PaedoSocrates (Kevin) discusses with Lukas Novak the truth of this at the Yahoo! group thomism:
message 1980:
The Notion of Consequencemessage 2093:
jamesmiguez scripsit:
> COMMENTS:
> This is not, in the least, a literally intuitive reconstruction.
> Klima is, by habit and training, using a modern logician's
> procedure, sometimes properly employed in branch chain dialectic. I
> remember quitting a modern symbolic Logic course, when this kind of
> procedure was introduced with the pious ejaculation that, quote:-
> Anything "follows-from" a contradiction.
While it is not my intention to make any comments on Klima's analysis of the Anselm's argument (although I indeed have an opinion), I would like to react to James's criticism of the standard notion of consequence.
"Followin-from" or logical consequence is defined thus: A conclusion follows from the premises iff it is impossible that all the premises be true and the conclusion false.
I hope it is clear to everyone that given this definition, it is indeed true that anything follows from a contradiction, since given that it is impossible that a contradiction be true, it is also impossible that the contradictory premises were true and the conclusion false, therefore, the definition is satisfied in case of any conclusion whatsoever.
Now the only objection against this can be that this notion of logical consequence is somehow wrong, useless or whatever.
Do decide whether it is, we must ask, whether the notion serves its purpose. What is the purpose of the notion of consequence? The relation of consequence is introduced as a "truth preserving" relation: as a relation that will guarantee to you that you can never arrive at a falsity when you start from truth. Since this is what logic is about: it is an instrument by means of which we are able to derive safely new propositions from known ones, and can be sure that IF the latter are true, then the former are as well.
The notion of consequence is just a precised expression of this requirement. Therefore, IF one wishes a universal truth-preserving relation, then the notion of consequence as defined above is THE relation wanted. Refusing to accept the notion of consequence as useful equals to refusing to accepting the notion of preserving truth in inferences as useful. Of course, noone is obliged to favour truth over falsity.
Rather than to object against this very central notion of logic (logic is nothing else than the theory of consequence), James perhaps meant to object agains the notion of material implication in modern logic, which admittedly does not express well the actual meaning of the natural-language phrase if-then (which fact, however, is no objection against using this logical constant; it is just objection against passing it for an adequate and exhaustive analysis of the menaing of "if-then) ?
Lukas
In a message dated 06/06/06 3:49:01 PM Mountain Daylight Time, lukas.novak@... writes:
> PaedoSocrates@... scripsit:
>
> In a message dated 05/06/06 12:35:46 AM Mountain Daylight Time, lukas.novak@... writes:
>
> LUKAS:
> "But in our homeland, where we will see his essence, it will be for us much more self-evident that God is, than it is now for us self-evident that affirmation and negation are not both true."
>
> Lukas
>
>
> KEVIN:
> REPLY:
> Thank-you, Lukas, for the prompt translation. Are you sure that St. Thomas meant "homeland" rather than "But in The Father (Patria) etc.?
LUKAS:
> Yes. "Patria" is Homeland, "Pater" is "Father". What is meant is Heaven, of course.
>
> Lukas
UNANSWERED PART OF PREVIOUS POST:
KEVIN (formerly):
WHOOPS! Dumb question!!!
Probably it is "homeland" for The Father would be "Pater"/father, rather than "patria" (father's homeland?). So that seems straightened out.
What do you think of the phrase, requote, "...than it is now for us self-evident that affirmation and negation are not both TRUE.", given "p" & "not-p" and the alleged principle of LOGIC that "anything-follows-from-a-contradiction".
After all affirmation and negation are just another grammatical way of saying "contradiction" for affirmations contradict negations AND negations contradict affirmations. I'll return to your original post on consequence shortly.
Good to hear from you Lukas and Thank-you for the translation. Also thanks for the "heads-up" on De Veritate. Obviously that treatise must have some insights by Aquinas into Major or material logic, as distinct from your expertise in formal logic.
Kevin
PRESENT COMMENTS:
I attended at our local Catholic University library, with the frustrating result of No "De Veritate" nor "Disputed Questions" by St. Thomas Aquinas. The Summa and Summa Contra Gentiles were there, but not-"ON TRUTH" by Aquinas.
So I picked up a different translation of De Ente & Essentia (very clear; compared to two other translations I have read---still, even with the better translation, Aquinas was still quite young when he wrote that work. In Aristotle's phraseology, he "lisps" a bit. How come nobody translates that title as Of Entities and Essences? That is what the title suggests to me.), of St. Thomas's Commentary on The Book of Causes (He was mature then; no "lisping" there; The Book of Causes seems to be an Arabic regurgitation of some of Proclus's Elements of Theology) and Gilson's Christian Philosophy of St. Thomas Aquinas.
Good books all, but not what I was looking for!!! How could a Catholic University not have ON TRUTH by St. Thomas? But they did have John Stewart Mill's entire works! How bizarre. Rant ended. From "the net" I learned that one can buy the De Veritate of St. Thomas, in 3 volumes, for around 150 dollars. 3 Volumes?!!! Holy Smokes.
As to the questions, above recited, there was NO REPLY. Conclusion:- Lukas likes "to correct" (previously-corrected errors), but not "to answer" questions. So, back to Lukas's notion of consequence:-
LUKAS:
"Followin-from" or logical consequence is defined thus:
[DEFINITION:]
A conclusion follows from the premises iff it is impossible that all the premises be true and the conclusion false.
COMMENT:
The term IFF (2 "f"s; no misspelling) apparently refers to the biconditional-hypothesis meaning "IF and only-IF" all the premises of an argument are true (THEN) it is impossible for the conclusion of a VALID argument to be false.
LUKAS:
I hope it is clear to everyone that given this definition, it is indeed true that anything follows from a contradiction, since given that it is impossible that a contradiction be true, it is also impossible that the contradictory premises were true and the conclusion false, therefore, the definition is satisfied in case of any conclusion whatsoever.
COMMENT and QUESTION:
It is not clear to me, Lukas. What do you mean by a contradiction???
ADDITIONAL COMMENT:
Contradictions are two propositions. They are not "a" thing, but, rather, "two" things, to wit, two propositions, with identical subjects and logically-opposed predicates.
LUKAS:
Now the only objection against this can be that this notion of logical consequence is somehow wrong, useless or whatever.
REPLY:
How about the notion of:- "a contradiction"? That is what I object to. I do not think that logical consequence is somehow wrong, useless or whatever. Your point depends on the consequence of A contradiction or "from" A contradiction
LUKAS:
Do (ie. TO) decide whether it is, we must ask, whether the notion serves its purpose. What is the purpose of the notion of consequence?
REPLY:
Rather, what is the puropse of the notion of "a contradiction", when contradictory propositions are 2 logically-opposed "notions" [affirmation vs. negation of the same predicate attributed to or denied of (not attributed to) a subject or substance].
LUKAS:
The relation of consequence is introduced as a "truth preserving" relation: as a relation that will guarantee to you that you can never arrive at a falsity when you start from truth. Since this is what logic is about: it is an instrument by means of which we are able to derive safely new propositions from known ones, and can be sure that IF the latter are true, then the former are as well.
QUESTION & COMMENT:
But if you start from a true proposition and its contrardiction which is a false proposition, how do you preserve truth, as a consequence of 1 true proposition and 1 false proposition?
