Logic Post 27

The following are proofs of validity for particular argument forms. You should be able to find the justification, i.e., the rule of inference for each statement that is not a premise. These examples are taken from Kegley and Kegley, Introduction to Logic, p. 276.

First argument:
1. A ⊃ B
2. B ⊃ C
3. A ∨ ~D
4. ~C / ∴ ~D
5. A ⊃ C
6. ~A
7. ~D

So, in the first argument, which is contained in lines 1-4, you want to find the justification used in lines 5,6, and 7.

Second argument:
1. M ⊃ N
2. N ⊃ O
3. P ⊃ Q
4. M v P/∴ O v Q
5. M ⊃ O
6. (M ⊃ O) · (P ⊃ Q)
7. O v Q

If you want answers to any of these exercises just send a message to my inbox.

Logic Post 26

Which rule of inference corresponds with the following argument forms? Do your homework!


1)
[(A ⊃ B) · C] v (D · S)
~ (D · S)
_____________________
∴ [(A ⊃ B) · C]

2)
(p · q)
__________
∴ (p · q) v (r ⊃ s)

3)
(r ⊃ s)
(s ⊃ t)
________
∴ (r ⊃ t)

4)
[(A · B) ⊃ (C · D)]
[(P v R) · (Q v T)]
__________________
∴ [(A & B) ⊃ (C· D)] · [(P v R) · (Q v T)]

5)
[(D ⊃ E) ⊃ (A v B)] · (P ⊃ C)
(D ⊃ E) v P
____________________________
∴ (A v B) v C

Logic Post 25

The Three Laws of Logic (Sometimes referred to as the Three Laws of Thought)

1) The Law of Identity:
a. A is A or Anything is itself.
b. If a proposition is true, then it is true, which means that every proposition of the form p ⊃ p is true. Therefore, it is a tautology.

2) The Law of Excluded Middle:
a. Anything is either A or not A.
b. Any proposition is either true or false, i.e., it makes the claim that every proposition of the form p v ~p is true. Therefore, it is a tautology.

3) The Law of Contradiction:
a. Nothing can be both A and not A.
b. No proposition can be both true and false, i.e., it makes the claim that every proposition of the form p · ~p is false, and in this case contradictory.

There are criticisms of these three laws. For example, one such criticism against the law of identity is that the world is constantly changing. Hence, things are never the same from second to second. However, there is a confusion in this kind of thinking, viz., there is a difference between logical identity and physical identity. If someone states that "X has changed," then that requires that X's identity remain the same throughout a series of changes, or it would not be possible to say that X changed. There is obviously constant change going on in the world, but that does not negate identity. Moreover, there remains constancy of the referent throughout our discourse, i.e., identity in our meanings. So, when we talk of a tree, we mean a tree, and not some other object.

There are obviously other criticisms.

Logic Post 24

Enthymemes

Enthymemes are arguments in which a premise or premises are left out. Sometimes even the conclusion is left out - it is supposedly understood.

Enthymemes are quite ubiquitous in discourse, so it is important to familiarize yourself with them. They are used because the premise or conclusion is understood, and stating them would be to state the obvious. However, sometimes people will leave out part of an argument, because to state the premise or conclusion would obviously make the argument false. So to avoid criticism sometimes people will purposely leave a premise or a conclusion unstated. I know it is hard to believe that people actually do this.

You need practice to get good at solving enthymemes. Understanding these concepts is one thing, but actually solving the problems is quite another. Do not assume that because you understand what I am writing that you automatically can solve the problems. Logic is like math you need practice. Without it you will not be able to reason well.

Logic Post 23

There is an important difference between the Rules of Replacement and the Rules of Inference. The Rules of Inference can only be used on entire lines of a proof. So, in a proof, X can be inferred from X · Y, if X · Y make up the entire line. You cannot infer X from W ⊃ (X · Y) using Simplification. When using the Rules of Replacement this is not the case, because logically equivalent expressions can replace other logical equivalent expressions even if they do not constitute a whole line in a proof.

You're going to need more information than what I've given you here to learn to use these correctly. You should find a book with exercises, and one that explains the Rules of Replacement more thoroughly. Hopefully, this will give you somewhat of a guide to know what to study. I also would recommend studying the categorical syllogism. There are videos on Youtube that will explain much of this in detail.

Logic Post 22

Rules of Replacement

You need to memorize these rules of replacement along with the rules of inference.


1) Absorption

(p ⊃ q) ≡ p ⊃ (p · q)

2) Double Negation

p ≡ ~~p

3) De Morgan’s Theorems

~(p v q) ≡ (~p · ~q)
~(p · q) ≡ (~p v ~q)

4) Commutation

(p v q) ≡ (q v p)
(p · q) ≡ (q · p)

5) Association

[(p v q) v r] ≡ [p v (q v r)]
[p · q · r] ≡ [p · (q · r)]

6) Distribution

[p v (q · r)] ≡ [(p v q) · (p v r)]
[p · (q v r)] ≡ [(p · q) v (p · r)]

7) Transposition or Contraposition

(p ⊃ q) ≡ (~q ⊃ ~p)

8) Material Implication

(p ⊃ q) ≡ (~p v q)

9) Material Equivalence

(p ≡ q) ≡ [(p ⊃ q) · (q ⊃ p)]
(p ≡ q) ≡ [(p · q) v (~p · ~q)]

10) Exportation

[(p · q) ⊃ r] ≡ [p ⊃ (q ⊃ r)]

11) Tautology

p ≡ (p v p)
p ≡ (p · p)


“It should be noted that the eight Inference Rules and the eleven Rules of Replacement constitute a complete system of truth-functional logic in the sense that the construction of a formal proof of validity for any valid truth-functional argument is possible. However, some of the rules are redundant. Thus, for example, Modus Tollens is redundant because every instance in which Modus Tollens is used, the Principle of Transposition and Modus Ponens can function equally well. Disjunctive Syllogism could also be replaced. But these two argument forms are easy to grasp and the use of all nineteen rules makes proofs considerably easier (Kegley and Kegley, Introduction to Logic, p. 280 and 281).”

