So, if we had 5 people who had a negative emotion and 5 people who had a positive emotion, would your post be good or bad? Or, would it be both good and bad?

The only thing that counts is how I feel, i.e., what emotion I feel. It's my emotions that determine what's good and bad. My emotional response is negative, therefore it is bad.

I had nothing but negative emotions while reading your OP, so it must be bad.

"Tim likes apples" is not objectively true? Isn't it a fact that Tim likes apples? Are you saying there are subjective facts? — jamalrob

No, it is not objectively true, it is subjectively true, it is dependent on the subject (the person). Yes, I am saying there are subjective facts.

And if "Tim likes apples" is subjectively true because it's "dependent on Tim for its truth or falsity", then "Slavoj Zizek is a philosopher" is also subjectively true, because it's similarly dependent on Slavoj Zizek. — jamalrob

Yes, that's true. However, I'm not saying the lines at times don't get blurred between these concepts. Many concepts are like this. However, when I say subjective, I'm referring to those things that are mind-dependent. For example, the apple that Tim likes is objective (mind-independent), but his likes and dislikes are mind-dependent.

I imagine you might go on to say that subjective truths are only about a person's "taste, likes or dislikes, etc", as if those were something private and inaccessible. But those things are expressed in a person's behaviour: Tim's taste for apples can be seen in his excessive consumption of apples, and Zizek's taste for thinking about Hegel is expressed in the fact that he's a philosopher who writes about Hegel. — jamalrob

No, the fact that they are subjective doesn't mean they are private and inaccessible. We can observe Tim's likes and dislikes based on what he does. He interacts with the objective apple, but this interaction reveals something about his personal tastes. There is a component of objectivity and a component of subjectivity to his actions.

For me it is clear that there are subjective truths and objective truths. There are contingent truths and necessary truths. I have no problem dividing these up.

Did anyone say what "objective truth" is? It seems to me that before anything can be said categorically, the thing spoken of ought to be reasonably well understood. I do not understand what is meant by "objective truth." Anyone take a moment and straighten me out? — tim wood

I'm taking your question to be tongue-in-cheek.

Objective truth should be contrasted with subjective truth. Objective truths are quite mind-independent. For example, the Earth has one moon reflects a state-of-affairs that exists apart from any mind. In other words, one could eliminate all minds, and the fact would still obtain. There might not be anyone around to apprehend the objective truth, but the fact would still exist.

Subjective truths, on the other hand, are mind-dependent. For example, "Tim likes apples," is dependent on Tim for its truth or falsity, i.e., it is either the case that Tim does or does not like apples. The truth of the statement, for Tim, is subjective, dependent on the subject, his taste, likes or dislikes, etc. Eliminate all minds and you eliminate all subjective truths.

The structure as I see it, consists of the world, minds, and language; and the relationship between these three. The world is the backdrop, and we find ourselves existing in it. Our mind helps us to interpret the world (it's in the relationship between our minds and the world that bedrock or basic beliefs form); so in a sense our mind is the center between the world and language. — Sam26

Keep in mind that I am not trying to give a Wittgensteinian view, although I am using some or many of Wittgenstein's ideas, i.e., I am trying to expand on the idea of bedrock beliefs.

I asked in the OP, "What is the structure of our beliefs?" In a later post I suggested that the structure consisted of three things (the world, minds, and language). There is a kind of scaffolding of beliefs that takes place between these three. However, if we look at history, the world and minds come first, i.e., before language (at least before a sophisticated language). This is why I have talked about prelinguistic beliefs, which I believe is essential to understanding bedrock beliefs, at least prelinguistic bedrock beliefs. My idea of beliefs is that beliefs extend beyond language, and that the best evidence of the existence of such beliefs is in our actions. Our actions show what we believe. If I open a door, this action, shows that I believe there is a door there. Our actions are a constant source of what we believe apart from statements or propositions. Wittgenstein suggests this very thing, "...we can see from their actions that they [primitive man] believe certain things definitely, whether they express this belief or not [my emphasis] (OC, 284)."

If we want a good representation of what a prelinguistic bedrock belief looks like, then we simply need to look at those actions that express beliefs apart from statements or propositions. Almost any interaction with the world (the background) shows that we believe certain things. Even moving from point A to point B within the world shows that we believe in certain relationships between ourselves and the world.