True and false propositions are contrary, according to Aquinas, and contradictory when single subjects and logically opposed predicates are evident. No mention of contradiction or contradictory propositions or "notions" here.
LUKAS:
The notion of consequence is just a precised expression of this requirement. Therefore, IF one wishes a universal truth-preserving relation, then the notion of consequence as defined above is THE relation wanted.
REPLY:
Prescinding from the obvious fact that you mention nothing of contradictions in your argument concerning consequence, let us examine the alleged precision of your definitions. What's so precise about a biconditional hypothesis, called, by you, a definition? Aren't biconditionals close to circular arguments, while also being close to convertibly predicable attributes---otherwise known as properties?
COMMENTS:
1. Some people know that false biconditional antecedent-consequent/consequent-antecedent propositions yield a true biconditional hypothesis, as also does a true antecedent proposition followed by a true consequent proposition and vice versa.
2. But, where 1 proposition of a biconditional hypotheses is true and the other is false the biconditional hypothesis, itself, is altogether false. It doesn't matter where the false and true biconditional propositions are placed in sequence, as long as the pairs of propositions disagree in truth value. (= 2 falsehoods in both directions)
3. Hmmm...Seems to be more falsity than truth in/with biconditionals.
LUKAS:
Refusing to accept the notion of consequence as useful equals to refusing to accepting the notion of preserving truth in inferences as useful.
REPLY:
I don't refuse to accept the notion of consequence as useful. I just refuse to believe that "Anything logically follows from a contradiction.", for if that notion is TRUE, then you probably have to accept that arguments which contain contradictory sets of propositions are VALID. YES or NO, Lukas?
LUKAS:
Of course, noone is obliged to favour truth over falsity.
REPLY:
Tell that to either Christ, at your final judgment, or the Judge at your local courthouse. eg. "I swear to tell the truth, the whole truth, and nothing but the truth, sir, but I am not obliged to tell the truth!" The judge may reply that you are obliged to tell the truth. The law "says so", but no one is obliged to obey the law, although the judge is obliged to jail people for perjury, otherwise known as giving, making or asserting contradictory propositions under oath.
Jail "follows-from" legally-proved contradictory-propositions, at court, unless you can find a "modern logician" seated upon "the Bench" of your local courthouse. And they're not too hard to find, since all modern lawyers were "educated" by modern philosophers and modern logicians, while modern lawyers are appointed to various "Benches", by, guess who---modern lawyers, who are so honest, just like their clients are so honest---and, "therefore"...(the farmer hauled another load away).
LUKAS:
Rather than to object against this very central notion of logic (logic is nothing else than the theory of consequence), James (ie. Kevin) perhaps meant to object agains the notion of material implication in modern logic, which admittedly does not express well the actual meaning of the natural-language phrase if-then (which fact, however, is no objection against using this logical constant; it is just objection against passing it for an adequate and exhaustive analysis of the menaing of "if-then) ?
Lukas
REPLY:
No, I didn't mean to do that. But you probably did. The modern notion of "p" as antecdent in a TEST situation and "not-P" as the non-antecedent in an existential CONTROL situation, is a wonderful way of proving EFFICIENT and MATERIAL causation in modern scientific experiments, using existential hypotheses as MAJOR premises. But "p and not-p" as a single ASSUMPTION is non-sense.
P and not-P are contradictory so-called "assumptions" (pleural)---NOT "an" assumption and NOT "a" contradiction, but rather 2 contradictory propositions.
I do think that the only thing that "logically follows from a contradiction"---in TRUTH meaning TWO contradictory propositions---is that someone, who contradicts him/her-self is a a liar OR plain ignorant, or, alternatively, anyone who contradicts "being", whether or not we think about, or speak about, "being" is also a liar OR, once again, both ignorant and forgetful about his or her previous assertions.
It is TRUE that modern symbolic logicians do object against the, requote "material implication (ie. TRUTH) in modern logic, which admittedly does not express well the actual meaning of the natural-language phrase if-then." (NOVAK).
But I am not a modern symbolic logician and am perfectly happy with affirming the antecedents of hypotheses OR denying the consequents of hypotheses to reach VALID conclusions about either the antecedents or the consequents of hypotheses.
But I would never call an hypothesis either true or false, being convinced by Aristotle's Metaphysics and his Posterior Analytics that mere hypotheses are NOT PRINCIPLES. But I can prove that at least one modern symbolic logician mentions the dissatisfaction of either people or symbolic logicians with calling hypotheses "truth functional", quote:
BONEVAC:
"People use modal logic to evaluate and justify reasoning about possibility and necessity. Aristotle and medieval logicians tended to think of possibility, actuality and necessity as modes of truth, that is, as ways in which sentences could be true or false. The study of the modes of truth became known as modal logic.
"Contemporary interest in modal logic stems partly from unhappiness with treating conditional sentences in terms of truth functions. [Very telling. Modern logicians are unhappy with truth functions!!!] Within the framework of truth- functional sentential logic, we can symbolize sentences of the form "IF A, THEN B.", only as formulas having the structure "A ----> B". As our definition of the conditional truth function indicates, these formulas are TRUE, whenever A is FALSE or B is TRUE."
COMMENT:
What Bonevac probably means is what I said above:- Denying consequents WARRANTS denying antecedents in conjunction with a HYPOTHETICAL MAJOR premise [Thus "whenever A is false"; But experimental scientists don't think that way---ie. in terms of A being FALSE---but rather in terms of A being non-existent as an EFFICIENT CAUSE, which necessarily precedes its EFFECT in both necessity and in time In short, NO EFFECT (not-B), then NO CAUSE (not-A)].
As to Bonevac's other expression "OR whenever B is TRUE", he arguably means that given an hypothesis of the form IF A, THEN B; a.k.a., A ----> B; [which is, once again, actually two propositions; (1) antecedent and (2) consequent propositions (pleural)], which he calls, once again, A SENTENCE (but any scientist knows that an hypothesis is 2 sentences referring to at least 2 existential OBJECTS called CAUSES and EFFECTS, in general), B is TRUE, whenever A causes B. In other words, when there is an actually CAUSAL RELATION between A and B, when A exists then B existentially follows from A. (ie. "Whenever B is TRUE" the hypothesis "If A, then B" is a "true" hypothesis; although no scientist considers hypotheses to be either true or false---just antecedents and consequents to be experimentally tested).
Of course, being a modern symbolic logician, perhaps Bonevac doesn't want to take such arguable CAUSAL relations seriously!
WATCH Bonevac try to make such CAUSAL RELATIONS appear "puzzling"!
BONEVAC (continues):-
But taking a truth-functional rendering of IF seriously, gives rise to a variety of puzzles. [COMMENT:- How "unsurprizing" when Bonevac fails to mention taking THEN to be equally as "serious" (or even as "frivolous") as IF. But he "explains"; or at least attempts to "explain" such puzzles, next paragraph. KB].
BONEVAC:
First a truth functional analysis leads to the "paradoxes of material implication." [COMMENT:- His "scare-quotes" indicate that he may not take such "paradoxes" very seriously. But he doesn't seem to take THEN into consideration at all! He is also regurgitating Lukas or Lukas him because they have the same teachers.]
