The NT Application of the Canaanite Conquest

Originally posted by DeepThought
I see what you're saying about probabilistic logic although I had to read it three or four times before I got what you were saying. But I'm wondering about the symbolic stuff at the top and it's translation into normal language.

You can replace (P & R) with (P v R), and then assert ¬P to draw the odd conclusion that if you didn't have a cup of tea ...[text shortened]... s a problem with the back reference "it", is this what you mean by strengthening the antecedent?

But you can't replace (P&R) with (PvR). From (P>Q) you can derive ((P&R)>Q), but you cannot derive ((PvR)>Q). You can check this with truth-tables. No assignment of truth-values to P,Q, and R makes (P>Q) come out true while also making ((P&R)>Q) come out false. But there is an assignment of truth-values [P=f, R=t, Q=f] that makes (P>Q) true while also making ((PvR)>Q) come out false. In plain English, the world could be such that "If P, then Q" is true while "If P or R, then Q" is false. But that means there's no valid inference from "If P, then Q" to "If P or R, then Q".

Originally posted by bbarr
Nice post.

From P > Q it follows that (P & R) > Q. So, apparently, the following is a valid inference:

1) If I had a cup of tea right now, then I would enjoy it very much.

2) Therefore, If I had a cup of tea right now and somebody put broken glass in it, then I would enjoy it very much.

Of course, we're not inclined to agree. This example of 'str ...[text shortened]... t set of worlds is a subset of worlds where Reagan does not win. In that subset, Anderson wins.

Yep, that looks right. McGee's purported counterexample loses its intuitive appeal if we clarify the distinction between a conditional probability and a probability of a conditional. And I would agree that our assessment of the conditional should be sensitive to context.

Originally posted by bbarr
But you can't replace (P&R) with (PvR). From (P>Q) you can derive ((P&R)>Q), but you cannot derive ((PvR)>Q). You can check this with truth-tables. No assignment of truth-values to P,Q, and R makes (P>Q) come out true while also making ((P&R)>Q) come out false. But there is an assignment of truth-values [P=f, R=t, Q=f] that makes (P>Q) true while al ...[text shortened]... is false. But that means there's no valid inference from "If P, then Q" to "If P or R, then Q".

There's something wrong with my understanding of conditional proof.

Originally posted by DeepThought
There's something wrong with my understanding of conditional proof.

We can walk through it, if you want.

Originally posted by bbarr
We can walk through it, if you want.

Thanks. The book I read on logic was EJ Lemmon (on loan and given back so I don't have access to it) so I'll follow his notation but give the dependencies on the right as typesetting on RHP forums isn't the easiest. Also he doesn't distinguish between assumptions and givens which I like to do.

M.P. = modus ponens
&I = conditional inclusion (A,B => A&B)
vI = or inclusion (A => AvB)
C.P. = conditional proof

A) P->Q, R => (P&Q)->R

1) P->Q given
2) R given
3) P assumption
4) Q M.P. 1,3
5) P&R &I 2,3
6) P&R->Q C.P. from 4 and 5 eliminating assumption 3. Based on 1, 2

This works because (P&R) depends on both of them being true. If R is false then Q can still be true since it's a material "if". On an if and only if (i.e. P<->Q, R => (P&R)<->Q) it fails unless we also have Q->R.

Now for the flawed proof:
B) P->Q => (PvR)->Q

1) P->Q given
2) P assumption
3) Q M.P. from 1, 2
4) PvR v.I. from 2
5) (PvR)->Q C.P. eliminating assumption 2, depends on 1

In a strange way PvR is stronger than P, since it's true when R is true and P isn't. I can always make step 4 assuming P is true, but somehow lines 4 and 5 conspire to produce the wrong answer. So it's kind of alarming that I can go through more or less the same steps in the second proof as the first but come out with a fallacy. I wonder if step 6 in "proof" A is flawed, but gets the right answer by legerdemain?

There's also the proof P&¬P => Q
1) P&¬P given
2) ¬Q
3) ¬¬Q 1,2 R.A.A eliminating 2
4) Q from 3 by D.N. assumes 1

But 2 and 1 aren't obviously connected. The plain language statement is that all things follow from a contradiction - but I'm wary of the proof, since I haven't derived a contradiction from ¬Q I've just asserted it. There's something similar going on with the problem I have with conditional proof, which is what I'm confused about.

Originally posted by DeepThought
Thanks. The book I read on logic was EJ Lemmon (on loan and given back so I don't have access to it) so I'll follow his notation but give the dependencies on the right as typesetting on RHP forums isn't the easiest. Also he doesn't distinguish between assumptions and givens which I like to do.

M.P. = modus ponens
&I = conditional inclusion ( ...[text shortened]... milar going on with the problem I have with conditional proof, which is what I'm confused about.

I don't really understand the last one you gave for the explosive principle.

But the principle should hold, as seen here:

(1) P & ~P (taken on supposition).
(2) So P (from 1, conjunction elimination).
(3) So P or Q (from 2, vI).
(4) But ~P (from 1, conjunction elimination).
(5) So Q (from 3 & 4, modus tollendo ponens).
(6) (P & ~P) -> Q.