I think the later Wittgenstein has contributed to a more careful linguistic analysis, which can lead to using language, especially in philosophy, in a more precise way. I think that we have to be careful about how we emphasize the phrase "use is meaning," because there are quite a few uses that are incorrect. In fact, Wittgenstein is criticizing philosophers for their use of words and/or propositions. Use has to be seen in the proper context, i.e., in the social context, but even this is easily misunderstood. I don't have any confidence that Wittgenstein will be clearly understood in a wider social context.

One area of criticism is that there is a limit to language in terms of metaphysics. He still held onto this idea in his later philosophy. I think this is and was a mistake.

For example, despite numerous reports of ball lightening and its seemingly inexplicable behavior, the sightings were dismissed as the delusions of incompetent or lying observers-- until physicists investigating nuclear fusion possibilities developed mathematical theories describing plasmas. Their theories clearly applied to ball lightening. Suddenly, people who reported ball lightening were not written off. — Greylorn Ell

The problem seems to be, as I've mentioned before in other threads, is that people seem to think that unless science proves X, then we can't know X. My claim is based on knowledge acquired in other ways. For example, I don't need science to tell me that the orange juice I drank this morning is sweet, I've tasted it, or that there is an oak tree in my back yard, I've seen it. And there are other ways that we come to have knowledge, for instance, much of what we know is based on testimonial evidence. While it is true that testimonial evidence can be very unreliable, it can also be very strong. I've put forth my argument in the thread https://thephilosophyforum.com/discussion/1980/evidence-of-consciousness-surviving-the-body/p18

Logic Post 21

Deductive Methods

When analyzing arguments you want to look for forms that correspond to valid rules of inference. For example, consider the following argument form:

Premise 1. [p v (~q ⊃ r)] ⊃ [~s v (r · t)]

Premise 2. ~ [~s v (r · t)]

Conclusion: ~ [p v (~q ⊃ r)]

First, just because the argument has a large number of variables don’t let that intimidate you. Second, you want to keep the conclusion in mind, since this is where you are heading. Next you want to take note of the major connective in the first premise, which is ⊃, it has the form p ⊃ q. Now notice that the second premise is a negation, and it has the form ~q. At this point if you have memorized the eight rules of inference you should be able to see where this is leading.

So let’s break premise one down so that we can see how it corresponds with p ⊃ q.

Premise 1. [p v (~q ⊃ r)] ⊃ [~s v (r · t)]

p = [p v (~q ⊃ r)]

then comes the major connective ⊃

q = [~s v (r · t)]

So premise one has the form p ⊃ q.

Now let’s look at premise two.

~ [~s v (r · t)]

Premise two is the denial of q in the argument form p ⊃ q, so it has the form ~ q.

We now have

p ⊃ q

~q

You should now be able to see that the example above matches the rule of inference called Modus Tollens. Premise two denies the consequent. If you kept your eye on the conclusion Modus Tollens is the obvious choice.

The conclusion is ~[p v (~q ⊃ r)], which is the denial of p.

Therefore, the argument form

Premise 1. [p v (~q ⊃ r)] --> [~s v (r · t)]

Premise 2. ~ [~s v (r · t)]

Conclusion: ~ [p v (~q ⊃ r)]

is the same as


Modus Tollens

p ⊃ q
~q
______
∴ ~p

We have now figured out a simple proof using one of the rules of inference.

True wisdom comes in questioning everything - in never being settled until all posdible questions have been asked and answered. — Harry Hindu

I don't know where you get the idea that "true wisdom comes in questioning everything," I don't agree with that either. As I said earlier, this thread is just a guide for people. If you think it's an important point, then start a thread and debate the issue with those who want to debate. I'm not going to debate the issue.

Ah, I see what you mean. Thanks.

_______________________________________________

It looks a bit better now I think.

Ya, they are hard to read. I'll try to line them up.

Hopefully there aren't too many errors. :gasp:

Logic Post 20

Rules of Inference

While it is true that truth-tables are very good at testing the validity of truth-functional arguments, they tend to be a bit cumbersome; especially if you have four or more variables (remember that truth columns grow exponentially at a rate of 2^n).

It is much easier to deduce validity using deduction, i.e., you use deductive argument forms that have already been shown to be valid to perform a sequence of elementary arguments which will then confirm the main conclusion. “The elementary arguments, in essence, are a set of rules, called transformation rules, for they specify which truth-functional statement forms may be inferred from which others. The transformation rules are then subdivided into inference rules and substitution rules. Systems made up such sets of rules are called natural deduction systems. The selection of the rules in these systems is relatively arbitrary; any set will do so long as it is complete (Kegley and Kegley, Introduction to Logic, p. 271).”