BONEVAC (continues):
These "paradoxes" though not really contradictions in a logical sense, show that our definition of the conditional can lead us to count some bizarre arguments as valid. Both these argument forms are VALID in classical sentential logic.
(1) a.
p
Therefore q -----> p
(1)b.
not-p
Therefore p ----> q
COMMENT:
Bonevac has a strange take on "classical sentential logic", which has nothing to do with hypotheses which relate to classical modal logic. But even with actual respect to classical modal logic, this "stuff" above is almost BACKWARDS compared with and contrasted to classical modal logic, which looks like this:
(1) a. (revisited)
q -----> p (IF q, THEN p.)
p (Affirms consequent)
"THEREFORE?" Nothing! [Fallacy:- Affirming the consequent]
eg. If quadruplets, then parents.
parents
Therefore quadruplets?
(Hardly! Singles, twins, triplets or "quints" equally imply parents
even when there are no "quadruplets")
BUT ON THE OTHER HAND:-
q -----> p (IF q, THEN p.)
q (Affirms the antecedent)
Therefore p (VALID Affirms consequent.)
eg. If quadruplets, then parents.
quadruplets
Therefore, parents
[VALID and SOUND]
As to Bonevac's "b." scenario, we have, according to classical modal logic
p ----> q
[eg. If penicillan, then quashed infection of non-penicillan-resistant "bug".]
not-p
[eg. no available penicillan]
"THEREFORE?" [Nothing follows; Invalid denial of antecedent.]
(A bug could also be quashed by another antibiotic
or by an infected person's own immune system---or NOT!)
BUT ON THE OTHER HAND:-
p ----> q
[eg. If penicillan, then quashed infection of non-penicillan-resistant "bug".]
not-q
[Nothing quashed a penicillan sensitive "bug"]
THEREFORE, not-p
[No one administered penicillan.
Either No antibiotic at all or they administered a wrong antibiotic to which
the "bug" was resistant. VALID and SOUND]
We continue with Bonevac's "puzzles" for his readership, quote, given his two examples of "truth functional" hypotheses, quote
BONEVAC:
However, arguments according to them sound strange [I hope my arguments didn't sound so "strange" as those Bonevac proceeds to use as "examples". KB]
BONEVAC:
(2) a.
The Colorado river is good for white-water canoeing (p)
Therefore IF a nuclear bomb just exploded over the rockies (q)
THEN the colorado river is good for white-water canoeing (p)
[COMMENT:- This is a serious hypothesis/argument? OH PLEASE!!!]
ANALOGY:-
IF a nuclear bomb just exploded over the rockies THEN the Colorodo
river may be 30 feet higher from excessive snow melt OR
non-existent if rock-slides DAM the river OR divert the river
in which POSSIBLE cases there will be NO "white water" at all.
SILLY example. UNSERIOUS example of a VALID argument.
It is still a VALID argument, but in this way:
Hypo:- IF a nuclear bomb just exploded over the rockies
THEN the Colorodo River is good for white-water-canoeing
for the next 15 minutes or awhile longer
Minor: A nuke just exploded (affirms antecedent)
Conclusion- Therefore the river is good for white-water-canoeing
for the next 15 minutes or awhile longer.
[VALID and SOUND; But POSSIBLY-not within a few hours!
In truth, it is just a hypothesis.).
Bonevac's second example:
b. NOT many Americans eat Thai food. (not-p)
Therefore IF many Americans eat Thai food (p),
THEN sales of antacids will soar (q)
NON-REBUTTAL:
IF many Americans eat spicey Thai food (p), THEN sales of antacids will soar (q).
Sales of antacids will not soar. (not-q)
Therefore NOT many Americans will eat spicey Thai food (not-p)
[VALID and SOUND; although only POSSIBLE, not necessary.]
Who knows the future or the effects of Thai food on American tummies? I don't. And do Thai people consume many antacids because of their spicey diets? Probably not. But Americans do---probably CAUSED by listening to silly arguments OR by eating TOO MUCH FOOD, at all hours, whether or not it's "spicey-food".
BONEVAC (contiunes):
Neither argument seems VALID (Bonevac's conclusion).
REBUTTAL::
But they are valid arguments---just not very serious arguments. In TRUTH, Bonevac only seriously looks at EITHER (2. a.) the consequent of his first examplary hypotheses, ALONE (Colorado "whte water canoeing"; but NOT at any possible effects of "nuking-the-Rocky-mountains"), OR at (2. b.) the antecedent of his second examplary hypothesis ALONE (The eating non-habits of Americans.).
And no American needs a symbolic logic course to know about "white-water-rafting" on the Colorodo river or about his own countrymen's eating habits. But, at least we know what Bonevac takes seriously from his examples of "seemingly INVALID" hypothetical arguments---(1) American recreational pastimes (when he is not further corrupting the minds of his students) and (2) American eating habits (between classes, involving the corruption of the minds of his S-L students with frivolous examples of "seemingly INVALID", but actually VALID, though silly, hypotheses).
Bonevac provides his students with further examples of silly HYPOTHETICAL arguments which are actually VALID according to both classical modal logic and what he calls "classical sentential logic", which has nothing to do with hypotheses and everything to do with CONTRARY and CONTRADICTORY propositions.
But then Bonevac says something interesting with respect to Novak's definition of consequence and Novak's "following" argument for "anything follows from a contradiction" being a, requote, very central notion of logic, although what Novak really means is that CONSEQUENCE is a central notion of logic, which has nothing to do with justifying "Anything follows from a contradiction." At least Lukas hasn't rationally explained what "anything" follows from a contradiction, nor has he explained WHY "anything logically-follows from a contradiction" (meaning two propositions which contradict each other.)
Consequence is a very central notion of logic. But that "Anything logically follows from a contradiction.", usually expressed as "Anything follows from a contradiction." is what I have characterized as an unproveable "pious ejaculation" which has much more to do with TRUTH and FALSITY than consequence or validity.
But here is what I find interesting by Bonevac:
BONEVAC:
The English connectives IF and only IF aren't truth functional, therefore, although the truth-functional CONDITIONAL (ie. an hypothesis) approximates them closely within a wide range of cases.
COMMENT:
This fella even writes sentences in the same style as his "p", therefore "If q, then p" hypothetical examples. What he seems to be saying above is:-
Biconditional hypotheses are not truth functional. But, truth-functional hypotheses approximate (are like?) biconditionals.
OR (perhaps)
Maybe he meant that biconditionals are like truth functional hypotheses in a wide range of cases, even though they aren't truth functional.
How "therefore" fits into the above quoted sentence of Bonevac is entirely "beyond" me. It sounds like he means:- Although biconditional hypotheses are not truth functional, biconditional hypotheses are like truth-functional conditional hypotheses (approximate them) in a wide range of cases.
What I say in response to that "revelation" and to parody that revelation, is that trure propositions are an awful lot like false propositions in a wide range of cases too. Take for example the simple proposition that, quote:-
Lukas Novak is inhaling some Czechoslovakian air.