Originally posted by DeepThought
Thanks. The book I read on logic was EJ Lemmon (on loan and given back so I don't have access to it) so I'll follow his notation but give the dependencies on the right as typesetting on RHP forums isn't the easiest. Also he doesn't distinguish between assumptions and givens which I like to do.

M.P. = modus ponens
&I = conditional inclusion ( ...[text shortened]... milar going on with the problem I have with conditional proof, which is what I'm confused about.

Great, let’s take the first case. From P>Q we want to derive (P&R)>Q. To the right of each line, in parentheses, I’ve included their justifications and line dependencies.

1. P>Q (Premise, 1)
2. P&R (Assumption, 2)
3. P (Conjunction Elimination, 2)
4. Q (Modus Ponens, 1,2)
5. (P&R)>Q (Conditional Proof, 1)

Just after stating our premise the proof begins, as conditional proofs invariably do, with the assumption of the antecedent of the conditional we aim to derive. When we get Q on line (4), we’ve done so using both lines (1) and (2). But when we “conditionalize out” in our conclusion, we thereby discharge the assumption in line (2). So, our conclusion rests solely on line (1). So, our conclusion follows from our sole given premise.

Contrast this with the attempt to derive (PvR)>Q :

1. P>Q (Premise, 1)
2. PvR (Assumption, 2)
3. P (Assumption, 2,3)
4. Q (Modus Ponens, 1,3)
5. R (Assumption, 2,4)
6. ????

As above, after stating our premise we begin with the assumption of the antecedent of the conditional we aim to derive. In this case, we’re assuming the disjunction (PvR). Disjunctions are true just in case at least one of their disjuncts is true. So, to derive something from a disjunction we have to show, in effect, that given our premise set our desired conclusion follows independently from each disjunct. In this case, that means we have to show both that Q follows from P>Q together with the assumption P, and also that Q follows from P>Q together with the assumption R. The problem, though, is that our premise allows us to derive Q from the assumption P, but neither it nor anything else allows us to derive Q from the assumption R.

In your proof, line (5) is invalid. You haven’t derived Q from (PvR), so you can’t “conditionalize out” and discharge the assumption upon which (PvR) is based. Line (5) is still predicated on the assumption in (2) and hence doesn’t follow merely from our premise.

Originally posted by Zahlanzi
genocide is evil.

how are you still not getting this?

there are no justifications, no "let's look at this from a different angle"

genocide is evil and those who engage in it are evil.

And it is total BS that a deity would command people to kill. It just shows how man made the whole affair is, introducing anthropomorphisms into religion is a sure sign it is not religious at all but political in nature.

I sneak in here a word on my OP.

In the Old Testament God brought the people OUT in order to bring them IN. The Exodus was more than a liberation. It was a liberation with a divine purpose to bring the former slaves INTO a promised land of luxurious supply for everything they needed.

It was also God's intention that on that good land they build God a temple for Him to dwell in. The typology refers to God's eternal purpose. He saves human beings to bring them OUT of the tyranny of Satan and the currupted humanity to bring them INTO a new humanity. That is a humanity saturated, permeated with all the riches of Christ.

The good land the Israelites came to was occupied by horrible cultures and even giants. They had to fight to chase them out or disperse them or kill them. There were times when God said that they should not pity. God seems to have made certain that these kingdoms were really worthy of total vanquish before He brought the Israelites in.

God had told Abraham in Genesis 15 that He would not bring the Hebrews into Canaan yet because the people had not gotten bad enough yet. Four hundred more years of decline and degradation would ripen the inhabitants of Canaan for divine judgment in the one time in history conquest by a theocratic nation. Then God added on another 40 years for these nations to witness "the army of Jehovah" wandering in the wilderness preparing to enter Canaan.

"And He said to Abram, Know assuredly that your seed will be sojourners in a land that is not theirs, and they will serve them; and they will afflict them FOUR HUNDRED YEARS.

And in the fourth generation they will come here again, FOR THE INIQUITY OF THE AMORITES IS NOT YET COMPLETE." (Genesis 15:14,16)

By the time the 400 plus 40 years were finished the Canaanite society had become so sinful that the divine command, a one time historical event, was for the genuine theocratic nation to disperse their societies and in some combatant situations show no pity.

The NT parallel is that the Christian church towards the old man, the fallen adamic humanity, should after being forgiven in Christ's redemption, ruthlessly live Christ to the termination of the Satanified and corrupted fallen man.

This is an action of each Christian in the "army" towards himself or herself and not towards others. With one's self the Christian should be ruthless ignoring self pity to let the new nature in Jesus Christ thoroughly replace the old way of living. That is an old man that has caused the Son of God to have to go to the cross for it is hopelessly corrupted before God.

Originally posted by bbarr
Great, let’s take the first case. From P>Q we want to derive (P&R)>Q. To the right of each line, in parentheses, I’ve included their justifications and line dependencies.

