If you find 'so' vulgar, start using 'ergo' AND seguitur. The sequi is exactly the 'follow' as in "That does not follow".
Grice (words): "He said that Jack was an Englishman, and he said that Jack was brave. And he said, 'therefore'. But surely he did not say that the latter -- i.e. that Jack was brave, FOLLOWED from Jack being an Englishman. Not that he even hinted that it did NOT follow, either."
Most likely he may have implicated. This is a conventional implicature. It detaches. "Therefore" carries it, but other words --e.g. "Tomatoes" -- do not. On the other hand, proper conversational implicatures (improper for my aunt) are always nondetachable or never detachable if you should.
Grice was a liberal. He thought, "He is an Englishman; he is, therefore, brave" is hardly a nonsequitur. In fact, Griceianism is all for "sequiturs". What follows is my adaptation of examples from wiki, 'nonsequitur'. I rewrite the examples using 'therefore' to mark the intention on the utterer that there is a valid supplementation of the enthymeme.
"If I buy this cell phone, all people will love me."
"You buy this cell phone; therefore, all people will love you."
Wiki: "However, there is no actual relation between buying a cell phone and the love of all people. This kind of reasoning is often used in advertising to trigger an emotional purchase."
"If you buy this car, your family will be safer."
You buy this car; your family will, therefore, be safer.
"(While some cars are safer than others, it is possible to decrease instead of increase your family's overall safety.)"
"If you do not buy this type of pet food, you are neglecting your dog."
Do not buy his type of pet food; you will be, therefore, neglecting your dog."
"(Premise and conclusion are once again unrelated; this is also an example of an appeal to emotion.)"
"I hear the rain falling outside my
window; therefore, the sun is
not shining."
"(The conclusion is a non-sequitur because the sun can shine while it is raining.)"
Fallacy of the undistributed middle
"The fallacy of the undistributed middle is a logical fallacy that is committed when the middle term in a categorical syllogism is not distributed. It is thus a syllogistic fallacy. More specifically it is also a form of non sequitur."
"The fallacy of the undistributed middle takes the following form:"
"1.All Zs are Bs.
2.Y is a B.
3.Therefore, Y is a Z."
"It may or may not be the case that "all Zs are Bs," but in either case it is irrelevant to the conclusion. What is relevant to the conclusion is whether it is true that "all Bs are Zs," which is ignored in the argument."
"Note that if the terms were swapped around in either the conclusion or the first co-premise or if the first premise was rewritten to "All Zs can only be Bs" then it would no longer be a fallacy, although it could still be unsound. This also holds for the following two logical fallacies which are similar in nature to the fallacy of the undistributed middle and also non sequiturs."
1.Men are human.
2.Mary is human.
3.Therefore, Mary is a man.
Affirming the consequent
Any argument that takes the following form is a non sequitur
1.If A is true, then B is true.
2.B is true.
3.Therefore, A is true.
"Even if the premises and conclusion are all true, the conclusion is not a necessary consequence of the premises. This sort of non sequitur is also called affirming the consequent."
1.If I am a human (A) then I am a mammal. (B)
2.I am a mammal. (B)
3.Therefore, I am a human. (A)
"While the conclusion may be true, it does not follow from the premises: I could be another type of mammal without also being a human. The truth of the conclusion is independent of the truth its premises - it is a 'non sequitur'."
"Affirming the consequent is essentially the same as the fallacy of the undistributed middle, but using propositions rather than set membership."
Denying the antecedent Another common non sequitur is this:
1.If A is true, then B is true.
2.A is false.
3.Therefore, B is false.
"While the conclusion can indeed be false, this cannot be linked to the premise since the statement is a non sequitur. This is called denying the antecedent."
1.If I am in Tokyo, I am in Japan.
2.I am not in Tokyo.
3.Therefore, I am not in Japan.
"Whether or not the speaker is in Japan cannot be derived from the premise. He could either be outside Japan or anywhere in Japan except Tokyo."
Affirming a disjunct
Affirming a disjunct is a fallacy when in the following form:
1.A is true or B is true.
2.B is true.
3.Therefore, A is not true.
"The conclusion does not follow from the premises as it could be the case that A and B are both true. This fallacy stems from the stated definition of or in propositional logic to be inclusive."
1.I am at home or I am in the city.
2.I am at home.
3.Therefore, I am not in the city.
"While the conclusion may be true, it does not follow from the premises. For all the reader knows, the declarant of the statement very well could have her home in the city, in which case the premises would be true but the conclusion false. This argument is still a fallacy even if the conclusion is true."
Denying a conjunct
Denying a conjunct is a fallacy when in the following form:
1.It is not the case that both A is true and B is true.
2.B is not true.
3.Therefore, A is true.
"The conclusion does not follow from the premises as it could be the case that A and B are both false."
1.It is not the case that both I am at home and I am in the city.
2.I am not at home.
3.Therefore, I am in the city.
"While the conclusion may be true, it does not follow from the premises. For all the reader knows, the declarant of the statement very well could neither be at home nor in the city, in which case the premises would be true but the conclusion false. This argument is still a fallacy even if the conclusion is true."
