Wittgenstein was addressing the various psychological scientisms that was the rage of his day; pointing out it's mostly just confusing and new knowledge beyond ordinary understanding of these things is impossible. — boethius

Where did you get this idea from? On Certainty, which was written in the last year and half of his life was addressing Moore's propositions.

I don't. Both "belief" and "knowledge" and "justified" are applicable to our "foundational belief"; our ordinary language has no normal utility to name what we won't normally ever inspect. — boethius

When it comes to bedrock beliefs or foundational beliefs, my point has been consistently that they are not beliefs that can be known, i.e., they are not epistemological. They are beliefs that are shown in our actions. The best way to understand this, is to think of them nonlinguistically, as I have already pointed out in other posts. The difference is connected with Wittgenstein's saying and showing.

In Wittgenstein's critique of Moorean propositions (common sense propositions) he points out over and over again that it is improper to use the word know in reference to them (at least generally). For example, it is generally improper to say "I know I have hands," but one can say, "I believe it." I need not give grounds when I say "I believe p," but when saying "I know p," it is normally a requirement, i.e., how do you know. Saying "I know..." often means that I have the proper grounds, as Wittgenstein points out.

What is known rests on a chain of reasoning, however, that chain comes to an end at some point. That end point is what is bedrock, viz., a bedrock belief. No further grounds can be given, no further doubts entertained.

What Wittgenstein found interesting about Moore's propositions is that they seem to play a peculiar role in our epistemological framework. Bedrock beliefs fulfill a special logical role in epistemology. They support the structure of epistemology. Understanding this, solves two problems, 1) the infinite regress problem, and 2) the problem of circularity.

I found an interesting video that is worth watching if you have an interest in this thread.

https://www.youtube.com/watch?v=sPGZSC8odIU

Ya, because it's so obviously wrong that it's like saying 2+2 doesn't equal 4.

:up:

Are you suggesting that someone who just jumped into ice cold water to save another person's life mightn't feel at all good about themselves? That they might just do so robotically, because it's the right thing and remain dispassionate in the face the praise they later receive (either from others or from their own self-appraisal)? — Isaac

No, that is not what I am suggesting. I am saying, as per the OP, my emotional state probably has nothing to do with risking my life to save another, i.e., I do it because it is the right thing to do. Moreover, we judge whether the action was heroic, not based on the emotions involved (and there are emotions involved), but based on the action itself. It is the act itself that we judge good or bad, right or wrong.

The belief that our "...[e]motions are the only perceptions of good, bad, beauty, etc." is just silly. Moreover, the reasoning is childlike.

So what would guide people's behaviour if not making them feel good, doing what's right? — Isaac

Sometimes doing the right thing has nothing to do with how you feel, and that's the point. It may not feel good to jump into ice cold water to save another person's life, but you do it in spite of how it makes you feel. Feeling good may be a byproduct of most good acts, but not all the time.

At least your sticking to your argument, that takes commitment. Why are you arguing for your conclusion, since it doesn't matter what others think. Your emotional response is the only thing that matters. Unless you want the reinforcement of other positive emotional responses to your post.

It would seem to me to be a sad thing if people just pursued those things that made them feel good.

Why shouldn't it be done, i.e., it is a good emotion for the serial killer. Moreover, if everyone understood positive emotions as good, then they would have to agree that it's good for the serial killer. All positive emotions are good, therefore the serial killer is correct to pursue this positive emotion, even if some people wouldn't see it as positive.

So, the good is equivalent to whether or not I have positive emotional response (rhetorical question). If a serial killer gets a positive emotional response from torturing someone, that emotion is good for him or her, i.e., he or she should continue to pursue that emotional response?

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: