An introduction to formal logic

[MRC_Hans]

I notice that half the threads Franko has "hijacked", are actually about him. Cant really blame him for that IMHO. But it takes at least two to hijack a thread: One to provoke, and one or more to let themselves be provoked. As long as we have been answering why should Franko stop posting?

Hans

 

[Tricky]

Well, I've been waiting patiently for Franko to answer these, but that seems unlikely, so I'd like to try my hand. I have never studied formal logic either, but this is the kind of puzzle I like.

Some Saudis are Muslims
Some Muslims are terrorists
Some Saudis are terrorists.

Valid or invalid? Show your work. Remember, both premises and the conclusion are true, and the terms in the conclusion appear in the premises, and there is "flow".

Conclusion is invalid. It could be that all Muslim terrorists are of some nationality other than Saudi. The sets do not necessarily overlap.

An object with covers, pages, and a binding is a book.
This is a book.
This has covers, pages and a binding.

Invalid. Although all items with covers, pages and bindings are books, not all books have these traits (books on tape, for example )

An object with covers, pages, and a binding is a book.
This is not an object with covers, pages, and a binding.
This is not a book

Invalid (convoluted version of last question). Although all items with covers, pages and bindings are books, there is no statement as to what things without these traits can be. They may or may not also be books.

How'd I do?

 

[Peter Soderqvist]

Formal logic cannot deal with our fuzzy world!!

Some Saudis are Muslims, Some Muslims are terrorists, Some Saudis are terrorists.

Soderqvist1: This sequence is truth in a fictional world there all humans are Saudis, but in our real world, it is quite possible that some Saudis are terrorists, for instance Usama Bin laden. But Saudis and terrorists are not a symmetrical relation there, because at least the IRA terrorists are not Saudis, therefore, some terrorists are not Saudis. Aristotelian logic is proper to use when it comes to symmetrical relations, for instance, number 5 belongs to the set of odd numbers, but not at all to the set of even numbers, but most relation in our world are not symmetrical, because Terrorists and Saudis doesn't always belong to the same set, or simply; it is a kind of fuzzy relation between them, and it follows from that, that we need fuzzy logic, when we deal with our real world, because the Aristotelian binary logic cannot deal with fuzzy values, but fuzzy logic can always deal with the Aristotelian binary values, or in simpler terms; two valued logic cannot deal with many values, but many valued logic can deal with two values.

Fuzzy Logic by Bart Kosko and Satoru Isaka
The binary logic of modern computers often falls short when describing the vagueness of the real world. Fuzzy logic offers more graceful alternatives. At the heart of the difference between classical and fuzzy logic is something Aristotle called the law of the excluded middle. In standard set theory, an object either does or does not belong to a set. There is no middle ground: the number five belongs fully to the set of odd numbers and not at all to the set of even numbers. In such bivalent sets, an object cannot belong to both a set and its complement set or to neither of the sets. This principle preserves the structure of logic and avoids the contradiction of an object that both is and is not a thing at the same time. Sets that are fuzzy, or multivalent, break the law of the excluded middle- to some degree. Items belong only partially to a fuzzy set. They may also belong to more than one set. Even to just one individual, the air may feel cool, just right and warm to varying degrees. Whereas the boundaries of standard sets are exact, those of fuzzy sets are curved or taper off, and this curvature creates partial contradictions. The air can be 20 percent cool-and at the same time, 80 percent not cool.

Fuzzy degrees are not the same as probability percentages, a point that has eluded some critics of the field. Probabilities measure whether something will occur or not. Fuzziness measures the degree to which something occurs or some condition exists. The statement "There is a 30 percent chance the weather will be cool" conveys the probability of cool weather. But The morning feels 30 per- cent cool" means that the air feels cool to some extent-and at the same time, just right and warm to varying extents. The only constraint on fuzzy logic is that an object's degrees of membership in complementary groups must sum to unity. If the air seems 20 percent cool, it must also be 80 percent not cool. In this way, fuzzy logic just skirts the bivalent contradiction-that something is 100 percent cool and 100 percent not cool-that would destroy formal logic The law of the excluded middle holds merely as a special case in fuzzy logic, namely when an object belongs 100 percent to one group.