Lukas is inhaling, therefore the proposition is TRUE! Whoops, now he is exhaling some Czechoslovakian air, therefore the proposition above cited is FALSE. Whoops, now it is TRUE (Lukas inhales.), then FALSE (Lukas exhales), then TRUE (Lukas is inhaling Czech air.), then FALSE (Lukas is exhaling; not-inhaling; Czech air.), etc., etc. And this is not any Bonovakian "case" of a biconditional non-truth-functional double-hypothesis approximating any truth-functional single hypothesis! No sirree, Bob.
It is, instead, a "case" of an IDENTICAL-proposition alternating between TRUE and FALSE about every 10-30 seconds, depending upon Mr. Novak's living metabolism, his physical condition and his present air requirements which depends upon what he is actually doing at present. eg. Running a marathon OR sleeping soundfully and peacefully in his bed.
SO HYPOTHETICAL QUESTION:
IF Daniel Bonevac is an actually competent logician and biconditionals are NOT "truth functional" THEN why does Lukas Novak employ non-truth-functional BICONDITIONALS for his so-called definitions???
Biconditionals are TRUE just so long as their antecedent and consequent propositions AGREE in truth value. In other words a FALSE antecedent and a FALSE consequent AND "vice versa" (switch antecedent and consequent around) make for a TRUE biconditional hypothesis---but not for very good definitions, which, when good, are always TRUE and convertibly-predicable.
But, of course, biconditionals composed of EQUALLY-FALSE propositions are TRUE biconditionals because FALSE propositions have been contradicting TRUE propositions since the garden of Eden AND if you have a FALSE antecedent you're going to get a FALSE consequent everytime. That's the TRUTH.
But St. Thomas Aquinas and Aristotle before him, take care of Bonevac's frivoulous examples with the simple assertion that, quote
AQUINAS:
"In the same way, the being of the thing, NOT its truth, is the CAUSE of truth in our intellect. Hence the philosopher says that an opinion or statement is TRUE FROM (consequent upon) the fact that a thing IS, not from the fact that a thing is true (Aristotle; The Categories; Ch. 5., 4b line 8).
[Summa I; Q. 15. Article 16. Reply Obj. 3.]
That is why the same proposition (forget biconditional "definitions", involving 2 propositions in a reciprocating antecedent-consequent || consequent-antecedent relationship) may be TRUE (Lukas is inhaling) for only a short time, then FALSE (Lukas is inhaling? No he is not inhaling because he is exhaling!) within a few seconds of time. Thus Aristotle's definition of the basic axiom of thought (Law of Contradiction) is that:- The same attribute (inhaling) cannot at the same time belong (to Lukas) and not belong (to Lukas) to the same subject (Lukas) in the same respect (in respect of the breathing of Lukas Novak).
And that definition of the Law of Thought involves no hypothesis at all, because it is the basic AXIOM of all thought.
WHAT, therefore, logically-follows from a contradiction?
Lukas Novak is inhaling (TRUE)
Lukas Novak is not inhaling (FALSE)
Therefore...
REPEAT:
What logically follows from the above contradiction?
(I won't hold my breath waiting for any answers.)
Kevin
2095
Message 2100(response to 2098):
Re. The Notion of Consequence
In a message dated 21/06/06 11:47:18 AM Mountain Daylight Time, lukas.novak@... writes:
> OK: so if you favour truth over falsity, you should be interested
> in the truth-preserving relation of consequence as defined.
>
> I apologise, I must end here, no time.
>
> Lukas
O.K. I do like your definition of consequence:- TRUE premises NEVER yield FALSE conclusions is fine. But that definition (however poorly I have rephrased it) does not explain, nor justify (IMMLTHO) the proposition:-
Anything follows from (consequence) A CONTRADICTION. (True? False?)
But Thank-you for your time, Lukas. To review:-
POINTS MADE & COMMENTS (since I have time):
Kevin's QUESTION & COMMENT (previously):
But if you start from a true proposition and its contradiction which is a false proposition, how do you preserve truth, as a consequence of 1 true proposition and 1 false proposition?
LUKAS:
In this case, you don't have anything to preserve. But it nevertheless holds that IF the premises were true (which is impossible) THEN the consequence would have to be true as well.
REPLY:
O.K. When premises are true, so are the conclusions of valid arguments. But Consequence (validity) and truth (soundness) are different "ideas". And, of course, one does "have something to preserve", in cases of true vs. false contradictory propositions, to wit, the truth of the true proposition which contradicts the falsehood of the false proposition! The true proposition is worth preserving and that is what I am interested in, while logicians are mostly interested in consequence (validity).
Novak's hypothesis about consequence is fine, but he still doesn't tell me how anything follows from a contradiction. However, when Lukas actually said, requote
LUKAS:
"In this case, you don't have anything to preserve.",
(Q.) wasn't he really saying that NOTHING follows from that kind of contradiction (ie. 1 true proposition vs. 1 false proposition)???
After all Lukas defined/described a contradiction as a proposition which is necessarily false, per, requote:
COMMENT and QUESTION (previous):
It is not clear (that Anything follows from a contradiction; True? or False?) to me, Lukas. What do you mean by a contradiction???
LUKAS:
A proposition that is necessarily false.
HMMM!!! So much for other "pious ejaculations" related to truth preserving, since a contradiction is a proposition that is necessarily false. [What necessitates the falseness of a proposition? Non-being.]. However I don't believe that propositions are necessarily true or necessarily false or necessarily contradictory, unless there are two propositions which, in truth, are contradictory-propositionS (pleural).
PROPOSITION (Oxford Concise):
n. (noun) Statement, assertion, as ~ too plain to need argument, especially (logic) form of words consisting of predicate & subject; (mathematics, abbreviated prop) formal statement of theorem or problem, often including the demonstration, as Euclid Bk. I ~ 5; proposal, scheme proposed; (slang) task, job, problem, objective, occupation, trade, opponent, prospect, etc. Hence ~al adjective [Medieval English, from Old French or Latin propositio (as following, see -ION)].
ARISTOTLE (on propositions):
Every sentence has meaning, not as being the natural means by which a faculty is realized, but, as we have said, by convention [The limitation "by convention" was introduced because nothing is, by nature, a noun or name---it is only so when it becomes a symbol ; inarticulate sounds, such as those brutes produce, are significant, yet none of these (inarticulate sounds of non-human animals) constitutes a noun.]. Yet every sentence is not a proposition; only such are propositions which have truth or falsity in them. Thus a prayer is a sentence, but is neither true nor false. (Aristotle's "Scheffer stroke"?)
Let us, therefore, dismiss all other types of sentences but the proposition, for this last (type of sentence) concerns our present inquiry, whereas the investigation of the others belongs rather to the study of rhetoric or of poetry.
Ch. 5.
The first class of simple propositions is the simple affirmation, the next, the simple denial; all others are only one by conjunction.--(snip)--We call those propositions single which indicate a single fact, or the conjunction of the parts of which results in unity; those propositions, on the other hand, are separate and many in number, which indicate many facts, or whose parts have no conjunction...
...To return:- of propositions one kind is simple, ie. that which asserts or denies something of something, the other composite, ie. that which is compounded of simple propositions. A simple proposition is a statement, with meaning, as to the presence of something in a subject OR its absence, in the present, past or future, according to the divisions of time.
CH. 6.
An affirmation is a positive assertion of something about something, a denial a negative assertion.