Rules of Inference

1) Modus Ponens (MP):

p ⊃ q
p
_____
∴ q

2) Modus Tollens (MT):

p ⊃ q
~q
______
∴ ~p

3) Disjunctive Syllogism (DS):

p v q
~p
____
∴ q

p v q
~q
____
∴ p

4) Hypothetical Syllogism (HS):

p ⊃ q
q ⊃ r
______
∴ p ⊃ r

5) Simplification (Simp):

p · q
_____
∴ p

p · q
_____
∴ q

6) Conjunction (Conj):

p
q
__
∴ p · q

7) Addition (Add):

p
__
∴ p v r

8) Constructive Dilemma (CD):

(p ⊃ q) · (r ⊃ s)

p ∨ r
______
∴ q v s

You can use these inference rules on entire lines only. Never use them on parts of a line. For example, do not use simplification in the following way: (p · q) ⊃ r to get p ⊃ r

Also these inference rules are used in one direction only. For example, you can work your way from p · q to p using simplification, but not from p to p · q.

You want to memorize the rules of inference.

Logic Post 19

Tautologies, Contradictions, and Contingent Sentences

Before going on we will examine the differences between tautologies, contradictions, and contingent sentences.

First, a tautology is a statement form that is true under all possible interpretations of its variables; or another way of saying it is that a sentence is tautological if and only if there is no interpretation of the sentence which produces a false truth-value. Keep in mind that it is under the main connective that one looks to find the appropriate truth-value. Here are some examples:

1)
2) 3)
In the above three examples we are using brackets around the main connective to ONLY illustrate our point. And the point is, that in each of these truth-tables under the main connective we have all Ts, that is to say, all substitution instances are true. Therefore, given our definition of a tautology each of the above examples are tautological.

There are a couple of feature that we need to be aware of when thinking about tautologies. First, tautologies tell us nothing about the world, i.e., they are noninformative. For instance, if I say "Either it is snowing, or it is not snowing"- this statement is necessarily true, but it tells us nothing about the whether.

Second, we can determine the truth-value of a tautology a priori, which simply means, that we can know the truth of the statement quite apart from the evidence.

A statement whose truth is logically impossible is called contradictory; or a statement is a contradiction if and only if there is no line of the truth table that shows a truth-value of true. The following statement forms are contradictory, and can be seen as such by their truth-tables:

Note the line of truth-values under the bracketed main connective.



Any statement such as "Triangles have three sides and triangles do not have three sides" is contradictory in virtue of its form, p · ~p.

We have discussed statement forms that have all true truth-values (tautologies), and statement forms that have all false truth-values (contradictions). We will now complete this section with a definition of contingent statement forms.

The first two statement forms had either all true truth-values, or all false truth-values. And as you would expect the final statement form has a mixture of both true and false truth-values. It is considered a contingent statement because its truth-values are not dependent upon logic alone, but are contingent upon some state-of-affairs. For instance, the statement "The glass is sitting on my desk, or it is not sitting on my desk" is contingent upon things other than the form of the statement. The following is an example of a contingent statement form, and its corresponding truth-table:

Note that under the bracketed main connective there is a mixture of true and false truth-values, which means that the statement form is contingent.

Logic Post 18

Biconditionals

Two statements are materially equivalent if they have the same truth-values. The symbol ≡ is the symbol we are using in this thread to stand for material equivalence. Thus, if we say that "Two times two equals four, if and only if, four times one equals four," then the two statements are materially equivalent, since both have the same truth-value.

Material Equivalence

p ≡ q

Truth-table
Given the definition of material equivalence - that two statements are materially equivalent if and only if they have the same truth-values - we can see this is so by looking at line one and line four.

These compound statements are more commonly referred to as material biconditionals or just biconditionals, because they are equivalent to material conditionals. For example, the material biconditional "Two times two equals four, if and only if, four times one equals four" in symbolic form looks like this, p ≡ q; and it is equivalent to the two-directional conditional "If two times two equals four, then four times one equals four, and if four times one equals four, then two times two equals four," which is symbolized as, (p ≡ q) · (q ≡ p).

Remember that the material biconditional only captures the minimal truth-functionality of the English biconditional. There is no connection implied by the component statements. It only states that they both have the same truth-value.

Logic Post 17

There is much that can be accomplished using the ⊃ symbol. For instance, using the definition of the ⊃ symbol we can get the valid argument form known as Modus Ponens.

Modus Ponens

p ⊃ q
p
_____
∴(Therefore), q

How do we know that this necessarily follows? Because using our truth-tables we know that any instance where the antecedent is true, the consequent is true. Hence, using Modus Ponens, we have constructed a valid argument form. Keep in mind the differences between validity and soundness, which we discussed earlier.

Another valid argument form that follows from the definition of the symbol ⊃ is called Modus Tollens, i.e., if we deny the consequent, we can conclude the denial of the antecedent.

Modus Tollens
p ⊃ q
~q
_____
∴ ~p

There are two corresponding fallacies that are derived from the Modus Ponens and Modus Tollens. First, the valid form...

Modus Ponens
____________
p ⊃ q
p
_____
∴ q

"If we have desegregation we will have some busing.
We have desegregation.
______________________________________
Therefore, we will have some busing(Kegley and Kegley, p. 240)."

The invalid form is the following:

p ⊃ q
q
_____
∴ p

If we have desegregation we will have some busing.
We have some busing
__________________________________
Therefore, we have desegregation.

The above invalid form commits the fallacy of affirming the consequent. We know this because the definition of a conditional for our purposes, states that we cannot have false consequent when the antecedent is true. We can see this in line two the following truth-table.

_______________________________________________________

The fallacy that corresponds to Modus Tollens is the fallacy of denying the antecedent. Let us first look at the valid form...

Modus Tollens
____________

p ⊃ q
~q
______
∴ ~p

"If the paper burns, there is sufficient oxygen present.
There is not sufficient oxygen present.
__________________________________________
Therefore, the paper does not burn (Kegley and Kegley, p. 240)."

This is obviously a valid form, however, if we deny the antecedent, then we commit the fallacy of denying the antecedent.

Fallacious Form
_____________
p ⊃ q
~p
______
∴ ~q

If the paper burns, there is sufficient oxygen present.
The paper does not burn.
__________________________________________
Therefore, there is not sufficient oxygen.

We can obviously see that this could be false, because it is certainly possible that oxygen is present and the paper still will not burn. Maybe the paper is wet, or maybe there is not enough oxygen.

Logic Post 16

Conditional Statements and Material Implications

A conditional statement is a statement that is composed of two component statements joined by the truth-functor if...then (⊃). These statements can also be referred to as hypothetical statements or material implications. Speaking of implication keep in mind that the English word implies more than one meaning, and these meanings can be conveyed using the connective if...then. Statements using the aforementioned connective can imply a logical, causal, definitional, or decisional relationship.

The component statements that make up the conditional are called the antecedent and the consequent. The antecedent precedes then, and the consequent follows then in the conditional. A simple conditional statement that implies a logical relationship is the following:

If John is a philosopher, then John is a thinker.

p = John is a philosopher.
q = John is a thinker.

p ⊃ q

The connective if...then is defined using truth-tables in the following way: If a substitution instance of p or any other variable is true, and a substitution instance of q or any other variable is false, then the substitution instance of p ⊃ q is false. For all other substitution instances of p and q, the statement form p ⊃ q is true. This can be clearly seen in the example that follows:

p--------q------------p ⊃ q
___________________
T--------T---------------T
T--------F---------------F
F--------T---------------T
F--------F---------------T

Therefore, the preceding truth-table says that it is never the case that p can be true and q false, which is to say, that it denies the conjunction of its antecedent with the negation of its consequent.

Conditional statements are unlike conjunction and disjunction in that the order of the statements in a conditional make a difference when constructing truth-tables. This can be clearly seen in the following instance:

If the car runs out of gas, then the car will stop running.
If the car stops running, then the car runs out of gas.

What falsifies each of these statements is different, i.e., in the first statement p ⊃ q, when p is true and q is false we get a false statement, but if we reverse the p and q, as in q ⊃ p, we get a false statement when p is false and q is true. Hence, the truth-tables will look like the following:

p ------- q ------------ p ⊃ q ------------ q ⊃ p
_________________________________
T---------T ---------------- T ----------------T
T -------- F ---------------- F ---------------T
F -------- T ---------------- T ----------------F
F -------- F ---------------- T ----------------T

If you look at the main connectives in each of the conditionals you can see that what falsifies one does not falsify the other.

Remember that in a conditional statement the antecedent implies the consequent, which means that if the antecedent is true, then the consequent is true. However, this is hypothetical, and as such, the conditional statement does not tell us anything about the truth of its component statements; it is only saying that IF the antecedent is true, the consequent is true.

There are more complicated issues when it comes to conditional statements. For instance, "If John or Mary go to the movies, then Mary or John will go to the movies", is a logical relationship. However, in the statement "If John goes to the movies, then Mary will go to the movies," is a factual relationship.

The most important way in which conditional statements differ, has to do with their truth-functionality, that is, most conditionals uttered in out daily lives are not as straightforward. Consider the following:

"If the Red Sox beat Pittsburgh, then the Red Sox will win the World Series" - symbolized R ⊃ W. Now let us suppose that someone places a bet that this conditional is true. We know that if the Red Sox beat Pittsburgh, and yet the Red Sox fail to win the World Series, then obviously R is true and W is false, and the person fails to win the bet. On the other hand, if the Red Sox beat Pittsburgh, and they also win the World Series, then the statement is true and the person wins the bet. Now let us look at this in a standard truth-table where the two results are represented.

R ------- W -------------R ⊃ W
_____________________
T -------- T -----------------T
T -------- F -----------------F
F -------- T
F -------- F

A problem arises if the Red Sox fail to beat Pittsburgh, because then it is not clear what to say about this conditional as it relates to the last two results. However, we cannot simply leave the last two results blank even though it would be odd to call the statement true, it would definitely not be false. Hence, when faced with a choice (for our purposes) we will construct our truth-tables so that all statements with false antecedents are true.

A conditional statement, as currently defined, provides us with a minimal common meaning for uses of the "if... then" statement, which once again means that the consequent cannot be false if the antecedent is true. So the point is, that since this is a minimal condition for the meaning of a conditional statement, it only partially satisfies the uses of the "if...then" statement in English, just as the disjunction only partially fulfills all the meanings of or in English.

Finally, it should be noted that there are other more powerful forms of logic that are better equipped to handle these kinds of problems. However, it is a good idea to get a good handle on sentential logic first before going on to master other forms of logic (like quantification theory).

I have not completely analyzed conditional statements and material implications. If you want a more complete analysis, you'll have to do some research. This thread is a guide, nothing more.