See also Ignoratio elenchi, Modus tollens, Modus ponens, Post hoc ergo propter hoc
Regression fallacy, Fallacy of many questions
References
1.Barker, Stephen F. (2003) [1965]. "Chapter 6: Fallacies". The Elements of Logic (Sixth edition ed.). New York, NY: McGraw-Hill. pp. 160–169. ISBN 0-07-283235-5.
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The inclusive "OR" fallacies are interesting, and fairly common. Or just call 'em "Either/Or" fallacies, or maybe false dichotomy. Of course, stated correctly, the disjunction tautology is... the Law of the Excluded middle (ie, your computer is either on, OR your computer is off ( not-on)). But often there is a false dichotomy--"he's either guilty, or not guilty"! well, there could be partial guilt, he's an accessory, insane, justifiable homicide, etc. The modus ponens forms fairly obvious to most in college town, but you will hear plebes (Gott bless the bloody bastards!) affirm the consequent--(imagine Roger Scrutonizer-like arrogant example here)
ReplyDeleteThe early ad-lingo examples are more like generalizations..."If you buy this car, your family will be safer." Well, that actually might hold. As with the "4 out of 5 dentists recommend Krelm", or "studies show Alpo leads to healthier dogs," sort of ad lingo. Not necessarily true, but perhaps inductively sound--would require some ...v-word (verification).
Yes. Of course, mine was an ad hominem. I was arguing that "therefore" is misused -- not a word I would use -- then the piece of reasoning is held to be invalid.
ReplyDeleteSurely, but D. Frederick disagrees with me there, when I present an argument, I present it as valid. So it's usually others who find it invalid and are thus able to say that I have misused, in their view, 'therefore'.
Frederick has argued that since one is never sure if one's argument is valid or not, it's best not to use 'therefore' or deny that 'therefore' implicates a valid argument.
But back to the formalisation into /=: this symbol is NOT taught at Logic 101. Rather, instructors spend most of the useful time teaching useless things like 'or'/v divergences and such. I would think that
p ) q
and
p /- q
or
p /= q
(where the two latter are syntactic and semantic entailment) are different on various fronts. Not only is the pair of 'entailments' metalogic, but it seem to work better for formula schemata, rather than formulae themselves.
In any case, the textbooks will tell you that there IS a way to reduce '/-' and '/=' in terms of the horseshoe ')', and that is the method of the associated conditional (wiki).
/- p
reads 'it is a theorem' that p.
Thus, if '/-' symbolises 'therefore',
'p /- q' cannot be read as "... is a theorem" because here the symbol of syntactic entailment holds between TWO things.
Incidentally, perhaps we could analyse,
"He concluded with a kind word to his uncle".
Versus:
"He concluded that his uncle was not THAT idiotic".
In the second use, which I'm going to overuse instead of 'reason', the idea is that, after some thinking, he reached a final stage.
Therefore, reasoning involves this and that. I once did the OED about this, and found an expression,
"woman's reason" -- this is the argument:
Peter is charming
-----
Therefore, Peter is charming.
Grice finds that 'trivial' -- but valid ('p; therefore, p") The OED has it as a 'woman's reason' (I like him because I like him).
Aristotle, who was a beggar, called it a question-begging strategy which he did NOT deem fallacious.
Similarly,
"It is soporific; therefore, it will induce sleep" (example in wiki under 'question-begging').
Note that the thesaurus online incorrectly -- for a pedant logician that is -- has 'therefore' as synonymous with 'if' and 'then', and 'if ... then'. Enough to have Kurt Goedel turn in his grave -- which incidentally, is somewhere in Germany, I surmise.
|- is shorthand for conclusion, no different than three dots ∴ (or THEREFORE, in older texts). BUT |= is a semantic operator, is it not--not really first-order logic, but meta-logic really. It means something like ...satisfiability (usually a set of statements or something...ie back of Boolo/Jeffrey's text on Computability and Logic). Your list of operators, or Grice's list works for first-order logic (though should include biconditional)
ReplyDeleteVery good point about what, with Jones, we call, with some reluctance "syntactic" (|-) and "semantic" (|=) entailment. Indeed, I´ve seen the inverted three dots -- and posted a short blog post on them -- to mean, not just "because", which is logical, but "this is a premisse", which I thought redundant.
ReplyDeleteThis way, Kramer´s paradox would be solved. For Barbara would read,
This is a premise A
This is a premise A
This is a conclusion A
--- Yes, Grice does not list "if and only if" or provides the biconditional. I like to use this as the sign of the biconditional:
p )( q
i.e. to horseshoes facing each other.
The topic has been taken up by recent pragmaticists (e.g. Horn in the Journal of Pragmatics, "From if to iff" -- which I´ve read and which reminded me of D. F. Pears on "Ifs and cans", now credited in the OED3 under "implicature" (1973) -- Pears notes, with Horn, that "if" conversationally implicates "iff".
This does not say much about "iff" per se, but I trust Grice thought that an indicative biconditional was like his indicative conditional, only double!