Now it is possible BOTH to affirm and to deny the presence of something which is present or of something which is not (present), and since these same affirmations and denials are possible with reference to those times which lie outside the present, it would be possible to contradict any affirmation or denial.
Thus it is plain that Every affirmation has an opposite denial and, similarly, Every denial an opposite afffirmation.
We will call such a PAIR of PROPOSITIONS a pair of contradictories.
Those positive and negative propositions are said to be contradictory which have the same subject and predicate. The identity of subject and predicate must not be 'equivocal'. Indeed there are definitive qualifications besides this, which we make to meet the casuistries of sophists.
[Aristotle; On Interpretation; Ch. 4. through Ch. 6.; 17a lines 1 to 37]
COMMENT:
Aristotle thinks that contradictories are pairs of propositions. In contrast, Lukas affirms that a contradiction is a necessarily-false-proposition. Both men are logicians. Aristotle is ancient. Novak is modern. Whom should we believe? Let us examine Novak, together, to see if what he says is true, because (formerly)...
ADDITIONAL COMMENT (previously):
Contradictions are two propositions. They are not "a" thing, but, rather, "two" things, to wit, two propositions, with identical subjects and logically-opposed predicates.
NOVAK:
This is just one kind of contradiction.
REPLY:
Thank-you. I happen to think that contradictory propositions (pleural) are important "kinds" of contradictions (pleural), and the second important "kind" of logically opposed propositions (pleural) are pairs of contrary propositions.
We'll see what other kinds there are...
NOVAK:
Furthermore, in order that it be a contradiction, it must indeed be a _one_ composite proposition.
REPLY:
WHY? A composite proposition may be two or more simple PROPOSITIONS (or 3, 4, 5, 6 etc.), according to Aristotle. Lukas continues to refer to "it" and "a contradiction", but his words do not make 2, or more, propositions (pleural) into "one" proposition (single/simple) at all. See Aristotle above.
LUKAS:
Either (re. COMPOSITE-PROPOSTION KB) (1) one where of subject a conjunction of contradictory predicates (note "pleural" predicates KB) is said (S a (P&~P)),---[ie. 2 propositions, 1. S a P, &, 2. S a ~P; Still 2 propositions KB] or a sentential conjunction of two kategorial propositions ((SaP)&(Sa~P)), [Still 2 propositions KB] or any other proposition [Back to 1 proposition (singular) KB] that cannot be true.
COMMENT:- "...that cannot be true."!!! So much for truth preserving. And Lukas clearly indicates 2 propositions with both of his "composite-proposition", no matter how hard he tries to write of a single proposition/contradiction or an "it".
(1) He writes of contradictory predicates (pleural) in his first "kind" of contradiction (Single predicates? No! Two predicates = 2 propositions.) and of (2) two kategorial (categorical) propositions (pleural) in his 2nd "other-kind" of contradiction (One categorical proposition? No! Two categorical propostions.).
Kevin (previously):
Your point (ie. Anything follows from a contradiction.) depends on the consequence of A contradiction or "from" A contradiction
LUKAS:
See above (for) the definition of a contradiction.
REPLY:
I have Aristotle's definition of contradictory propositions (pleural). Why is Novak's definition better (or even different upon close scrutiny), for definitions always relate to questions of sameness and difference, according to Aristotle. To show a difference is enough to overthrow one "definition" with a better definition. Lukas's definition of a contradiction is a proposition that is necessarily false.
Aristotle's definition of a contradiction, by contrast, is two propositions with the same subject and the same predicate, where the identical predicate is affirmed of an identical subject in one case and the identical predicate is denied of the identical subject in the opposite or contradictory case.
Aristotle says nothing of truth or falsehood in defining contradictory propositions, although he clearly says that propositions are the only kinds of sentences with either truth or falsity in them. As to contradictory-propositions, as contradictories (whether true or false) he simply says that every affirmative proposition has a contradictory negative proposition as its logical opposite and that every negative proposition has a contradictory affirmative proposition as its logical opposite.
That is a difference between Novak and Aristotle, although, upon close examination, Lukas seems to be indicating 2 propositions, despite his insistence on 1 contradictory proposition. So Aristotle and Novak arppear the same, as logicians, upon close examination. But they are definitely different, for nowhere does Aristotle ever say that a contradiction is necessarily-FALSE.
Aristotle clearly asserts, along with St. Thomas, that true propositions have necessarily-false propositions as their contradictions and, conversely, that false propositions have necessarily-true propositions as their contradictions. That seems to be a big difference between Novak and Aristotle, whereas their notions of consequence seem to be the same, when Novak's allegedly single propositions are closely examined, thereby indicating 2 propositions.
But to show sameness is not enough to establish a definition, for a standing-philosopher, like Socrates (when standing), is the same as a standing-stooge, like Meletus (when standing), as standers. But a stooge is not a philosopher no matter the same arrangement of each man's limbs.
Kevin (formerly): I don't refuse to accept the notion of consequence as useful. I just refuse to believe that "Anything logically follows from a contradiction.", for if that notion is TRUE,
LUKAS:
This is not a notion but a statement.
ARGUMENT:
Lukas interrupts or contradicts my antecedent as a "notion". So, "Anything logically follows from a contradiction. is not a notion, but a statement. (O.K. for the sake of argument.) It is a true statement, according to Lukas.
BUT:
NOTION (Oxford Concise):
noun 1. General concept under which particular thing may be classed (in phil. first, second ~ = first, second INTENTION) 2. Idea, conception, (The notion of my doing it is absurd ; What he means I have not the haziest notion); view, opinion, theory, vaguely held or insecurely based (has a notion that ; such is the common notion). 3. Faculty, capability, or intention of (has no notion of obeying, obedience, discipline, letting himself be made a fool of) 4. Something in the way of miscellaneous wares, esp. cheap useful ingenious article. pl. || (not American) Traditional special vocabulary of Winchester College.
[From Latin notio (NOTICE, -ION)]
NOTIONAL (Oxford Concise):
adjective (Of knowledge etc.) speculative, not based on experiment or demonstration, whence notionalist(2) noun; notionally 2. adverb; (of things, relations, etc.) existing only in thought, imaginary; (of persons) fanciful.
[from medieval Latin notionalis (NOTICE, -AL)]
THUS:
Notion and Notional don't sound like very strong terms above, given phrases like "vaguely held or insecurely based" theories or opinions and "not based on experiment or demonstration" (adjective notional), especially when LOGIC is demonstrative, given true and primary premises.
So to rephrase my hypothesis with Lukas's correction:
I just refuse to believe that "Anything logically follows from a contradiction.", for if that statement is TRUE, then you probably have to accept that arguments which contain contradictory sets of propositions are VALID.
LUKAS:
Yes. A deduction of the kind "If 2+2=5, then I am the Pope" is valid.
CRITICISM:
"IF 2+2=5, THEN Lukas is the Pope" is neither a deduction, nor valid, nor a contradiction, so far as I can see. It is, I think, a hypothesis and a sequence, otherwise known as a conditional proposition, where one proposition depends upon another proposition in a sequence. But is it a consequence or a contradiction?