Logic Post 15

Disjunction

We will now consider a statement joined by the truth-functor symbol v. When two statements are connected by the truth-functor or it is called a disjunction. Each component statement is called a disjunct. An instance of a disjunction is: "Either Plato was a philosopher, or he was a physician." This disjunctive sentence is symbolized as:

p v q

When we state a disjunct we are putting forth two possibilities. These two possibilities have two senses - one is called inclusive, and the other is called exclusive. The inclusive sense is when we are admitting the possibility that each of the disjuncts may be true. The inclusive sense is represented by the following truth-table:

p--------q--------p v q
_______________
T--------T----------T
T--------F----------T
F--------T----------T
F--------F----------F

As we can observe by this truth-table the only case in which the inclusive disjunction is false is when both disjuncts are false.

As for the exclusive sense of the disjunction, this is a case where we may want to rule out both disjuncts as true. As in the following example:

George is in France, or George is in Italy.

This is clearly an exclusive sense, because George cannot be in both places at once. The exclusive sense says either p is true or q is true, but not both. However, this sense is not used in truth-functional logic, although, the two senses share a common trait, that is, that at least one of the disjuncts must be true (Kegley and Kegley, Introduction to Logic, p. 232).

Given the definition of a disjunctive statement, that it is false only when both disjuncts are false, and true in each of the other three alternatives, we get the following two valid disjunctive argument forms:

p v q
~p
____
Therefore, q.

p v q
~q
____
Therefore, p.

These are called disjunctive argument forms, and they are another one of the rules of inference that you will learn later.

There is an interesting case of the disjunction involving a negative, which can be written in two ways. The first is ~(p v q), and the second is (~p · ~q) - these are equivalent forms, and can be seen as such in the following truth-tables.

p--------q--------p v q-------- ~(p v q)
_________________________
T--------T----------T----------------F
F--------T----------T----------------F
T--------F----------T----------------F
F--------F----------F----------------T


p--------q-------- ~p-------- ~q-------- ~p · ~q
________________________________
T--------T----------F-----------F-------------F
T--------T----------T-----------F-------------F
T--------F----------F-----------T-------------F
F--------F----------T-----------T-------------T

Those of you who remember Ephilosopher (the philosophy forum), most of this comes from a thread I did back then on logic. I'm revising parts of it as I go along, but much of the work is already done. This is why I'm able to post so fast. I'll also be adding to the original thread, so that part will take more work.

Logic Post 14

Conjunction

When two sentences are joined together by the truth-functor and, they are called conjuncts; and compound sentences formed by the truth-functor and are called conjunctions. An example of a simple conjunction is "Wittgenstein was a philosopher, and he was also an engineer"; and since we are using the truth-functor symbol · this sentence would be symbolized in the following way:

Using p to refer to "Wittgenstein was a philosopher", and q to refer to "He was also an engineer" - the sentence would be symbolized p · q.


There are a number of ways to express a conjunction in English.

1) "Mathew stays but Jane leaves."
2) "Mathew stays, however Jane leaves."
3) "Mathew stays, moreover Jane leaves."
4) "Mathew stays although Jane leaves."
5) "Mathew stays yet Jane leaves."
6) "Mathew stays even though Jane leaves."

The previous six examples taken from Kegley and Kegley's book, Introduction to Logic, p. 228.

In order for someone to commit himself to the truth of a conjunctive statement, that person would have to accept that both p and q are true. Otherwise, the conjunctive statement is false. This is clearly seen in the following truth-table:

p--------q--------p · q
_______________
T--------T----------T
F--------T----------F
T--------F----------F
F--------F----------F

As can be seen in this truth-table a conjunctive statement is only true if both of its component statements are true. So, what this means is that if we are committed to the truth of p, and committed to the truth of q, then we are committed to the truth of p · q. The argument p · q is therefore valid for the conjunctive form.

p
q
__
Therefore, p · q.

Later you will come to know this as one of the rules of inference known as conjunction.

Logic Post 13

Continuing with truth-tables

Negation

The · symbol that is used as a sentence connective is a kind of operator, that is, it can operate on two separate sentences to produce a third compound sentence A · B. For instance, the operator "It is well known that" operates on the following sentence "Abraham Lincoln was the sixteenth president of the United States" to construct the compound sentence "It is well known that Abraham Lincoln was the sixteenth president of the United States".

The tilde symbol ~ that is used to symbolize negations is an operator of this kind, i.e., it can generate a new sentence out of just one starting sentence. The negation operator is the only one that does this in standard sentential logic. All (or almost all) others including "It is well known that" are non-truth-functional operators.

Negation is one of the easiest truth-functional operators to learn, because it only operates on individual sentences. The operation of the negation symbol is straight forward, because if you negate a true sentence, you get a false sentence, and if you negate a false sentence, you get a true one.

We negate sentences in English in a variety of ways. Examples are given in the following sentences:

1) "Eleven is not even."
2) "Eleven is uneven."
3) "It is not the case that eleven is even."
4) "It is false that eleven is even."
5) "Eleven is odd."

The above five examples were taken from Kegley and Kegley's Introduction to Logic p. 226.

