... from that thing which is false what truth can come?
Ecclesiasticus 34:4
Time for something of more substance.
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The truth table for the material conditional is:
p | q | p ⊃ q |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
One can readily understand the first line and even the second line after a moment's reflection, but why does the conditional yield a True value for False antecedents? What follows are my attempts to make sense of it.
First try:
Eminent logicians agree that these are the values, so we are justified on relying on their opinion. This being logic however, we demand proof, so:
Second try:
"P then Q" is shorthand for ~P v Q or alternatively ~(P ^~Q). If one works out the truth table for these, the result is our starting table. However, still, at a gut level feels wrong: Out of Falseness, Truth?
Third try:
If one takes a step back, it makes sense that by starting with Falseness anything goes. One can draw anything from a False starting point, for instance "In a hole in the ground there lived a hobbit". From here one can go anywhere, to Mt Doom, there, and back again. Now, more seriously:
Fourth try:
Going line by line one can see: first, having the condition of the antecedent met, we arrive at the consequent; that is the least we can expect of of any self-respecting conditional. Second, we don't want having the antecedent met, and not getting the consequent; the second line covers that. The last two lines makes clear that it doesn't care, it has no provision, when the antecedent is False. The consequent may be True or it may be False, it makes no judgment other than let it pass. Parallely:
Fifth try:
So what is actually True for a conditional when its antecedent is False? Not the conclusion, but the expresion itself, the whole conditional. Having the antecedent False doesn't spring its magic , P then Q, leaving it aloof and unblemished ("You go do your nonsense; I'll keep to myself"). From another viewpoint:
Sixth try:
Russell said that for any two propositions p and q either p⊃q or q⊃p. If you try a few in your head you'll see he's right in some sense. I think there's some controversy here.
Seventh try:
If you're still not convinced maybe you're right after all. Let's look at the alternatives and see if there' something better. Alternatives have been offered to fix the conditional and I'll look at just two of them here: giving the Truth table other values to better reflect natural language conditionals and getting into modal.
Other values:
There's an interesting article by Fulda0 which analyzes different possible truth values for the conditional making it seven sets in all+ old material conditional. For my part implication #4 looked at first like the best candidate since its gives a False values for the False antecedents. It quickly becomes apparent that adopting it would bring even graver problems . Here's its Truth table:
p | q | p ⊃ q (4) |
T | T | T |
T | F | F |
F | T | F |
F | F | F |
For starters, the third line apparently makes the fallacies of denying the antecedent and affirming the consequent disappear. Then, we'd get some funky cause and effect relationships:
I aced Logic last year. The grape harvest in France shot up that same year. Then, If I ace Logic, the grape harvest will shoot up.
If next year the harvest looks chancy, I can picture France's grape growers in session drafting a letter earnestly asking me to hit the books and save their livelihood. I claim no such power.
What if we change back the third line?
p | q | p ⊃ q (2) |
T | T | T |
T | F | F |
F | T | T |
F | F | F |
If I shoot a man through the heart, then I'll kill him on the spot.
If I were to shoot him and he dies for sure; so far so good.
If I shoot him through and he survives, that cannot be.
If I abstain from from shooting and he dies anyway; ok (maybe he had a medical condition)
It is not the case that I don't shoot him, and he doesn't die; that cannot be. Wait! Someone else could shoot him down for me. So having the fourth line all False is absurd.
Let's see if flipping the Truth value with the third helps:
p | q | p ⊃ q (3) |
T | T | T |
T | F | F |
F | T | F |
F | F | T |
The third line says that it can't be that he can continue living even when I abstain from shooting. The table is tantamount to affirming that I have power of life and death over him and no one else. Once more, that's a bit much beyond me.
Going Modal:
Some philosophers were bothered at the time by the paradoxes material conditional (which are a different can of worms) or by how it could be strengthened. Notably, Lewis came up with the modern strict conditional and opened the field of modal logic. Are strict conditionals better?
According to Stanford1 paradoxes, now modal, popped up anyway and according to Girle2 "none of these [material] 'paradoxes' are problematic in and of themselves".
Also according to Girle3, to some, the material conditional appears to be no problem for math, so why bother?
Judging from Konyndyk4 what modal logicians regard as strict Truths, it appears as if their field is very restricted especially since it doesn't include the usual suspect (ie physical necessity). Since logical Truths are included and material conditionals are tautologous, It appears to me that necessary Truths support the material rather than step away from it. I need someone shed some light on this point.
According to Barker5, a) Lewis overstated his case since Russell never drew anything in the Principia Mathematica from a False antecedent, ie he only drew Truth from Truth ; and b) Russell used the word 'implication' because he couldn't come up with a better term.
0 Fulda, J 2010, 'The Full Theory of Conditional Elements: Enumerating, Exemplifying, and Evaluating Each of the Eight Conditional Elements', Acta Analytica, 25, 4, pp. 459-477, Academic Search Complete, EBSCOhost, viewed 1 June 2015.
1 Sanford, D 1989, If P, then Q, Routledge, p73
2 Girle, R 2009, Modal Logics and Philosophy, Acumen, Durham, GBR. Available from: ProQuest ebrary. [2 June 2015] p89.
3 Girle, R 2003, Possible Worlds, Acumen, Durham, GBR. Available from: ProQuest ebrary. [2 June 2015] p33.
4 Konyndyk, K 1986, Introductory Modal Logic, University of Notre Dame Press; 1st edition
5 Barker, SF 2006, 'Lewis on Implication', Transactions Of The Charles S. Peirce Society, 42, 1, pp. 10-16, Academic Search Complete, EBSCOhost, viewed 1 June 2015.
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