Arguments

The three classic elements of the argument are the modus ponens (also called detachment):
(p & (p → q)) → q

pq p → qp & (p → q)(p & (p → q)) → q
TTTTT
TFFFT
FTTFT
FFTFT

the modus tollens:
(~q & (p → q)) → ~p

pq~p~q p → q~q & (p → q)(~q & (p → q)) → ~p
TTFFTFT
TFFTFFT
FTTFTFT
FFTTTTT

and the syllogism:
((p → q) & (q → r)) → (p → r)

pqrp → q q → rp → r(p → q) & (q → r) ((p → q) & (q → r)) → (p → r)
TTTTTTTT
TTFTFFFT
TFTFTTFT
TFFFTFFT
FTTTTTTT
FTFTFTFT
FFTTTTTT
FFFTTTTT

Note that the syllogism is essentially the property of
transitivity of the logical operator IMPLIES.

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

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:

  1. If the BIOS test runs fine, the CPU and motherboard must be OK.
  2. 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.
  3. The BIOS test runs fine and the memory is OK.
  4. Therefore there must be a flaw in the OS.
You assign logical variables to the
propositions as follows: so that your argument is represented as:
  1. p → q
  2. (q & s) → r
  3. p & s
  4. therefore r
This representation points out a fact of life: any reasonably interesting argument involves more than three logical variables; this is the exception to our general statement earlier that we would usually confine ourselves to logical expressions involving at most three variables. In the forgoing, we will refer to the propositions as premises, as is usually done with the propositions representing assumptions in an argument.

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) → conclusion
so in this case, we must construct a truth table for:
((p →q) & ((q & s) → r) & (p & s)) → r
If the column for that expression indicates that it is a tautology, then the argument is valid:

pqrs p → qq & s(q & s) → r p & s(p →q) & ((q & s) → r) & (p & s)((p →q) & ((q & s) → r) & (p & s)) → r
TTTT TTTTTT
TTTF TFTFFT
TTFT TTFTFT
TTFF TFTFFT
TFTT FFTTFT
TFTF FFTFFT
TFFT FFTTFT
TFFF FFTFFT
FTTT TTTFFT
FTTF TFTFFT
FTFT TTFFFT
FTFF TFTFFT
FFTT TFTFFT
FFTF TFTFFT
FFFT TFTFFT
FFFF TFTFFT

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:

  1. if the slowness is system-wide, then it is either a hardware problem or an OS problem
  2. if new hardware was just installed, it is either a hardware problem or a configuration problem
  3. the slowness is system-wide and new hardware was just installed
  4. therefore it is a hardware problem
You assign the logical variables as follows: so that your argument is represented as:
  1. p → (q | r)
  2. s → (q | t)
  3. p & s
  4. therefore q
Constructing a truth table to test the validity of the argument, we see:

pqrst q | rq | tp → (q | r) s → (q | t)p & s(p →(q | r)) & (s → (q | t)) & (p & s) ((p →(q | r)) & (s →(q | t)) & (p & s)) → q
TTTTT TTTTTTT
TTTTF TTTTTTT
TTTFT TTTTFFT
TTTFF TTTTFFT
TTFTT TTTTTTT
TTFTF TTTTTTT
TTFFT TTTTFFT
TTFFF TTTTFFT
TFTTT TTTTTTF
TFTTF TFTFTFT
TFTFT TTTTFFT
TFTFF TFTTFFT
TFFTT FTFTTFT
TFFTF FFFFTFT
TFFFT FTFTFFT
TFFFF FFFTFFT
FTTTT TTTTFFT
FTTTF TTTTFFT
FTTFT TTTTFFT
FTTFF TTTTFFT
FTFTT TTTTFFT
FTFTF TTTTFFT
FTFFT TTTTFFT
FTFFF TTTTFFT
FFTTT TTTTFFT
FFTTF TFTFFFT
FFTFT TTTTFFT
FFTFF TFTTFFT
FFFTT FTTTFFT
FFFTF FFTFFFT
FFFFT FTTTFFT
FFFFF FFTTFFT

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:

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:

  1. If we burn the oil from the Canadian tar sands, we will raise the concentration of carbon dioxide (CO2) in the atmosphere to over 500 p.p.m.
  2. During the early Pliocene (about 4.5 million years ago, the most recent period when the Earth was warmer than it is now), the CO2 concentration was around 400 p.p.m.
  3. During the early Pliocene, sea levels were 25 meters higher than today.
  4. If increased CO2 caused sea levels to rise during the early Pliocene, it will do so now.
  5. If sea levels rose 10 meters, approximately 600 million people would be displaced.
  6. 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 "CO2 levels are high" in place of phrases such as "we will raise the concentration of carbon dioxide (CO2) in the atmosphere to over 500 p.p.m." and "the CO2 concentration was around 400 p.p.m."; similarly for the sea levels (i.e., "above normal").
  1. There are variations in the Earth's orbit around the Sun.
  2. Natural variability in the Earth's climate is caused by variations in the Earth's orbit around the Sun.
  3. The orbital variations currently indicate that the Earth should be cooling.
  4. The Earth has been warming since 1880.
  5. 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:

Now try your hand at this one:

  1. If Alberich forsakes love, he can forge the magic ring.
  2. If Alberich has the magic ring, he can build an army and destroy Valhalla.
  3. If the ring is taken from Alberich (by Wotan), he will curse its future owners to die.
  4. If Wotan takes the ring, he will give it to Fasolt in partial payment for the construction of Valhalla.
  5. If Fasolt gets the ring, Fafner will kill him and take it for himself.
  6. If Fafner has the ring, (two operas later) Siegfried will kill him and take it for himself.
  7. If Siegfried gets the ring, Hagen will kill him, but Brunnhilde will get the ring.
  8. If Brunnhilde gets the ring, she will immolate herself on Siegfried's funeral pyre, and the flames will destroy Valhalla.
  9. Alberich forsakes love, and Wotan takes the ring.
  10. Therefore Valhalla is destroyed.
Hint: Remember that "but" is semantically identical to "and".

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 as
BooleanTable[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.
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.

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


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