(p & (p → q)) → qthe

pqp → qp & (p → q)(p & (p → q)) → qT T T T T T F F F T F T T F T F F T F T

(~q & (p → q)) → ~pand the

pq~p~qp → q~q & (p → q)(~q & (p → q)) → ~pT T F F T F T T F F T F F T F T T F T F T F F T T T T T

((p → q) & (q → r)) → (p → r)Note that the syllogism is essentially the property of transitivity of the logical operator IMPLIES.

pqrp → qq → rp → r(p → q) & (q → r)((p → q) & (q → r)) → (p → r)T T T T T T T T T T F T F F F T T F T F T T F T T F F F T F F T F T T T T T T T F T F T F T F T F F T T T T T T F F F T T T T T

There are two others, whose proofs we leave to the reader:

**Disjunctive Syllogism**: ((p | q) & ~p) → q**Constructive Dilemma**: ((p | q) & (p → r) & (q → s)) → (r | s)

These argumentative constructions (!) are the bases for the arguments we will discuss below.

Consider the following scenario:

You just upgraded to a new motherboard in your computer, but because the new motherboard did not support your old CPU and memory, you had to replace those as well. Now the system is crashing randomly and you don't know what to do. The store where you purchased the new hardware tested your memory and pronounced it OK. You attempt to reason as follows:

- If the BIOS test runs fine, the CPU and motherboard must be OK.
- If the CPU and motherboard and memory are all OK, then there must be a flaw in the way your OS handles the new motherboard.
- The BIOS test runs fine and the memory is OK.
- Therefore there must be a flaw in the OS.

- p = the BIOS test runs fine
- q = the CPU and motherboard must be OK
- r = there must be a flaw in the OS
- s = the memory is OK

- p → q
- (q & s) → r
- p & s
- therefore r

In order to test the validity of an **argument**, you must construct a
truth table for the following
logical expression:

(premise 1 & premise 2 & .... & premise n) → conclusionso in this case, we must construct a truth table for:

((p →q) & ((q & s) → r) & (p & s)) → rIf the column for that expression indicates that it is a tautology, then the argument is valid:

p | q | r | s |
p → q | q & s | (q & s) → r |
p & s | (p →q) & ((q & s) → r) & (p & s) | ((p →q) & ((q & s) → r) & (p & s)) → r |

T | T | T | T | T | T | T | T | T | T |

T | T | T | F | T | F | T | F | F | T |

T | T | F | T | T | T | F | T | F | T |

T | T | F | F | T | F | T | F | F | T |

T | F | T | T | F | F | T | T | F | T |

T | F | T | F | F | F | T | F | F | T |

T | F | F | T | F | F | T | T | F | T |

T | F | F | F | F | F | T | F | F | T |

F | T | T | T | T | T | T | F | F | T |

F | T | T | F | T | F | T | F | F | T |

F | T | F | T | T | T | F | F | F | T |

F | T | F | F | T | F | T | F | F | T |

F | F | T | T | T | F | T | F | F | T |

F | F | T | F | T | F | T | F | F | T |

F | F | F | T | T | F | T | F | F | T |

F | F | F | F | T | F | T | F | F | T |

Since the final column is all Ts, the argument is valid.

It is amusing to note that this problem was encountered by the author during the writing of this text. It turns out that while this argument was valid, and a bug report was submitted to the authors of the OS, the actual problem was bad memory after all! The vendor's memory tester was woefully inadequate for finding the kind of errors that were causing the problem; a more intensive test produced in excess of 4,000 errors in a 17 hour test of the 512 MB of memory.So while the argument is valid,

there is no guarantee that the conclusion is true. The difference between the validity of an argument and the truth of its conclusion is often very difficult for students, and it is of primary importance. We established the validity of our logic, but that does not prove the truth of the conclusion.