1. P>Q (Premise, 1)
2. P&R (Assumption, 2)
3. P (Conjunction Elimination, 2)
4. Q (Modus Ponens, 1,2)
5. (P&R)>Q (Conditional Proof, 1)

Just after stating our premise the proo ...[text shortened]... ) is still predicated on the assumption in (2) and hence doesn’t follow merely from our premise.

I think I had the "proofs" organised wrongly, which sort of lead to the confusion. My assumption was P, but yours P&R (PvR), and so your Q actually follows from the assumption (or doesn't in an obvious way) whereas mine doesn't properly. So my proof A only accidentally gets the right answer. Instead of starting with what's on the L.H.S. of the '=>' I needed to be assuming what is on the L.H.S. of the '->'. Which lead me to have R as an extra and unneeded given.

My vague aim with logic is to try to understand Gődel's completeness/incompleteness theorems which aren't the easiest.

Originally posted by DeepThought
There's also the proof P&¬P => Q
1) P&¬P given
2) ¬Q
3) ¬¬Q 1,2 R.A.A eliminating 2
4) Q from 3 by D.N. assumes 1

But 2 and 1 aren't obviously connected. The plain language statement is that all things follow from a contradiction - but I'm wary of the proof, since I haven't derived a contradiction from ¬Q I've just asserted it. There's something ...[text shortened]... milar going on with the problem I have with conditional proof, which is what I'm confused about.

Right, this is one of the proof strategies for the Principle of Explosion. The problem with your reductio is that you've missed a couple steps.

Consider:

1. P&~P (Premise, 1)
2. ~Q (Assumption, 2)
3. ~Q&(P&~P) (Conjunction Introduction, 1,2)
4. P&~P (Conjunction Elimination, 1,2)
5. ~~Q (R.A.A., 1)
6. Q (Law of Double Negation, 1)
7. (P&~P)>Q (Conditional Introduction, {0} "empty set of premises"😉

The trick here is making sure that the contradiction you want to derive from the assumption ~Q actually depends on ~Q. So, above, I used a contamination strategy to yield that dependence.

Originally posted by LemonJello
I don't really understand the last one you gave for the explosive principle.

But the principle should hold, as seen here:

(1) P & ~P (taken on supposition).
(2) So P (from 1, conjunction elimination).
(3) So P or Q (from 2, vI).
(4) But ~P (from 1, conjunction elimination).
(5) So Q (from 3 & 4, modus tollendo ponens).
(6) (P & ~P) -> Q.

See my above post to bbarr, I think my third proof was suffering from the same problem as the first two. R.A.A. is something on the lines of:

Q->(P&¬P) => ¬Q

my "proof" doesn't really have (P&¬P) as a conclusion from Q so it probably only accidentally works because of the way I was misusing conditional proof. I now think that all I'd really proved trying to do it that way was the law of non-contradiction and had Q floating around as a confounder. If I write it out more fully:

1) ¬Q
2) P&¬P
3) ¬Q->(P&¬P) (can I do this?)
4) ¬¬Q
5) Q

I'm thinking I can't do line 3. Whereas since => Q->Q is a theorem:

1) P&¬P
2) (P&¬P) -> (P&¬P)
3) ¬(P&¬P)

is fine because I can clearly get from line 1 to line 2 as all I'm doing is inserting a theorem.

Originally posted by bbarr
Right, this is one of the proof strategies for the Principle of Explosion. The problem with your reductio is that you've missed a couple steps.

Consider:

1. P&~P (Premise, 1)
2. ~Q (Assumption, 2)
3. ~Q&(P&~P) (Conjunction Introduction, 1,2)
4. P&~P (Conjunction Elimination, 1,2)
5. ~~Q (R.A.A., 1)
6. Q (Law of Double Negation, 1)
7. (P&~P) ...[text shortened]... ctually depends on ~Q[/i]. So, above, I used a contamination strategy to yield that dependence.

Neatly answering the question in my post to LemonJello before I'd finished it. So the proof's not so much wrong as missing a few steps, but is wrong in the post to LJ as I more clearly haven't established that ¬Q leads to (¬P&P).

Incidentally if you're putting parentheses around inverted commas adding a full stop between the " and the ) prevents smiley faces - so that ("this".) doesn't come out as ("this"😉.

Back to the OP.

The whole history of Israel in the OT is a type of the New Testament church. This history consists of some matters as typology -

the Passover lamb,
the Exodus,
the journey through the wilderness,
the eating of manna,
the water from the rock,
the building of the tabernacle,
the entering into the promised land,
the defeating of the enemies in the land,
the gaining ground for the establishment of God's kingdom,
the building of the temple,
the filling of the temple with the glory of God

All these matters have symbolic significance for the new covenant church.

While many Christians pay attention to the Passover lamb and the falling manna to be types of Jesus Christ, many do not realize the whole Good Land, the promised land of Canaan is also a type of Christ.

What then do the enemies in the land that needed to be conquered have to do with Christ? This is the Christ dispensed into man. Christ is imparted into the new covenant saints. Therefore, since they are a combination of the old creation indwelt with the victorious Godman Jesus, they must "fight" their way into this new obtained reality.

It is a fight for the enjoyment of all that Christ is as the believers learn to live a new life in Christ who has been born within them.