The modern study of fuzzy logic and partial contradictions had its origins early in this century, when Bertrand Russell found the ancient Greek paradox at the core of modern set theory and logic. According to the old riddle, a Cretan asserts that all Cretans lie. So, is he lying? If he lies, then he tells the truth and does not lie. If he does not lie, then he tells the truth and so lies. Both cases lead to a contradiction because the statement is both true and false. Russell found the same paradox in set theory. The set of all sets is a set, and so it is a member of itself. Yet the set of all apples is not a member of itself because its members are apples and not sets. Perceiving the underlying contradiction, Russell then asked, "Is the set of all sets that are not members of themselves a member of itself ?" If it is, it isn't; if it isn't, it is. Faced with such a conundrum, classical logic surrenders.
http://www.fortunecity.com/emachines/e11/86/fuzzylog.html

Soderqvist1: Btw, our world is not an additive affair, because when we add two poisons together like sodium, and chlorine, we will not end up with some double poisonous mixture, because chemical interaction between these compounds, will give us something not poisonous, namely; salt!

 

[Titanpoint]

I notice that half the threads Franko has "hijacked", are actually about him.

That's inaccurate. There have been a few (in comparison) which name him specifically. The fact is that his spamming is designed to get all of the them to be about him.

Cant really blame him for that IMHO. But it takes at least two to hijack a thread: One to provoke, and one or more to let themselves be provoked.

Again, is it the behavior of one troll that is the problem or the behavior of naturally-inquisitive people? What is the R&P forum for if not to question things?

As long as we have been answering why should Franko stop posting?

Hans

Unfortunately Hans, Franko has become both cause and effect.

TP

 

[CWL]

Well it is a sad thing. I was one of the people who welcomed Franko back after his disappearing act. I was hoping for some open and honest debate. Unfortunately I was disappointed.

Franko, although obviously capable of intellectual thought, is also obviously stuck within his own dogmatic frame. He is unable to accept any alternative standpoints.

He is unique on this forum in that he claims to hold "the Truth". Yet he is unable to prove or explain it.

There is no point in trying to debate anyone holding a sermon.

Sad thing.

 

[MRC_Hans]

Unfortunately Hans, Franko has become both cause and effect.

I'm not trying to remove blame from Franko, although it might seem so. The way he has acted lately, he's a genuine PITA. What I'm trying to say is that we should take a look in the mirror as well. But maybe we should stop this discussion; the less attention -----

Hans

 

[Peter Soderqvist]

Mathematics is a fictional truth!

TO FRANKO

Franko wrote on page 2, 10-31-2002 04:29 PM: 2 + 2 = 4 demonstrate the validity of that whitehead ... I suspect there is an invisible flaw point it out for us!!!

Soderqvist1: This is an abstract truth, because mathematics deals only with undefined terms, but it is not always truth in the real world, namely, 2-liter water + 2-liter alcohol is not 4-liter, because chemical reaction between these compounds will give us a mixture little less than 4-liter. But if we add 2 apples and 2-apples together, its weights has the approximated additive value, because the weight will rise additively when we add apples together, but its temperature is not an additive affair, because the apples' temperature have not been risen. But the opposite is more truth, when we add nitro to glycerin!

Is it truth that; if B is brother to S, is S also brother to B? Not always; because S is sometimes sister to B!

Quotations by Albert Einstein
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Quoted in J R Newman, The World of Mathematics (New York 1956).
http://www-groups.dcs.st-and.ac.uk/~history/Quotations/Einstein.html

 

[whitefork]

(this is not an attack on anyone)

Nice work Trickster, you got it.

Let's see.

This thread is called An Introduction to Formal Logic.
I posed a couple of questions about validity, and you answered them correctly.

For the purposes of getting this back on track, I'll complete the drill.

1. Some Saudis are Muslims
Some Muslims are terrorists
Some Saudis are terrorists.

This is well-formed, but the so-called MIDDLE TERM (Muslims) is not "distributed" among subject and predicate. Venn diagrams (the "Ballentine Ale" symbol - 3 overlapping circles) would show this clearly but I don't know how to put them up here. What is means is that the set which is the intersection of Saudis and Terrorists may be null under the premisses (point being, all the terrorists, may in fact be non-saudis under the premisses).