Finally, let us use the following truth-table to define the truth-functor negation:

The statement "The earth has one moon" has two possible truth-values, and the statement "It is not the case that the earth has one moon" has two possible truth-values. Hence, letting p stand in for each of the aforementioned statements we get the following truth-table:

p------- ~p
_______
T---------F
F---------T

There can also be statements that involve more than one negation. Consider - "It is false that it is not the case that Abraham Lincoln is not tall." Let p represent "Abraham Lincoln is tall" and we will construct the following truth-table.

p--------- ~p---------- ~~p---------- ~~~p
____________________________
T-----------F--------------T----------------F
F-----------T--------------F----------------T

It is best when using the negation symbol to express it by the statement, "It is not the case that," which reverses the truth-value of the statement. While it is true that the ~ symbol is equivalent to most English uses of the word not, it doesn't always convey the correct meaning. In logic, ~ always means contradictory. However, there are uses of the word not in English that do not convey a contradiction. For example, "Some males do not smoke pot" does not contradict the statement that some males do smoke pot. In other words, both statements can be true, so the not in this case doesn't involve a contradiction.

I'm skipping over categorical deductive logic, which was originated by Aristotle in the Organon.

Logic Post 12

Continuing with symbolization...

What is a statement-form? A statement-form is a proposition that consists of logical symbols (∨ · ⊃≡ ~) and statement variables (p, q, r, etc.). For example,

p · (q ∨ r), which can be derived from the statement "If William is a liar, then either he is stupid or he is crazy."

Any substitution instance of a statement-form is any statement with that form. For example, the substitution instance of ~ p would stand in for any statement with the form "It is not the case that George Washington was our 4th president." Statement forms (~ p) are not intrinsically true or false, only substitution instances are true or false. In other words, only where ~ p represents a particular statement, is it said to be true or false.

Truth-Tables

Truth-tables allow us to determine the truth-values of a particular statement given certain input values. For example, if we allow the letter p to serve as a marker for a given statement, and let the letter T stand for its truth-value true, and the letter F for the truth-value false, we can then show that any simple statement has two possible truth-values.

The following is an example:

p
---
T
F

Any compound statement p and q has only four possible sets of truth-values. The following is an example:

p-----q
_____
T-----T
F-----T
T-----F
F-----F

If we used truth-functors that involved three different statements, then we would need three statement-variables (p, q, and r). In the above example using two statement-variables it required four lines. If we used three variables it would involve eight lines. The following is another example:

p-----q-----r
_________
T-----T-----T
T-----T-----F
T-----F-----T
T-----F-----F
F-----T-----T
F-----T-----F
F-----F-----T
F-----F-----F

As you can see this can be quite cumbersome, because a table with one variable has 2^1= 2 lines; a table with two variables has 2^2 = 4 lines; a table with three variables has 2^3 = 8 lines; and so on.

Thanks.

Ya, well, we would disagree there.

Yes, that's true, and I mentioned this above to Harry. You can always rephrase a question and turn it into a statement. However, the question by itself is not true or false.

You should start your own thread on why questions should be considered propositions. The question is settled for me, I'm not going to debate it.

Logic Post 11

Before I go any further, I should give some of the symbols used in logic, and their meanings. I don't want to assume that everyone knows the symbols. I'm going to use the symbols used in the Principia Mathematica. However, Hilbert's symbols are probably used more often by mathematicians.

Negation (not) ~
Conjunction (and) ·
Disjunction (or) ∨
Material Implication (if, then) ⊃
Material Equivalence (if and only if) ≡
Therefore ∴

Why are symbols used? Logicians probably wanted a language that was free (as free as possible) from the abiguity, vagueness, and some of the defects of language. It was also a way for logicians to demonstrate the logical structures of statements/propositions.

To designate statements abstractly we will be using p and q as markers, i.e., p and q mark the position of statements. Next, we need symbols for truth-functional connectives, which I gave above. For example, "We will be buying a home and we will be buying a car next week." The individual statements are:

We will be buying a home. (We symbolize the first statement using p.)
We will be buying a car next week. (We symbolize the second part of the statement using q. If there were more statements involved, we would continue using r, s, t, etc.

The truth-functor: and
The statement form: p and q
It's symbolized using the symbol for and. p · q

Logic Post 10

There are two kinds of arguments in logic, deductive and inductive arguments. We will first discuss deductive arguments.

Deductive Arguments

A good deductive argument must be

(1) valid
(2) sound
(3) cogent

As I mentioned already, validity is a quality of good deductive arguments. Validity means that the form of the argument forces you to the conclusion. The correct form is crucial. Therefore, if the evidence is accurate, then the conclusion must follow. Note the following forms:

Premise 1: All X are Y.
Premise 2: b is an X.
Conclusion: Hence, b is a Y.

An argument of this form will lead you to a conclusion that is true provided the evidence, which is in the form of premises, are true. The following is an argument using the above form.

Premise (1) All cats are animals.
Premise (2) Morris the Cat is a cat.
Conclusion: Hence, Morris the Cat is an animal.

The following are more examples of valid deductive argument forms.

Modus Ponens: If X, then Y.
----------------------X.
----------------------Therefore, Y.

The following is an argument using this form:

Premise (1) If George is human, then George is a person.
Premise (2) George is a human.
Conclusion: Hence, George is a person.

Modus Tollens: If X, then Y.
-------------------- ~Y.
---------------------Hence, ~X.

Example:

Premise (1) If Harry is a cat, then Harry is an animal.
Premise (2) Harry is not an animal.
Conclusion: Hence, Harry is not a cat.