And, since when, is a hypothesis either a deduction or valid/invalid reasoning?
eg. HYPOTHESIS (Oxford Concise):
Supposition made as a basis for reasoning, without assumption of its truth or as a starting point for investigation; groundless assumption. So hypothetical(al) a.a. hypothetically adv. [From Late Latin, from Greek hupotheke]
QUESTIONS:
1. Can "groundless assumptions" or "suppositions as bases for reasoning" be either deductions or valid (deductions)? [I doubt it.]
2. Can a mathematical sum, whether correctly or incorrectly calculated, contradict or confirm whether Lukas Novak is, or is not, the pope? [I don't see how.]
3. IF "Anything follows from a contradiction" is a FALSE statement THEN what follows or does not follow (since it is only a statement and not a notion)??? What is the consequence if it is a TRUE statement (since it is only a statement and not a notion)? So, let us examine together certain falsehoods and consequences, followed by certain truths and consequences, according to Lukas's hypothesis which he describes as a valid deduction.
CLASSICAL MODAL LOGIC
1.
Hypothetical Major: IF 2+2=5, THEN Lukas Novak is the Pope. (Hypothesis)
Categorical Minor: 2+2 is not equal to 5 [denying antecedent] (True; Categorical)
Conclusion: Therefore Lukas Novak is not the Pope (True; Categorical)
[INVALID!!! Fallacy; denial of antecedent]
---on the other hand---
2.
Hypothetical Major: IF 2+2=5, THEN Lukas Novak is the Pope. (Hypothesis)
Categorical Minor: Novak is not the Pope [denying consequent] (True; Categorical)
Conclusion: Therefore 2+2 is not equal to 5 (True; Categorical)
[VALID!!! No fallacy; denial of consequent]
---on the third hand---
3.
Hypothetical Major: IF 2+2=5, THEN Lukas Novak is the Pope. (Hypothesis)
Categorical Minor: 2+2 =5 [affirms antecedent] (False; Categorical)
Conclusion: Lukas Novak is the Pope (False; Categorical)
[VALID!!! No fallacy; affirmed antecedent]
---on the fourth hand---
4.
Hypothetical Major: IF 2+2=5, THEN Lukas Novak is the Pope. (Hypothesis)
Categorical Minor: Novak is the Pope [affirms consequent] (False; Categorical)
Conclusion: Therefore 2+2 is equal to 5 (False; Categorical)
[INVALID!!! Fallacy; affirms consequent]
ARGUMENT:
Supposing that everyone knows that 2+2=5 is false and 2+2=not-5 (ie. 4) is true AND also that Lukas is the Pope is false and Lukas is not the Pope is true, what about the truth preserving function of consequence from the 4 arguments above
1. The invalid/fallacious "consequence" results in a true conclusion.
2. The valid consequence establishes a true conclusion from false premises.
3. The valid consequence preserves a false premise and a false conclusion.
4. The invalid/fallacious "consequence" preserves a false premise and conclusion.
According to Lukas Novak's definition of a contradiction being necessarily-FALSE, all 4 arguments involved necessarily-false propositions = contradictions. But the valid consequences preserved both true and false conclusions as did the invalid bogus "non-consequences". So what does consequence have to do with preserving truth? What does contradiction have to do with preserving truth, on Lukas's view of contradiction involving necessarily false propositions? Contradiction cannot preserve truth on Lukas's view of necessarily-false propositions. Let's see...
Validity (consequence) and Soundness (truth) seem to be different things. And contradiction seems to be a different thing from validity and soundness. So what is really "truth preserving", if anything? Is it really consequence that preserves truth?
Once again, let us examine an assertion of Lukas to see if it is true:
According to Lukas, requote (original post):
NOVAK:- "...we must ask, whether the notion serves its purpose. What is the purpose of the notion of consequence? The relation of consequence is introduced as a "truth preserving" relation: as a relation that will guarantee to you that you can never arrive at a falsity when you start from truth.
ANALYSIS:
The above indicates that you can never arrive at falsity when starting from the truth, since the relation of consequence is truth preserving when employed correctly. So, since I think I employed the notion of consequence corrrectly in two of Lukas's hypothetical examples and incorrectly for two of the same examples, let us reexamine those two examples where the notion of consequence was employed correctly, with respect to the alleged truth preserving function of consequence.
But don't forget. These exampls all begin with doubly-false antecedent-consequent propositions (pleural) in one hypothesis.
2.
Hypothetical Major: IF 2+2=5, THEN Lukas Novak is the Pope. (Hypothesis)
Categorical Minor: Novak is not the Pope [denying consequent] (True; Categorical)
Conclusion: Therefore 2+2 is not equal to 5 (True; Categorical)
[VALID!!! No fallacy; denial of consequent]
Here we begin with a false antecedent and a false consequent in an initial hypothesis. By validly contradicting a false consequent (in Aristotle's sense of contradictions involving affirmative and negative propositions), with the minor premise, we arrive at a true conclusion with respect to the antecedent. Thus we validly deny a false antecedent proposition. We have validly moved from falsity to truth as the CONSEQUENCE of contradicting a falsehood.
Second VALID example:
3.
Hypothetical Major: IF 2+2=5, THEN Lukas Novak is the Pope. (Hypothesis)
Categorical Minor: 2+2 =5 [affirms antecedent] (False; Categorical)
Conclusion: Lukas Novak is the Pope (False; Categorical)
[VALID!!! No fallacy; affirmed antecedent]
In this example, where no Aristotelian style of contradiction was employed, we validly affirmed a false antecedent and, hence, validly affirmed a false consequent, employing a VALID sequence and a logically warranted consequence. In this case, without an Aristotlelian style of contradiction, we preserved falsity with a warranted CONSEQUENCE---an allegedly "truth-preserving" function. But, to the contrary, we preserved a known falsehood as a VALID consequence.
In sum, a valid CONSEQUENCE seems to preserve falsehoods (eg. 3), when simple falsehoods are not contradicted. By contrast, the valid CONSEQUENCE is to refute falsehoods (eg. 2) when simple falsehoods are contradicted, according to Aristotle's description of contradictory propositions.
What about the invalid non-CONSEQUENCES mentioned?
The first invalid non-CONSEQUENCE (non-sequitur) was, requote:-
1.
Hypothetical Major: IF 2+2=5, THEN Lukas Novak is the Pope. (Hypothesis)
Categorical Minor: 2+2 is not equal to 5 [denying antecedent] (True; Categorical)
Conclusion: Therefore Lukas Novak is not the Pope (True; Categorical)
[INVALID!!! Fallacy; denial of antecedent]
By invalidly contradicting a known-false antecedent, we bogusly arrive at a true conclusion, employing an Aristotelian-style contradiction of a false statement, as a known non-Sequitur (illogical non-consequence).
And the final example:-
4.
Hypothetical Major: IF 2+2=5, THEN Lukas Novak is the Pope. (Hypothesis)
Categorical Minor: Novak is the Pope [affirms consequent] (False; Categorical)
Conclusion: Therefore 2+2 is equal to 5 (False; Categorical)
[INVALID!!! Fallacy; affirms consequent]
Here we invalidly affirm a known false statement and do not contradict it, according to Aristotle's definition of contradictions and invalidly conclude a known false statement as a bogus non-CONSEQUENCE.
