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 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 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 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 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

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 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 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 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 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.