The following argument may help make that more clear. Your system is running slowly, and wishing to analyze the problem logically you make the following argument:

- if the slowness is system-wide, then it is either a hardware problem or an OS problem
- if new hardware was just installed, it is either a hardware problem or a configuration problem
- the slowness is system-wide and new hardware was just installed
- therefore it is a hardware problem

- p = the slowness is system-wide
- q = it is a hardware problem
- r = it is an OS problem
- s = new hardware was just installed
- t = it is a configuration problem

- p → (q | r)
- s → (q | t)
- p & s
- therefore q

p | q | r | s | t |
q | r | q | t | p → (q | r) |
s → (q | t) | p & s | (p →(q | r)) & (s → (q | t)) & (p & s) |
((p →(q | r)) & (s →(q | t)) & (p & s)) → q |

T | T | T | T | T | T | T | T | T | T | T | T |

T | T | T | T | F | T | T | T | T | T | T | T |

T | T | T | F | T | T | T | T | T | F | F | T |

T | T | T | F | F | T | T | T | T | F | F | T |

T | T | F | T | T | T | T | T | T | T | T | T |

T | T | F | T | F | T | T | T | T | T | T | T |

T | T | F | F | T | T | T | T | T | F | F | T |

T | T | F | F | F | T | T | T | T | F | F | T |

T | F | T | T | T | T | T | T | T | T | T | F |

T | F | T | T | F | T | F | T | F | T | F | T |

T | F | T | F | T | T | T | T | T | F | F | T |

T | F | T | F | F | T | F | T | T | F | F | T |

T | F | F | T | T | F | T | F | T | T | F | T |

T | F | F | T | F | F | F | F | F | T | F | T |

T | F | F | F | T | F | T | F | T | F | F | T |

T | F | F | F | F | F | F | F | T | F | F | T |

F | T | T | T | T | T | T | T | T | F | F | T |

F | T | T | T | F | T | T | T | T | F | F | T |

F | T | T | F | T | T | T | T | T | F | F | T |

F | T | T | F | F | T | T | T | T | F | F | T |

F | T | F | T | T | T | T | T | T | F | F | T |

F | T | F | T | F | T | T | T | T | F | F | T |

F | T | F | F | T | T | T | T | T | F | F | T |

F | T | F | F | F | T | T | T | T | F | F | T |

F | F | T | T | T | T | T | T | T | F | F | T |

F | F | T | T | F | T | F | T | F | F | F | T |

F | F | T | F | T | T | T | T | T | F | F | T |

F | F | T | F | F | T | F | T | T | F | F | T |

F | F | F | T | T | F | T | T | T | F | F | T |

F | F | F | T | F | F | F | T | F | F | F | T |

F | F | F | F | T | F | T | T | T | F | F | T |

F | F | F | F | F | F | F | T | T | F | F | T |

The presence of even a single F in the final column indicates that this argument
is **not valid**, even though it may have been a reasonable approach in troubleshooting.
The flaw in the reasoning was that we didn't account for the possibility that it could
be BOTH an OS bug and a configuration error: the row containing the F in the final column
began with all of the variables being T except q. This corresponds to the situation:

- the slowness was system-wide
- new hardware was just installed
- it was
**not**a hardware problem - but it was
**both**an OS problem **and**a configuration problem

In that row, all of the premises are true, but the conclusion is false.
So we see that validity of an argument means that it is impossible to have a false conclusion when all of the premises are true.
That is, if the premises are all true, the conclusion *must* be, if the argument is valid.

But as we saw above, the validity of an argument does not prove that its conclusion is true, because one or more of the
premises *might be false*.

We suggest the reader analyze the following two arguments concerning global warming:

- If we burn the oil from the Canadian tar sands, we will raise the concentration of carbon dioxide (CO
_{2}) in the atmosphere to over 500 p.p.m. - During the early Pliocene (about 4.5 million years ago, the most recent period when the Earth was warmer than it is now),
the CO
_{2}concentration was around 400 p.p.m. - During the early Pliocene, sea levels were 25 meters higher than today.
- If increased CO
_{2}caused sea levels to rise during the early Pliocene, it will do so now. - If sea levels rose 10 meters, approximately 600 million people would be displaced.
- Therefore, if we burn the oil from the Canadian tar sands, we will displace over 600 million people.