What is the significance of these beliefs? The significance is that these beliefs arise quite apart from any conceptual or linguistic framework, which, I believe, is partly why Wittgenstein imparts a certain status to Moorean propositions. The belief, "This is my hand," although expressed linguistically using a statement, arises quite apart from language, i.e., my actions everyday demonstrate or show my belief that I have hands. A further point of significance is that such beliefs (bedrock) are outside any epistemological justification. Why? Because these kinds of beliefs arise quite apart from epistemology. Epistemology is linguistic, and the concepts used in epistemology are linguistic. There is no sense of justification when I open the door, I just do it. Many actions are like this, especially when looking back before language.

Wittgenstein talks about bedrock or foundational beliefs within the scope of language (for the most part), and it is with this scope that many other kinds of bedrock beliefs are formed. For example, the game of chess is based on rules, and these rules are foundational to the game. However, the difference between these foundational or bedrock rules, and the ones I demonstrated above, is that the rules of chess are expressed in language. The context drives the difference between these kinds of bedrock or foundational beliefs. This is partly what I mean by a scaffolding of these beliefs.

Logic Post 34

Fallacies Continued...

Fallacies of Neglected Aspect:

1) Hasty Generalization: One reaches a conclusion based on very little evidence (insufficient statistics), or one reaches a conclusion based on an atypical sampling (biased statistics).

2) False Cause: assuming something is the cause of X when it is not, or if it is the cause, it is only one of the causes.

3) Accident: one ignores an exception to a generalization. A generalization becomes true by excluding counter-examples or counter-evidence (no true Scotsman or the self-sealing argument).

4) Black and Whtie: one commits this fallacy when one accepts false alternatives. In other words, one reduces the alternatives to either X or Y, not allowing the possibility of other outcomes.

5) The Beard: One assumes that because it is difficult to draw a distinction, that no line or distinction can be made.

Logic Post 33

Inductive Reasoning:

Inductive arguments do not guarantee the conclusion, as do deductive arguments. If a deductive argument is sound (i.e., it is valid and the premises are true), then the conclusion follows with logical necessity. That is to say, if the premises are true and the deductive argument is valid, then the conclusion must be true. However, an inductive argument does not guarantee the truth of the conclusion, i.e., it goes beyond the evidence given in the argument to make a new assertion of knowledge. Since inductive arguments advance beyond the evidence to make new claims, the claims, or the conclusions can only be probable. So, it is in this sense that inductive arguments amplify what is contained in the evidence, or the premises. Thus, instead of speaking of inductive arguments as true or false, we say that they are strong or weak based on the strength of the evidence.

Good inductive arguments must include the following:

1) number
2) variety
3) scope of the conclusion
4) truth of the premises
5) cogency

Number refers to the cases cited in the premises, the greater the number the stronger the conclusion. High numbers do not necessitate the conclusion. High numbers only make it more probable that the conclusion follows. For example, compare two witnesses seeing Mary shoot John, as opposed to five witnesses seeing the same event. All things being equal, the latter is stronger than the former.

Variety refers to the variety of cases cited in the premises, i.e., the greater the variety, the stronger the conclusion. For instance, if we have five witnesses see Mary shoot John from one vantage point, i.e., all standing in the same place, it is not as strong as having five witnesses see the same shooting from five different vantage points.

Scope of the conclusion refers to how much your conclusion claims. The more the conclusion claims, the weaker the argument, the less the conclusion claims the stronger the argument. So, the more conservative your conclusion, the stronger the argument.

Truth of the premises obviously means that the supporting evidence must be true. Note that this is also true of deductive arguments. It goes without saying that if your evidence is not true, then the argument is suspect, to say the least.

Finally, cogency, viz., the argument's premises are known to be true by those to whom the argument is given. Any argument will be strengthened if the people to whom the argument is given know or agree that the premises are true.

Logic Post 32

Fallacies continued...

We have already discussed some fallacies that can take place in valid deductive forms, but there are many other kinds of fallacies that fall under general types. For example, there are fallacies of irrelevance, i.e., fallacies that are not relevant to the conclusion of the argument. Instances of such fallacies would include the following:

Appeal to Force: Accept the conclusion or else.
Appeal to Pity: This is an appeal to emotion.
Appeal to the People: It is an appeal to prejudice and majority opinions.
Against the Man: It directs the argument against the person giving the argument, rather than the argument itself.
Appeal to Authority: It is an appeal to an authority, viz., when the authority is not an expert in the field.
Irrelevant Conclusion: It is when the premises are not relevant to the conclusion, and in fact, may support a completely different conclusion.
Red Herring: This is used to steer the argument away from the main thrust of the argument.
From Ignorance: This fallacy draws a conclusion based on the absence of evidence. In other words, it assumes falsely that because there is no proof, then this proves something either true or false.