There are other deductive forms. For instance,

Hypothetical Syllogism: If X, then Y.
---------------------------------If Y, then Z.
---------------------------------Hence, if X, then Z.

Transposition: If X, then Y.
--------------------Hence, if not Y, then not X.

So, again, validity has to do with the structure or the form of the argument


The next criteria of a good deductive argument is soundness, which means that the argument is valid, plus the premises are true. The following argument is valid and sound.

Modus Ponens:

If I think, then I exist.
I do think.
Hence, I exist.

The next argument is valid, but not sound.

Modus Ponens:

Premise (1) If humans are dogs, then dogs are humans.
Premise (2) Humans are dogs.
Conclusion: Hence, dogs are humans.

This is a valid argument, but is it sound? No. It is not sound, because the premises are not true.

Our third criterion is cogency.

Now there are going to be some that disagree with this criterion. However, I believe it is very important.

Cogency means that the premise's of a deductive argument are known to be true by the person to whom the argument is given. What this means is that not only is the argument sound, but the premises are known to be true. There are proofs, i.e. deductive arguments that are sound; however, you don't know they are sound, because you don't know if the premises are true. The following is an example:

"The base of a souffle is a roux.
This salmon dish is a souffle.
Hence, the base of this salmon dish is a roux.
(Dr. Byron Bitar)"

Therefore, in order for a proof to work for you, you have to know the premises are true. If you do not know the premises are true, it will not convince you, even if the argument's conclusion is true.

These posts may help some of you with the basics in logic, but it's not like taking a course. However, you don't need to take a course in logic to become good at evaluating arguments, you just need a lot of practice. Logic isn't difficult, at least the basics aren't difficult, you just need lots of practice. Moreover, reading a logic text isn't enough, no more than reading a math book is enough to learn math.

Logic Post 9

Logical vs Non-logical Material

One of the problems in analyzing many arguments is separating the logical material from the non-logical material. Keeping in mind that logic is concerned with the informative side of language, i.e., with what is being asserted. You need to be able to distinguish between the emotive content and factual assertions; and to be able to translate emotive content into neutral content. Consider the following two propositions:

1) "Trump is a liar!"

2) "Trump was mistaken."

The first statement is likely to be from someone with a negative attitude, while the second one might be from someone with a more positive attitude. We are not concerned with the attitudes of people. We are more concerned with the factual content. Expressions of attitude indicate a positive, negative, or neutral evaluation of someone or something. As we said earlier in the discussion we want to focus on the cognitive use of language as opposed to the evaluative use.

You should get some practice reading articles and picking out and separating propositions into the five general categories that we have discussed.

I will conclude this section with the three basic kinds of disagreements.

There are disagreements in attitude, in belief, and verbal disagreements.

Ones attitude has more to do with one's state of mind or feeling about an event or fact, and less to do with what is claimed or asserted.

Disagreements about beliefs, on the other hand, are arguments over the supposed facts. These can be classified in two ways. First, the disagreement can be a real disagreement, in that there is a logical inconsistency in one of the arguments. Second, there can be an apparent disagreement, i.e., both arguments are consistent, which means the arguments can be resolved, at least in theory.

Finally, there can be a verbal disagreement. That is to say, the people arguing are using words or phrases with different meanings.

Buy a used logic book, that should help. Here is a cheap used one.

https://www.thriftbooks.com/w/introduction-to-logic_jacquelyn-ann-k-kegley_charles-william-kegley/2627456/item/2242315/?mkwid=%7cdc&pcrid=395840866446&pkw=&pmt=&slid=&plc=&pgrid=83212263032&ptaid=pla-439650351594&gclid=CjwKCAjwnIr1BRAWEiwA6GpwNVPQBkzwqUEAh8VRv7zy1SrcrnRWQ5SsxUCsrNwIFoyhPzE8pHi4HhoCv8MQAvD_BwE#isbn=0675083583&idiq=2242315

Logic Post 8

The five general categories of language are the following:

1) Cognitive (Informative) Function.

Language is used to convey information. As the following statements demonstrate.

"There are two desks in my room."

"The Japanese bombed Pearl Harbor on December 7th, 1941."

"Triangles have three sides."
(That triangles have three sides is also an example of an analytic statement. An analytic statement is one in which concept of the predicate is included in the concept of the subject.)
"Bachelors are unmarried men."
"All bodies are extended in space."
"All wives are female."

The one characteristic of these statements is that they can be spoken of as either true or false, i.e., they declare that something is or is not the case. That is not to say that other criteria cannot be applied to statements of information, since one can also ask if the statement is significant or not, or one can ask if it is useful or not.

Logic is not concerned with establishing truth or falsity. Logic simply asks if the conclusion follows from the truth of the premises. Another way to put it is that logic is concerned with the internal relationship between or amongst propositions.


2) Expressive
In these examples language is used to express feelings or emotions.

"I am having a great time at the beach."

"The portrait is beautiful."

"You idiot."

The important point here is that these statements express a feeling, emotion, or an attitude. These phrases are also considered evaluative, i.e., they reveal a positive or negative judgment by the speaker.



3) Evocative or Directive
Language is also used to arouse feelings, emotions, attitudes, and certain kinds of responses or actions in others. Examples of these kinds of statements are as follows:

"Duck!"

"Attention!"

"Please wash your hands before eating."

"Brush your teeth three times a day."

"John Doe for president."