I disagree that non-Aristotlean logic is required to solve this. It may be able to, but the fallacy has been known since the beginning and is easily dealt with by traditional methods. (Sorry Peter)

LW gave the equivalent Predicate Calculus demonstration elsewhere.

No Syllogism of the SOME, SOME, SOME variety is ever valid.

2. An object with covers, pages, and a binding is a book.
This is a book.
This has covers, pages and a binding.

Form If A then B.
B
Therefore A

Fallacy of affirming the consequent. Always invalid (Use of truth tables demonstrates that the conclusion can be false while the premisses are true - the definition of implication (if, then)

T implies T = true
T implies F = false
F implies T = true
F implies F = true

Hence, if you can demonstrate a condition where the conclusion is false and the premisses are true, you have an invalid argument. Conversely if assuming the falsity of the conclusion leads to a contradiction, the argument is valid (reductio).

For instance - this is hard to grasp but....

IF Today is November 5
AND Today is November 6

Then I am the emperor of China.

Assume that I am not the emporer of China.
By our assumptions, today is both November 5 and not November 5.
Since the assumption (I am NOT, etc) leads to a contradiction, we must conclude the negation of that assumption.

What this means is that from CONTRADICTORY premisses, anything at all follows.

(one point of the truth function of implication - false premisses implying a false conclusion has a truth value of TRUE)

Opposite side of the coin - any premise at all implies a tautology (statement that is always true). In the truth function of implication, when the conclusion is TRUE, the implication is always true, regardless of the whether the premisses are true or false.

3. An object with covers, pages, and a binding is a book.
This is not an object with covers, pages, and a binding.
This is not a book

If A then B
not A
Therefore not B

Denying the antecedant. Always invalid.

4. All US presidents are native born Americans
Some Catholics are native born Americans
Therefore some Catholic was a US president.

In this case, we also have a Middle term (native born american) that has not been distributed, etc.

But wait! This one is valid:

All native born Kansans are Americans.
Some Catholics are native born Kansans
Therefore some Catholics are Americans

(Middle term (Kansans) distributed)

I will not belabor this by giving all the rules for validity. See this site if you care http://www.philosophypages.com/lg/e08a.htm

I have beaten this thing to death by now, but my point is real simple.
If some argument of form X has true premisses and a true conclusion, we must not conclude that all arguments of Form X have a true conclusion when the premisses are true.

The rules of classical syllogism have been restated in the form of predicate logic (that thing where (x) means "for all x" and the backwards-E x means "for at least one x").

I personally prefer the syllogistic approach because it doesn't require the intimidating math-like symbols, and has a nice, intuitive feel to it that is sometimes deceptive.

All examples given here sound a little bit reasonable since they contain true statements, but not one of them is valid. The logical flaws can only be exposed by looking beyond the semantics (meaning) of the terms and seeing the underlying formalism.

Fork go back to work now.

thank you.

 

[Tricky]

Thank you, Whitefork. That was a better explanation of formal logic than I have ever heard before. I have a feeling I may want to read a little more on this.

 

[Willy]

Peter Soderqvist,

2 + 2 = 4 having an "invisible flaw" was sarcasm. I think you missed the point ...

 

[whitefork]

You must not get out much....
Problem with formal logic is that it's generally given a dry-as-dust presentation, and it doesn't have to be that way. If you can dress up the formalism in normal language, it's more fun. I keep pushing Pospesel's Arguments: Deductive Logic Exercises because it has examples, valid and invalid, from everyday discourse.

In my opinion, if one doesn't have a solid grasp of the bare basics of logic, there's absolutely no point in discussing guys like Cantor and Godel, because one will not understand what they're arguing about. Godel in particular is so susceptible to misinterpretation that as soon as I hear the name brought into a discussion, I assume that the speaker is about to emit a huge cloud of smoke. Formal undecidability is about formalism. The paper after all is called "On Formally Undecidable Propositions of Principia Mathematica and Related Systems."

That's why I flog the formalism hobbyhorse all the time.

Oh, yeah, I gave a "proof" of 2+2=4 on page 3 of the Logical Deism thread. (do a find on SUCCESSOR).

