Clutch

I really can't make heads or tails of that, John, but it has been a long weekend now.

Notice, though, that nothing caused 2. And nothing caused >, and nothing caused 1. Yet if there's anything that better be amenable to logical characterization, it's that 2>1.

You can see the same point (that -> does not mean 'causes') even with physico-causal examples. If my car starts, then the battery was charged; but this hardly means that my car's starting caused the battery's being charged. The antecedent is a sufficient condition for the consequent, but the sufficiency is truth-functional, not causal.

If you're late again, you'll be fired -- under what circumstances would the boss who uttered this be said to have uttered a lie? Only when you're late again, but are not fired. In the circumstance that you're late and are fired, the above conditional is obviously true. And your boss certainly can't be said to have lied if you're fired for some other reason altogether despite not being late, indeed even if you're never late again. In short, the conditional is false only when the antecedent is true but the consequent false; in all other circumstances it's true, and in particular, therefore, whenever the antecedent is false the conditional is true.

ex-xian

OK, I think I may know where the confusion is coming from.

When analyzing a conditional, these are the rules, keeping in mind that S=>P strictly means ~(S & ~P).

S=>P
S___
P

If it rains, the street is wet. It rained, therefore the street is wet.

and

S=>P
~P__
~S

If it rains, the street is wet. The street is not wet, therefore it did not rain.

These rules are known as modus ponens and modus tollens. I forget which is which. Clutch can probably tell us, his he has a philosophy degree (I think).

The two common fallacies are:

S=>P
~S__
~P

This is called denying the antecedent.
If it rains, the street will be wet. It did not rain, therefore the street is not wet. Obviously there are other ways the street can be wet.

S=>P
P___
S

Called affirming the consequent.
If it rains, then the street will be wet. The street is wet, therefore it rained. Again, there are other ways for the street will be wet.

In these situations, something can be said about the antecedent or the consequent depending on the available information. However, you analyzing a conditional in the sense of part of an argument, you take the truth value of the antecedent and consequent and make an evaluation of the truth value of the conditional as a whole. You can't take the result of an argument and try to analysis a small part of it based on the conclusion.

If you could restate the argument succinctly as possible, then maybe off us could put our heads together, get it symbolic form, and find out if it's true.

Clutch:
Do you do a lot of work in logic? Ever use reduction trees? If the answers are yes, I'd like to PM for advice about graduate schools and programs

John Page

Thansk for your diligence in replying after your true day!
1. In the context fo the human mind, the symbol 2 triggers the concept 2 (that we severally but intersubjectively share). Same goes for the "greater than" sign and the number 1.
2. Perceiving the collection of symbols 2>1 triggers an assembly of the above concepts into a compound concept which can be tested against axioms - e.g. a number that follows another number in the sequence of countable numbers is always larger than that other number.
3. Through these various comparisons of concepts we arrive at a conclusion we consider to be true.

Simply because a symbol written on a piece of paper represents a highly abstract entity doesn't mean it wasn't caused. Every time you think of the concept 1 there is an instance of it in your mind caused by your thought processes.
Without a causal relationship, meaning dissolves. I'm not proposing that -> means "causes", I'm proposing that every logical operator stands for a cause and effect process, not an ethereal absolute ideal. (Not that I'm accusing you of that!).
Thanks - I see the practical application in this example.

However, as I've pointed out with other examples, the boss' statement makes no claim as to what will (or will not happen) if I'm not late. As far as I can make out, this is an issue with two-valued logics that assume a statement must be either true or false and, as a consequence, when the the results of falsity in the antecedent are not specified the logic says the consequent can either be true or false. In some cases it can or some cases it can't.

Example: If the cow moves, then this statement is true.
A: The cow moves
B: This statement is true.

This gives me the following truth table:
A B A=>B
-----------------
TTT
FTF
FFF
TFF

Which is neither if nor iff. I had to return my Copi to the library so now I'm wondering what kind of relation this is supposed to be....