So, in sum, for these 2 fallacious non-examples of "CONSEQUENCE", the contradiction of a known false statement (eg. 1) invalidly (inconsequentially according to logical doctrine) "results" in a known-true conclusion. But, invalidly and fallaciously affirming a false proposition (eg. 4) results in a known-false conclusion, when we do not fallaciously, or otherwise (validly) contradict known falsehoods.
Preliminary Conclusion
It seems clear that contradicting known false statements is the key to "preserving-truth", no matter whether or not we employ the logical notion of CONSEQUENCE correctly or incorrectly. But recall, we began with BOTH a known-false antecedent and a known-false consequent set of propositions in our original hypothesis and Lukas tells us that the NOTION OF CONSEQUENCE is a "truth preserving" function.
So, we should also examine a similar hypothesis, where we, according to Aristotle's thesis concerning contradictions, contradict both known-falsehoods, so that we begin with truth and attempt to preserve the truth with only the notion of correct consequence, employing Lukas's basic example. However, we'll have to change the name and the arithmetical sum to begin with truth and to attempt to preserve the truth with correct logical (albeit hypothetical) consequences.
IF 2+2=4, THEN Benedict/Ratzinger is the pope (Hypothesis)
2+2=4 (Minr; Affirms antedent; TRUE)
Hence, Benedict/Ratzinger is the pope. (TRUE)
[Conclusion:- Valid-consequence preserves truth affirmatively]
-On The Other Hand-
IF 2+2=4, THEN Benedict/Ratzinger is the pope (Hypothesis)
Benedict/Ratzinger is NOT the pope.
(Minor; Validly denies consequent; with FALSE contradiction)
Hence, 2+2 = not-4
[Conclusion:- Valid-consequence destroys truth negatively]
NOVAK (requote): The relation of consequence is introduced as a "truth preserving" relation: as a relation that will guarantee to you that you can never arrive at a falsity when you start from truth.
Above we arrived at a falsity, by starting with the truth, employing the notion of CONSEQUENCE validly, but by contradictiong the truth with a false premise, which had to be necessarily false given Lukas's definition of a contradiction. The notion of CONSEQUENCE logically validated arriving at a falsity when starting from truth.
It shouldn't be necessary to go over the INVALID-non-CONSEQUENCE examples of invalidly affirming the consequent (Benedict is Pope) to bogusly arrive at the truth with a non-sequitur (2+2=4) OR invalidly contradict a true antecedent (2+2=4) with a false denial (2+2=not-4) to fallaciously arrive at a false statement (Benedict is not the pope.) as a bogus conclusion and inconsequentially-FALSE result.
In all cases, it is the CONTRADICTION of true propositions by false propositions which, primarily, causes falsehoods and the CONTRADICTION of false propositions by true propositions which, primarily, "preserves truth", in accordance with Aristotle's notion of contradictory propositions (pleural; affirmation vs. negation).
Lukas's Notion of Consequence, on the other hand, preserves falsity, just as well as it preserves TRUTH, given any false and uncontradicted proposition in a properly sequenced argument.
So the CONSEQUENCE of an uncontradicted false proposition in any logical argument is the "preservation of falsity", but the logical CONSEQUENCE of a contradicted false proposition in any logical argument "refutes falsity" and CONSEQUENTIALLY "reestablishes the truth" by controverting a falsehood.
FINALLY, with respect to Novak's "notion of consequence" we have these two propositions to examine, again, with respect to allegedly preserving truth, requote:
NOVAK:-
Anything follows from a contradiction. (TRUE?)
A contradiction is 1 necessarily-false proposition. (TRUE?)
Therefore...
If a contradiction is 1 necessarily false proposition, then a necessarily false conclusion seems to be the logical CONSEQUENCE of employing that necessarily false proposition in any logical argument, unless it is contradicted. But, according to Lukas, a contradiction is necessarily false. [Gridlock?]
Of course, on Aristotle's and Aquinas's view of contradiction, a contradictory pair of propositions involve one TRUE proposition, logically opposed to one FALSE proposition, by logical-necessity. Hence contradictories are necessarily opposed as TRUE vs. FALSE propositions. In sum a contradiction is not necessarily false, nor is a contradiction a single proposition, except by conjunction. And who conjoins true and false propositions to arrive at the truth???
The two contradictory propositions are neither necessarily true nor necessarily false, just necessarily opposed in "truth value", which suggests another "notion" than consequence to determine which is which. That notion is the self-evident proposition, which is the basic notion of material or major logic. Major logic has truth, rather than consequence (formal logic's basic notion) as its basic notion and induction (in contrast to deduction) as its fundamental principle.
I wish Jacques Maritain had been given the time or the grace to actually write his promised treatise on Major Logic. It would have been interesting, although arguably not up to St. Thomas's standards of truth. Otherwise he probably would have been given both the grace and the time. I guess it wasn't necessary. Aristotle's formulation of contradiction is probably still the basic standard.
Kevin
Plus:
1975:
In a message dated 09/05/06 10:57:04 PM Mountain Daylight Time, uncljoedoc@... writes:
> In a message dated 5/8/2006 10:15:58 P.M. Eastern Daylight Time, jamesmiguez@... forwards from Kevin:
>
>> That "pious ejaculation" entails an egregious perversion of the logical meaning of the expression "It follows..."!!! The only thing which logically "follows" a contradiction is that 1 proposition, of a pair of contradictory-propositions, is false & the other true, when the propositions are existential.
Hi,
You suggest that what follows from a contradiction is of rather limited value to analytic cognition.
Hi again Joe:
No! I asserted that nothing logical "follows from" an existential contradiction save one true premise and one false premise. But I suppose if you think egregious perversions are of limited value to analytic cognition, you may be right.
Just kidding.
Existential "contradictions" (metaphorically-speaking), on the other hand from analytic cognition (whatever that is), are the basics of the scientific method. Such metaphorical existential "contradictions" are the crux of the experimental science "game"! Let me explain...
Take, for example, the contradictory conjunction "P & not-P", from Daniel Bonevac's example of an indirect proof. Can you construct an analytic cognitive truth table for that conjunction, Joe, using your analytic cognitive skills or, alternatively, your symbolic logic habit, if you have been trained in that art or acquired habit?
"Not" is what symbolic logicians call a truth functional connective, in SL, according to our friend Bonevac. The letter P is a formula in the SL "metalanguage", corresponding to some proposition in English, which is called a "natural language" to symbolic logicians (even though all languages are conventional).
So whether an SL proposition is "P" or "not-P", I think that they are actually talking about one and the same sentence in speaking of "P" and of "not-P"! By contrast in so-called "classical" logic, "P" vs. "not-P" would be two distinct and contradictory propositions, using the same subject and, THEN, the same predicate in logical opposition. In short two contradictory propositions.
So, with "NOT", however symbolized, being called a truth functional CONNECTIVE, they must (I suppose) be talking about connecting propositions to real states of affairs or being. Maybe. Maybe not. I dunno. But it sounds suspiciously like what the SL "guys" may be talking about---connecting propositions with being/reality.
Unlike classical logic, certain SL symbols stand for entire sentences, whereas in classical logic their symbols (when not using "natural language"; but only mere letter symbols) stand for subject and predicate. I gave an example of Algazali using such "classical" symbolic notation when he wrote to his muslim friends
If every a is b, then some b is a!
eg. If every man is an animal then (conversely) some animal is a man.