 

[Victor Danilchenko]

Whitefork

i disagree with you on the utility of classical logic. In order for classical logic to be useful for examining the sorts of things you are examining, you have to bring in set theory. You can do so intuitively and thus not appear to complicate things, but it is well-known that "intuitive" set theory is inconsistent.

therefore, i think predicate logic is much better for such things, as many of the implicit set operations in boolean argument are replaced with well-defined quantifiers; and you don't have to use the intimidating symbols -- "FORALL" of "F" work just as well as "∀", and "EXISTS" or "E" work as well as "∃". besides, "forall" and "exists" are pretty comprehensible intuitively as well.

In fact, speaking about the flaws of classical logic... I was going to bring up a point that your earlier examples were unclear -- when you say "An object ... is a book", do you mean that some such objects are books, or all such objects are books? Simply having to frame your arguments in predicate-logic terms automatically does away with such ambiguity and imprecision.

Oh yeah, quick addition for others:

predicate logic is a powerful superset of boolean (AKA classical) logic; in fact, when you hear a mathematician or a logician speak about "logic", they will be usually speaking about predicate logic; it's the logical system to know. preedicate logic comes in multiple orders, and the first-order predicate logic (AKA simple "first-order logic") is just like boolean logic, with addition of two operators, called "quantifiers" -- universal quantifier ("forall" = "F" = ∀) and existential quantifier ("exists" = "E" = ∃). Predicate logic also allows you to specify predicates (additional operators), and the "orders" of predicate logic are distinguished by how and what sort of operators can be defined.

I will use alphabetic symbols, since the proper symbols (∀ and ∃) will not render correctly in all browsers. Also, "~" means logical negation.

Fx C(x)->M(x)
means "For all x, C(x) implies M(x)" -- for example, "all cats are mammals" ("for all x, x being a cat implies x being a mammal"), where "M(x)" means "x is a mammal" and C(x) means "x is a cat".

Ey M
means "There exists x such that M(x) is true" -- for example, "some entities are mammals".

Quantifiers can be stacked:

FxEy y=x+1
means "For all x, exists y such that y=x+1"; this is a statement of infinitude of integers.

see http://www.wikipedia.org/wiki/Predicate_logic for more info.

 

[whitefork]

Victor, I agree with you that if you wish to get really rigorous about formalism you need all that heavy lifting, but the examples I gave were not intended to do more than give a broad-brush view of validity and how to start thinking about it.

The original discussion from which this emerged had to do with the nature of syllogism itself, but went immediatly off in other dircetions, and I don't really want to go back there right now.

So, yeah, if you're going to do it right, get down into the set theoretical stuff, but for (ahem) pedagogic purposes, I've found that ordinary (if flawed) language examples work best at the beginning. Engage the interest first, then if people want to pursue it, hit them with the heavy machinery later.

Ever try teaching an intro to logic course to freshmen at Ohio State? (oh, the pain)

You are of course correct....

An speaking of "Oh the pain" Doctor Smith of "Lost in Space" - Jonathan Harris - died.

"Danger, Will Robinson"....

 

[Victor Danilchenko]

whitefork

Victor, I agree with you that if you wish to get really rigorous about formalism you need all that heavy lifting, but the examples I gave were not intended to do more than give a broad-brush view of validity and how to start thinking about it.

I can understand that; but don't you agree that even merely being aware of the formalisms' nature helps a lot? Such as in my example of an ambiguous statement -- "an object with leaves and cover is a book" is not precise enough to determine whether you are quantifying existentially or universally. Such ambiguities are often used exactly for the purpose of sneaking in unsound arguments.

So, yeah, if you're going to do it right, get down into the set theoretical stuff, but for (ahem) pedagogic purposes, I've found that ordinary (if flawed) language examples work best at the beginning. Engage the interest first, then if people want to pursue it, hit them with the heavy machinery later.

but... but... formal logic is so much fun!

I will always remember...

"Everybody loves my baby,
but my baby loves nobody but me"

implies that

"I am my baby"...

Ever try teaching an intro to logic course to freshmen at Ohio State? (oh, the pain)

One of the reasons I bailed out of the PhD program with MS -- teaching unwilling freshmen sitting there merely to fulfill a requirement is not my idea of fun.

 

[whitefork]

Such ambiguities are often used exactly for the purpose of sneaking in unsound arguments.