Cheers, John

John Page

ex:

Thanks. Yes, seems I'm guilty of looking at the propositional logic "if then" construct and trying to fault it by finding A's and B's that don't fit, rather than trying to find the logical relation that better describes the connection between A and B.

But then I believe in a system of (potentially) unlimited valued logic in which the truth values for different relations are merely digital milestones along the way from TTT through TFT and FTF to FFF.

Cheers, John

ex-xian

The logic you're using isn't powerful enough to analyze your argument. You'd have to do something like this (I'm using the reduction tree method):

1. X <=> (C=>X)-------1.Premise
2. C ----------------------2. Premise
3. ~X --------------------3. Denial of conclusion
----/-----------\
4. X----------~X---------4. from 1, material equivalence
5.(C=>X)---~(C=>X)--5. from 1, ME
----x------6.---C --------6. from 5, denied implication
---------------~X
-----------------o

(the x means the path is closed, the o means that path is open)


The reduction tree method works by denying the conclusion of an argument, building the tree and finding contradictions. In line 4. X contradicts ~X in 3, so the branch is closed. But in the other path, no contradictions are found. Therefore the argument is invalid. A counterexample can be found by looking at the open path and check the truth value of the popostions.

In this case, a counter example is
X C
F T

So the argument is invalid if X is false and C is true.

Scorpion

As I said above, you can't read B AND ~B from the table - only B OR ~B, which is not only non-contradiction but a tautology.

In other words, that table doesn't tell you anything about the truth of B when A->B and ~A are true.

-S-

jpbrooks

While I don't understand John's analysis of logical implication in terms of causality (causal relationships seem to be instantiations of the logical principles regarding implication), I, like him, see the instances where the antecedents of conditional statements are false as troubling. It may be possible to "demonstrate" logically that the truth value of a conditional is true whenever its antecedent is false. But wouldn't such a "demonstration" be unavoidably circular? If so, then why does formal logic (arbitrarily) stipulate a form of implication that seems to be rather different from the way we use implication in ordinary language? And if not, why not use the form of implication used in the logic of the "demonstration" rather than the form in question?

Interesting discussion! I'll be back in a few hours.

Clutch

No. In effect it depends upon Double Negation Elimination: ~~P->P. That's what tells you that in all the circumstances where the conditional is not false, it's true. It's not arbitrary.

First, it models many ordinary uses of implication perfectly, as in the lying example I gave above. Second, it gives a nice tidy truth-table -- which one might find an indecisive reason on its own, but is surely not arbitrary. And third, it is explicitly advanced as but one formal notion of implication itself. Hence it is distinguished from strict implication, from the subjunctive conditional, and so forth.

jpbrooks

Hello Clutch!


I follow you.
The problem, however, is that assigning a truth value of ~F to a conditional whose antecedent is false is still a stipulatory assignment. There seems to be no reason why one could not simply assign a value of F to the conditional when its antecedent is false. The only (possible) reason that I can think of for not doing so would be that on the stipulation that the truth value of the conditional [P -> Q] is ~T when its antecedent [P] is false, its truth table would become identical to that of [P ^ Q].

Granted. But ordinary language need not assume that [P -> Q] is false only when P is true and Q is false.

On the assumption that implication should be a logical "operation" that is distinct from other logical "operations" such as "AND" ("^") and "OR" ("v"), I would agree with you. However, there seems to be no reason why this assumption must be accepted.

Please forgive my ignorance (I have only completed one undergrad course in Logic), but by "subjunctive conditional", I'm assuming that you are referring to "counterfactual conditionals". But perhaps I'm wrong.

The problem here seems to be that, in ordinary language, the conditional statement itself does not tell us what formal notion of implication is intended. And such considerations rarely arise in normal everyday conversation.

John Page

I'm trying to understand the material implications of using this on the OP.

Was I correct in concluding it is false (invalid) that "If the moon is made of green cheese, then I'm the king of France."?

Cheers, John