These statements are a bit different from the ones I gave earlier, in that they are designed to produce an effect or an action from two perspectives: First, the purpose or purposes of the user of the sentences, and two, the effect the user of language wishes to have or not have ( Introduction to Logic, Kegley and Kegley, p. 34).

It is important in the study of logic to distinguish between informative statements and evocative statements. After all logic is concerned with what is being asserted, not with how it is being asserted. As Spock might say, emotional appeals are irrelevant.

Now I am not saying that we should eliminate all emotion from our language, but only that we should be careful when formulating an argument that we do not include appeals to emotion, and that we stay away from personal attacks.


4) The Evaluative Use of Language

The complexity of the language used to make 'value judgments' is mind-boggling. The contexts of such language includes just about every context imaginable. As you can imagine there is still much disagreement over how to characterize such language. In this brief introduction I cannot even begin to do justice to this use of language, so I will only make a few remarks.

As you probably already know evaluative language makes judgments of what is of value, i.e., what is good, just, and beautiful. All you have to do is to look at some of the arguments on the internet, and you will see the wide variety of views in relation to ethics, religion, politics, language, etc. Some people believe that value judgments are subjective, while others believe they are objective, and still others hold some middle ground. Some examples are as follows:

"Knowledge is good in and of itself."

"Trump is stupid."

"This is a good book."

"This song is beautiful."

"The Iraq war is just."



5) Finally, the Ethical and Aesthetic (pertaining to a sense of beauty, or having a love of beauty) use of language.

This use of this kind of language raises very important questions about how to interpret statements like, "Torture is wrong." Is it merely expressive, which translates into something such as, "I do not like torture" or "Torture - yuck!" Can it be that these statements are simply directive in nature - for example, "Do not torture!" And finally maybe statements like "Torture is wrong" are assertive-type statements that require us to give good reasons for accepting them.

These are very controversial topics, and will not be settled in my short musings.

I came across a brilliant paper published in a philosophical journal. I was thinking of the reasons behind philosophical disagreements and why there isn't some sort of consensus among philosophers regarding philosophical ideas. For anyone interested, l have attached a link to the article written by Prof Christopher Daly. — Wittgenstein

This problem has always fascinated me. The answer, I believe, lies in the complexity of language, the complexity of the human condition, psychology of belief, causes versus reasons for belief, intelligence (ability to reason), etc, etc. I don't think there is a way to solve this in the near future, maybe in the distant future. if we gain the ability to communicate mind-to-mind, it might clear up some of the fog.

Logic Post 7

Dimensions of Language

Since logic is concerned with both communication and understanding it will be important to sort through some of the functions of language. This will enable us to focus on the actual argument being presented; and it will help us to seek clarity and precision.

Formal analysis of an argument is not an easy task, since arguments in everyday life are rarely put in a form that is easy to analyze. However, if we want to think rationally we need to be able to think clearly about what we intend to say; and once we know what we intend to say, then we can concentrate on saying it well.

Keep in mind that language can also be used to persuade without concern for rational arguments. We see this all the time. Sometimes people will appeal to opinions, prejudices, and emotions without concern for rationality.

Remember some of the distinctions that I have already pointed out about sentences, and the different ways in which they can be used. Not all sentences make statements. For instance,

1. Is your name John?
2. Stand there!
3. Please don't do that.

These interrogative (sentences that ask a question) or horatory (sentences that exhort or encourage) sentences are not the kind of sentences we will be concerned with. We will be concentrating on statements or propositions that are declarative. For example, "The moon is approximately 240,000 miles from earth", or "It is snowing." What sets these sentences apart from the ones listed above is that you can properly ask about their truth or falsity.

Hence, logic again is concerned with statements or propositions, and thus with sentences that assert or deny something.

Language is very flexible, and in spite of the fact that language has many functions it can probably be classified under five general categories. The philosopher Ludwig Wittgenstein identified many of the uses of language in the Philosophical Investigations. The following are some examples:

Giving orders, and obeying them.
Describing the appearance of an object, or giving its measurements.
Constructing an object from a description (a drawing).
Reporting an event.
Speculating about an event.
Forming and testing a hypothesis.
Presenting the results of an experiment in tables and diagrams.
Making up a story; and reading it.
Play-acting.
Singing catches.
Guessing riddles.
Making a joke; telling it.
Solving a problem in practical arithmetic.
Translating from one language into another.

We are not talking about meaning, but whether the sentence can be said to be true or false. A question just isn't considered a proposition that asserts that something is or is not the case.

It seems that we can assert that something is the case in each of these examples:
1. That the person doesn't know who the third president of the United States is.
2. That the person wants us to be seated.
3. That the person wants us to keep quiet. — Harry Hindu

The point is that the question "Who is the third president of the United States?" is not a true or false statement/proposition. It doesn't make sense to say it's true or false. All you're doing is drawing an inference based on the question. That inference, may be true or false, but you're changing the sentence in order to do that.

Logic Post 6

Here are five steps to help you analyze an argument.

1) First find the conclusion, that is, what is the point of what is being claimed.
2) Once you have located the conclusion, then locate the supporting data - the reasons or evidence given in support of the conclusion.
3) Next rule out repetitious statements and emotional content.
4) If there are missing pieces of evidence, then supply what is needed to make the argument a good one. Ask yourself what additional evidence is needed for support. You may also need to ask yourself - what must be assumed in order for the conclusion to follow.
5) Finally, you may also what to look for additional arguments within the context of the main argument. That is, there may be smaller arguments within the larger context of the statements.