No one that I know would ever stoop so low.

but... but... formal logic is so much fun!

Pays real good, too, don't it?

 

[Mossy]

bump

(this one is even educational)

 

[whitefork]

Easy Propositional Logic

What you have here is the logic normally taught in introductory courses (and used in WFF 'n' Proof) - Logical connectives between "atomic sentences", which are declarative statements with a truth value, TRUE or FALSE.

This type of logic does not deal with the internal structure of statements but the relations between them.

The logical operators are truth functions, AND, OR, IF, IF and only if (IFF), NOT.

AND and OR are pretty intuitive. If you have atomic sentences A and B, the truth value of (A AND B) - conjunction - is true when both A and B are true, and false otherwise.

NOT is the negation of the following term.

With OR, the disjuction is true if either or both of the terms are true.

With IF, the implication is false if the first term is true and the second false. Otherwise the implication is true.

With IFF, the equivalence is true if the two terms have the same truth value.

If you have EXCEL or some other spreadsheet, you can set up a truth table for two variables (statements) and various combinations of operators.

Open a new spread sheet and in cells A1 through A4 and B1 through B4 type
TRUE TRUE
TRUE FALSE
FALSE TRUE
FALSE FALSE

That is all the combinations of truth value of two variables.

Then in C1 type =AND(A1,B1), in D1 =OR(A1,B1), in E1 =IF(A1,B1,TRUE), in F1 =NOT(A1), in G1, =NOT(B1), in H1, =OR(NOT(A1),B1), and in I1, =AND(IF(A1,B1,TRUE),IF(B1,A1,TRUE))

Then copy the cells C1 - H1 to the other three rows.
You should see the full values for all the functions.

(there is no IFF function in excel as far as I know. IFF would be represented as
AND(IF(A1,B1,TRUE),IF(B1,A1,TRUE) read as If A then B and If B then A (Logical equivalence - both must be true or both must be false)

Notice that IF is very weak, that IF(A1,B1,TRUE) is TRUE three out of four times. Read this as IF A then B. The function evaluates the truth of A1, and returns B1 is A1 is TRUE, and TRUE (the third operator) if A1 is false)
Note also that IF A then B is equivalent to OR(NOT(A1,B1), read as NOT-A or B.

Anyway, you can do this with any number of variables, the number of rows being a power of 2 (three variables, eight rows, etc).

Playing around with various combinations of operators shows certain equivalencies

You can show NOT(A and B) = Not-A or Not-B (deMorgan's theorem) by setting up NOT(AND(A1,B1) and OR(NOT(A1),NOT(B1))

Important to this whole process is that all logical equivalences that have the same truth value no matter how complex, can be derived from one another by logical, formal, operations. The system is complete, the truth value of any proposition is decidable, although there are no general rules for forming a proof.

But this tool allows the evaluation of very complex arguments and I might put some up here if anyone shows interest, or maybe if nobody does. Serves you right.

 

[whitefork]

In order to demonstrate validity under this system, you may construct an expression where the conjunction of the premisses is the antecendant of an IF statement and the conclusion is the consequent, as follows

IF (A [and B and C...]) then Conclusion.

Evaluate this expression under the standard rules for implication, and if all values are TRUE, the expression is a tautology. If all values are FALSE, it's a contradiction, and if both TRUE and FALSE appear, it's a contingency. Only tautologies are valid inferences.

In our example, if you construct in cell J1 the expression =IF(AND(E1,A1),B1,TRUE) - equivalent to

IF((IF A then B) and A) then B (known as Modus Ponens - remember that E1 is the expression If A then B)

and copy it to the other 3 rows, you'll have values equal to TRUE. A valid inference.

In K1, set up =IF(AND(E1,B1),A1,TRUE) - equivalent to

IF ((IF A then B) and B) then A - affirming the consequent, you'll have mixed values. The inference is false when A is false and B is true.

In L1 set up =IF(AND(E1,NOT(A1)),NOT(B1),TRUE) - equivalent to

IF ((IF A then B) and NOT-A) then not-B) - denying the antecedant, false when A and B are both false (has the same truth value as affirming the consequent, thus they are in fact the same fallacy, as Tricky pointed out above).