I will add to the list of fallacies as I go along, since there are literally hundreds of fallacies. However, next I will be saying something about inductive reasoning.

Logic Post 31

We have already talked about how invalid formal arguments are fallacious. We also mentioned that fallacies go beyond the scope of validity, i.e., a deductive argument can be valid and still be fallacious. How? First, if the premises used in a valid formal argument are contradictory, then validity would be useless in establishing the truth of the conclusion. So, based on the fallacy of inconsistency the argument would fail.

The second way a valid argument can be fallacious has to do with a case of begging the question. This means that the conclusion is simply a restatement of what is already assumed in the premises. You are not proving anything, if you are repeating your premises in the conclusion.

The point is that validity is a formal property of deductive arguments, and that it alone does not guarantee that the formal argument is not fallacious. Moreover, these two examples, are examples of informal fallacies. Informal fallacies involve considerations other than validity.

Logic Post 30

Fallacies:

There are two kinds of fallacies, formal and informal. Formal fallacies are associated with deductive argument forms, i.e, they are invalid forms of deductive arguments. In other words, whether a deductive argument is valid or not is partly what determines if it is fallacious or not. The reason that validity only partly determines whether a deductive argument is fallacious, is that the idea of a fallacy is much broader in scope than validity. So, although invalidity is enough to determine that a deductive argument is formally fallacious, it is not the sole criteria by which we determine if the argument is fallacious. Remember that formal fallacies are just a subset of all fallacies.

We know that Modus Ponens and Modus Tollens are valid deductive forms.

Modus Ponens:
p ⊃ q
p
∴ q

An invalid form of this argument is known as affirming the consequent:

p ⊃ q
q
∴ p

Thus, this invalid form is what makes it fallacious. Another example of an invalid form is seen using Modus Tollens.

Modus Tollens:
p ⊃ q
~q
∴ ~p

The following is an invalid form, called denying the antecedent:

p ⊃ q
~ p
∴ ~q

Again, any invalid form of a deductive argument is considered a formal fallacy.

Logic Post 29

It's best when constructing a formal proof to find the general form of the argument, and not let the complexity of the proof confuse you. Next, you want to look for propositions that occur in the premises, but not in the conclusion. Propositions that do not occur in the conclusion may be extraneous to the conclusion. Third, it's best to break down compound statements into their various parts, because it's much easier to work with singular statements.

Now let us consider an example:

1)
[A · (B v C)] ⊃ [(D v E) ⊃ (F ⊃ G)]
~[(D v E) ⊃ (F ⊃ G)]
∴ ~[A · (B v C)]

If we look at the first premise [A · (B v C)] ⊃ [(D v E) ⊃ (F ⊃ G)] we see that even though it has seven letters it has the form p ⊃ q. And if we look at the second premise ~[(D v E) ⊃ (F ⊃ G)] it is simply a negation of the consequent of the first premise, so it has the form ~q. Hence, the argument form is a substitution instance of the rule of inference known as Modus Tollens (reviewed in post 20).

p = [A · (B v C)]
q = [(D v E) ⊃ (F ⊃ G)]

Note the main connective between p and q in the first premise. This gives you a clue to which rule of inference to be looking for.

Modus Tollens (MT)
p ⊃ q
~q
_____
∴ ~p

:grin: You could do that StreetlightX.

Work on what Wiki articles?

Logic Post 28

Construct proofs for the arguments that follow. These arguments were taken from Kegley and Kegley, Introduction to Logic, pp. 277 - 278).

1. If God is loving, then if he condemns sinners to eternal damnation, God is unjust. He is not unjust. Therefore, he does not condemn sinners to damnation. (Use G, S, and U for letters in your symbolized argument.)

2. If Jane gets an A in logic, then she will not have to give up the scholarship. But if Jane does not get an A in logic, then she will stay in the Honors Club if and only if she will not have to give up the scholarship. But if either Jane will not be an A student or she will not stay in the Honors Club, then it is not the case that she will not have to give up the scholarship. Either Jane will not be an A student or she will not stay in the Honors Club. Therefore, she will stay in the Honors Club if and only if she will not have to give up the scholarship. (Use the following letters: L, S, H, and A)

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.