Hint: Be careful with the numbers in that one; it is probably best to "simplify" the argument using phrases such as "CO_{2}levels are high" in place of phrases such as "we will raise the concentration of carbon dioxide (CO_{2}) in the atmosphere to over 500 p.p.m." and "the CO_{2}concentration was around 400 p.p.m."; similarly for the sea levels (i.e., "above normal").

- There are variations in the Earth's orbit around the Sun.
- Natural variability in the Earth's climate is caused by variations in the Earth's orbit around the Sun.
- The orbital variations currently indicate that the Earth should be cooling.
- The Earth has been warming since 1880.
- Therefore global warming is not due to natural variability in the climate.

Hint: There are three implications in that argument, even though the word "if" doesn't appear.Comparing the last two arguments, we can make some general observations which might help you as you translate a "verbal" argument into logical symbols:

- Some premises can be simple statements of fact (i.e., premises 1 and 4 in the second argument). Some are obvious implications (although beware that the word "if" is used frequently in a rhetorical sense, which makes translation much more difficult). In general, any premise that is not a simple statement (or a conjunction or disjunction of simple statements) is probably an implication, even if the word "if" does not appear in it. Look for "causal" words and phrases, such as "caused" or "indicates" or "is due to". Another common phrasing appears in premises 2 through 4 in the first argument: "during the Pliocene" can be translated as "if it is the Pliocene".
- If you have the chance to choose two variables to represent phrases which have opposite meaning, use (for example) r and ~r instead.
- If you don't understand a premise, try to find different ways to state it
*that mean the same thing*. - As we will see with the next argument, simply translating an argument into symbolic terms can often make it easier to understand.

Now try your hand at this one:

- If Alberich forsakes love, he can forge the magic ring.
- If Alberich has the magic ring, he can build an army and destroy Valhalla.
- If the ring is taken from Alberich (by Wotan), he will curse its future owners to die.
- If Wotan takes the ring, he will give it to Fasolt in partial payment for the construction of Valhalla.
- If Fasolt gets the ring, Fafner will kill him and take it for himself.
- If Fafner has the ring, (two operas later) Siegfried will kill him and take it for himself.
- If Siegfried gets the ring, Hagen will kill him, but Brunnhilde will get the ring.
- If Brunnhilde gets the ring, she will immolate herself on Siegfried's funeral pyre, and the flames will destroy Valhalla.
- Alberich forsakes love, and Wotan takes the ring.
- Therefore Valhalla is destroyed.

Hint: Remember that "but" is semantically identical to "and".It often happens that non-equivalent translations of an argument can both by valid. It is important that the argument be stated clearly, and that the translation be as literal as possible without being too complex. For example, in the last argument, it is probably unnecessary to distinguish between someone taking the ring and that same person possessing it.As written, that argument involves (after a small amount of simplification) 14 propositions; however, the validity of the argument is far easier to establish than you might guess. If you know how to use Mathematica, you can use the "BooleanTable" function to facilitate determination of the validity of this argument.

In Mathematica, AND is represented by &&, OR is represented by ||, NOT is represented by !, EQUIVALENCE is represented by ==, and IMPLIES and XOR are represented by the functions Implies and Xor. For example, our first argument (about the crashing system), ((p →q) & ((q & s) → r) & (p & s)) → r, would be expressed asBooleanTable[Implies[Implies[p, q] && Implies[q && s, r] && (p && s), r], {p, q, r, s}]The output of this BooleanTable function is a list of 16 Trues, corresponding to the final column of the truth table above.

In the next section we move on to a generalization of logic called Boolean Algebra.

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©2013, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.

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