In M1 set up =IF(AND(E1,NOT(B1)),NOT(A1),TRUE) - denying the consequent (Modus Tollens), equivalent to

If A then B is equivalent to If Not-B then Not-A,

you'll have another tautology.

Lastly, in N1 set up =NOT(OR(A1,NOT(A1))) - meaning

(Neither A nor not-A) or (Both A and not-A) - law of the excluded middle -

you'll have a case where all values are false, a contradiction.

I think I've typed all this correctly.

 

[whitefork]

Inference in Sentential Logic

Since the truth value of any well-formed expression in this logic can be determined by the use of truth-tables, the rules of inference aren't strictly necessary. However, the proof of the completeness and consistency of Sentential Logic requires that the system be reduced to a set of axioms and rules. The lack of space in the margins prevents me from offering a full proof of the completeness.....

Anyway, every author seems to offer a slightly different, although complete, set of inference rules, and you can use these to construct proofs in the system.

Kahane (Logic and Philosophy) gives:

Modus Ponens: (If A then B) and A implies B.
Modus Tollens: (If A then B) and Not-B implies Not-A
Disjunctive syllogism: (A or B) and Not-A implies B
Simplification: (A and B) implies A
Conjunction: A, B implies (A and B)
Hypothetical Syllogism: (If A then B) and (If B then C) implies (If A then C)
Addition: A implies (A or B) - a really useful one
Constructive Dilemma: (If A then B) and (If C then D) and (A or C) implies (B or D)

All these can be shown to be tautologies under the definitions of the operators.

One more:

Substitution: Within any expression, a term like A (representing an atomic sentence) may have all of its occurrence replaced by ANY well-formed expression.

The entire machinery of sentential logic may be derived from these three axioms (this from Thomason, Symbolic Logic):

1. A implies (If B then A) - IF(A,IF(B,A,TRUE),TRUE) - meaning if A is assumed, we can validly derive A from anything.

2. (A implies (B implies C)) implies ((If A then B) implies (If A then C)) - Sort of a transitivity of implication.

In excel speak
=IF(IF(IF(A,B,TRUE),IF(B,C,TRUE),TRUE),IF(IF(A,B,TRUE),IF(A,C,TRUE),TRUE),TRUE)
- you need 8 rows to handle the three variables.

3. If Not-A implies Not-B, then B implies A - Modus tollens.

Two rules of inference:

1. Modus Ponens
2. Substitution of expressions for variables.

Don't pay any attention to the above unless you agree with Victor that rigor is fun. There's a good semester of work implied.

Otherwise stick to truth tables - they're adequate for our purposes of showing validity.

Although the truth value of any expression in sentential logic can be evaluated, the process of constructing a proof for the expression using only the rules of inference is by no means trivial - like Mr. Ellis used to say in Calculus "Differentiation is a science, Integration is an art". There are no general rules for forming proofs.

 

[whitefork]

Simple syllogistic logic

Syllogism deals with categorical propositions. These relate classes of entities (referred to as Subject and Predicate). The classes are assumed to be non-empty. The following comes straight out of the Middle Ages. Syllogism has been replaced by predicate logic, but it can serve as a simple introduction.

There are four kinds of propositions:

All S are P (type A)
No S is P (type E)
Some S is P (type I)
Some S is not P (type O)

Draw a square. On the upper left corner write A. On the upper right, E. On the lower left I, and on the lower right, O.

Draw the diagonals.

The diagonals represent contradictories. (If All S is P, then it is not the case that Some S is not P, and conversely) (If no S is P then it is not the case that some S is P, and conversely). A and O proprositions, and E and I are contradictory.

The relation between A and E is called Contrary. Both cannot be true, but both can be false (If all S are P, then it is not the case that No S is P, but it may be the case the neither is true)

The relation between I and O is called Subcontrary. Both may be true, but both may not be false. (Either some S is P or some S is not P)

The relationship between A and I, and E and O is called Subalternate. We may infer from All S is P that Some S is P, and from No S is P that Some S is not-P (This is the assumption of Existential Import, that the classes are non-empty. If the classes are empty, the inference is not valid.)

A and I are affirmative, E and O are negative.
A and E are universal, I and O are particular.

More to follow.