['Ghazali's example from "natural language"]
I think that SL people may come from the so-called "coherence theory of truth" school of modern philosophers, as distinct from the so-called correspondence and pragmatic "theories" of truth schools. An analog of coherence with respect to truth, is consistency in so-called "sets" of sentences or propositions.
So the SL "guys" seem to be more interested in how sets of propositions "agree" or "disagree" with each other, in either consistency or coherence, than in how human beings connect or disconnect subjects with predicates. They seem to be working off of Uncle Bertie Russell's "set theory", from mathematics and his theory of "signs".
However, no classical logician ever "thought" that the truth was any sort of theory, in the modern sense of theories. With classical logicians the truth is axiomatic, as opposed to "hypothetical", where modern theories are considered to be repetitively confirmable or verifiable hypotheses.
By contrast:- Theoria in the ancient sense meant the kind of knowledge and truth that Theos/God had of things---immutable, eternal, unchangeable---sorts of knowledge. Thus we have Aristotle saying in the Metaphysics that if "the divine" is anywhere in human things, it must be in the study of abstract mathematics, physics or theology---for in those subjects we apprehend so much of "the divine" which is possible for human beings to grasp. Or in Aquinas, you have the entirely "cocky" (according to modern "truth-theoreticians"!) assurance that THE CONTRARY of Scripture's truth can never be demonstrated, since Scripture rests upon infallible truth and the contrary of infallible-truth can never be demonstrated.
Truth, according to Aquinas's definition, isn't "correspondence with reality" (way too general; reality being the sum of all real things), but, instead the "conformity of an intellect with a thing." In other words one "thing" at a time, or one step at a time. No one is going to know or understand "reality", or have a mind which "corresponds with reality", at one time, or even with much of "reality", in any amount of time.
"Reality" is just TOO MUCH and TOO BIG.
But "P" and "not-P" are neither too much nor too big to handle. They're simple.
The SL "p" and "not-P" stuff probably comes out of logical positivism, and its descent from Hume. Positivists were not going to take courses in BARBARA-CELARENT logic from a bunch of "obsolete/irrelevant" monks in Catholic institutions. That "logic" was passe. And since Hume posed as a sort of "Newton-of-Philosophy", while guys like Jeremy Bentham were also posing as Newton-Clones and "thinking-up" things like the "Felicifical-Calculus" (I love the British because they pretend to be staid and serious---when, in truth, they have more WILD 'N CRAAAZY GUYS than even Steve Martin could "dream-up"), Hume's descendants just, naturally, had to "dream-up" some wild calculus symbols for what any ancient scholastic could do with MNEMONICS like Barbara-Celarent-Darii, etc.
Of course, it was also the "new age" of experimental science. So "chuck" those outmoded categorical syllogisms and TURN EVERYTHING into HYPOTHESES because that's what the "scientists" do. So that's what positivists tried to do. But it didn't work because those SCIENTISTS were not "reasoning" according to the rules. The illogical clowns were apparently "reasoning" by AFFIRMING THE CONSEQUENTS of their hypotheses---which is irrational. However they were getting results, like dynamite as well as anti-septic-surgery.
So they must be doing something INDUCTIVE with their experiments. Inductive sounds more logical than Irrational any day of a logical positivist's week. So the positivists started "throwiing around expressions" such as inductively valid, but deductively invalid and confusing themselves along with all their former students who DROPPED LOGIC and went into "science/math". At the same time, people were looking for an equivalent set of RULES OF INDUCTION to go along with their RULES OF DEDUCTION, but never found such expected "rules".
But what the scientific guys were actually DOING was taking any old garden variety hypothesis such as----IF "P" THEN "?" and also doing the same hypothesis with--- IF "not-P" THEN "?" In other words they were DOING "P" and "not-P" in an existential mode, which is illogical to DO in a mental mode.
Scientists do not misreason (as a general rule; although any individual scientist can be as "cracked" as the next guy) from AFFIRMED CONSEQUENT to AFFIRMED ANTECEDENT (contrary to logic's rules) but from AFFIRMED ANTECEDENT & not-AFFIRMED ANTECEDENT to "?"-CONSEQUENT, normally called TEST vs. CONTROL in a modern scientific experiment.
In short, scientists can existentially do, validly, what logicians are forbidden to mentally do in terms of logic. However, nothing succeeds like "success", so, apparently without understanding the WHY of the scientific method's success, logicians started to come up with "goofy sayings" such as "anything follows from a contradiction" in order to "justify" a logical imitation of the scientific method, such as the INDIRECT PROOF method with "assumptions" such as "P & not-P".
That "stuff" perfectly imitates the TEST vs. CONTROL "system" of experimental science. But logicians know that it is, at best, an "iffy" logical procedure, as Bonevac frankly confessed with his "flying pigs" example. So they only use indirect proofs when all their other actually-LOGICAL rules leave them "dead-ended".
And, of course, since many of the SL guys are actually LOGICAL, they have to put their little boxes around everything and make sure they cancel their "shows" and rub-out their intentionally introduced contradictions, in mathematically calculated ORDER and precision, so that one ERROR cancels-out a balancing-ERROR.
Unfortunately, no one can put "little boxes" around existential errors and cancel out one existential error with a counter-balancing existential error, then ERASE the whole intermediate MESS, while being SATISFIED with a final correct answer. Reality doesn't work that way. Stick one ERROR into an existential system and the whole system begins to react, in various unpredictable ways, to that error and NONE of the unpredictable reactions are either BOXED or ERASIBLE. eg. Ignoring the 95 theses of an Augustinian monk, by calling the "95" a monkish "quibble". Oh! That's where Apolonio may have gotten the term "quibble" from. I wondered about that...
At any rate, Joe, that's the gist of the "P" & "not-P" thesis in science. Let P stand for Pasteurization.
If P, then kids don't die from drinking bad milk
"P" in every "civilized" country of the world
ERGO kids don't die from drinking bad milk.
That is called AFFIRMING the antecedent, which is perfectly logical to do. The observations, and work by Louis Pasteur, which led to that sort of actual logic and actual science did have some apparently illogical (to positivists) steps. For example, poor Louis had to take tons of illogical criticism from "positivists" after basically pioneering sterilization and antisepsis, despite irrational opposition. But finally the "successes" convinced even the most hardened sceptics.
However when Louis began to inject supposedly "infectious matter" (which Louis had carefully rendered "inert" by sterilizing/pasteurizing the formerly infectious stuff) into patients another furor and scandal erupted, because what he was doing (immunization) went against "sterile" medical procedures and practices. However, curing rabies soon silenced more irrational critics, who failed to distinguish sterilization from immunization. What a guy! Even the atheists and positivists had to pay him homage.
PASTEUR:
The more I learn and understand the more my Faith becomes that of a Breton peasant. If I continue to learn more and understand more, perhaps I may gain the Faith of a Breton peasant's wife.
Atta Boy, logically-scientific-Louis!!!
Kevin
The Notions of validity in Anselm, Aquinas, and Novak:
1986, 1987, 1988, 2038, 2049, 2052, 2053, 2054, 2056, 2059, 2062
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