbols and logical symbols as interchangeable.For example, math has parentheses, and also has a negative symbol, - that looks a bit like logics negation symbol ~, so since changing (2 + 3) to -2 + -3 is a mathematically valid procedure, changing ~(p & q) to ~p & ~q should be logically valid, right?

Equivalence of ~a & g, ~(a & g), ~a & ~g

ag~a~ga & g~a & g~(a & g)~a & ~gTTFFTFFFTFFTFFTFFTTFFTTFFFTTFFTT

Exercise XII15. A v ((~B & C) v ~(~B v ~(Z v B))) T v ((~T & T) v ~(~T v ~(F v T))) T v ((~T & T) v ~(~T v ~T)) T v ((~T & T) v ~(~T v ~T)) T v ((F & T) v ~(F v F)) T v ((F & T) v ~(F v F)) T v (F v ~F) T v (F v ~F) T v (F v T) T v (F v T) T v T T v T T