2. Argumentation and Logic Flashcards
Come now, let us argue it out. Isaiah 1:18
What, in the philosophical sense, is an argument?
An argument in the philosophical sense is a set of statements that serve as premises leading to a conclusion.
1. If today is Sunday, the library is closed.
2. Today is Sunday.
3. Therefore, the library is closed.
Sentences (1) and (2) are the premises of the argument, and sentence (3) is the conclusion. You are saying that if premises (1) and (2) are true, then the conclusion (3) is also true. It is not just your opinion that the library is closed; you have given an argument for that conclusion.
Deductive vs Inductive
Deductive arguments: If the premises are 100% true and the argument is valid, then the conclusion is 100% guaranteed to be true. A valid deductive argument is one whose logical structure or form is such that if the premises are true, the conclusion must be true: valid or invalid, sound or unsound, if the premises are true, the conclusion must be true.
Inductive arguments: Even if the argument is valid, and even if the premises are 100% true, the conclusion is still only likely, not guaranteed. It still must pass the additional test of beating its competitors, in which case, it should be believed, yet, it never escapes the realm of propabilities: strong or weak, cogent or not cogent, a strong inductive argument with true premises renders the conclusion likely to be true, a more probabalistic argument. A cogent argument is a strong argument with true premises.
The differences between deductive and inductive arguments concern structure, independent of whether the premises of an argument are true.
Arguments may be either deductive or inductive. In a good deductive argument the premises guarantee the truth of their conclusions. In a good inductive argument the premises render the conclusion more probable than its competitors. What makes for a good argument depends on whether that argument is deductive or inductive.
A good deductive argument is not based on 100 percent certainty of the truth of the premises but rather on the conjunction of the premises being collectively more plausible than any contradictories (shows you what a fine line there is between deductive and inductive upon closer examination). Still, it remains the case that on the assumption that the premises of a deductive argument are 100 percent true, the conclusion is also 100 percent true. That’s what separates it from being an inductive argument. With an inductive argument, the premises can be 100% true and the conclusion is still just in the realm of probabilities, ideally, however, representing the highest probability amongst its competitors.
From IEP:
Deductive arguments may be said to be valid or invalid, and sound or unsound. A valid deductive argument is one whose logical structure or form is such that if the premises are true, the conclusion must be true. A sound argument is a valid argument with true premises. Inductive arguments, by contrast, are said to be strong or weak, and, although terminology varies, they may also be considered cogent or not cogent. A strong inductive argument is said to be one whose premises render the conclusion likely. A cogent argument is a strong argument with true premises. All arguments are made better by having true premises, of course, but the differences between deductive and inductive arguments concern structure, independent of whether the premises of an argument are true.
Deductive arguments are sometimes illustrated by providing an example in which an argument’s premises logically entail its conclusion. For example:
Socrates is a man.
All men are mortal.
Therefore, Socrates is mortal.
Assuming the truth of the two premises, it seems that it simply must be the case that Socrates is mortal. According to this view, then, this would be a deductive argument.
By contrast, inductive arguments are said to be those that make their conclusions merely probable. They might be illustrated by an example like the following:
Most Greeks eat olives.
Socrates is a Greek.
Therefore, Socrates eats olives.
Assuming the truth of those premises, it is likely that Socrates eats olives, but that is not guaranteed. According to this view, this argument is inductive.
It is not uncommon to be told that in order to assess any argument, three steps are necessary. First, one is to determine whether the argument being considered is a deductive argument or an inductive one. Second, one is to then determine whether the argument is valid or invalid. Finally, one is to determine whether the argument is sound or unsound.
What makes for a good deductive argument?
A good deductive argument will be one that is formally and informally valid, that has true premises, and whose premises, taken together, are more plausible than their contradictories.
Formally Valid / Formal Validity
The conclusion follows from the premises (truth aside) in accord with the rules of logic / no formal fallacies.
In philosophy, what is logic?
Logic is the study of the rules of reasoning (consisting of various subfields such as sentential logic, first-order predicate logic, many-valued logic, modal logic, tense logic, and so forth).
What is an invalid argument?
An argument whose conclusion does not follow from the premises in accord with the rules of logic is said to be invalid, even if the conclusion happens to be true. For example,
1. If Sherrie gets an A in epistemology, she’ll be proud of her work.
2. Sherrie is proud of her work.
3. Therefore, Sherrie got an A in epistemology.
All three of these statements may in fact be true. But because (3) does not follow logically from (1) and (2), this is an invalid argument. From the knowledge of (1) and (2), you cannot know that (3) is also true. The above is therefore not a good argument.
Informal Validity
The avoidance of informal fallacies.
A good argument will be not only formally valid but also informally valid. As we shall see, there is a multitude of fallacies in reasoning that, while not breaking any rule of logic, disqualify an argument from being a good one—for example, reasoning in a circle. Consider the following argument.
If the Bible is God’s Word, then it is God’s Word.
The Bible is God’s Word.
Therefore, the Bible is God’s Word.
This is a logically valid argument, but few people will be impressed with it. For it assumes what it sets out to prove and therefore proves nothing new. A good argument will not only follow the rules of formal logic but will also avoid informal fallacies.
A Sound Argument
An argument that is both logically valid (informally and formally) and has true premises is called a sound argument. An unsound argument is either invalid or else has a false premise.
The premises in a good argument must be true. An argument can be formally and informally valid and yet lead to a false conclusion because one of the premises is false. For example,
Anything with webbed feet is a bird. A platypus has webbed feet. Therefore, a platypus is a bird.
Plausibility
The principle which states that if the conjunction of the premises is more plausible than not, then the conclusion of a deductive argument is guaranteed to be more plausible than not. Also, so long as the conjunction of the premises is more plausible than not, we should believe in the conclusion of a valid deductive argument. Plausibility is more about contrasting the premises than conclusions.
Therefore, plausibility is the logical principle which states that a good deductive argument is not based on 100 percent certainty of the truth of the premises but rather on the conjunction of the premises being collectively more plausible than any contradictories (shows you what a fine line there is between deductive and inductive upon closer examination). Still, it remains the case that on the assumption that the premises are 100 percent true, the conclusion is also 100 percent true. That’s what separates it from being an inductive argument. Technically, plausibility is more about contrasting the premises than conclusions. Competiting conclusions are contrasted on the bases of their premises, not on their own. Obviously.
A good argument has premises that are collectively more plausible than their contradictories (or denials). For an argument to be a good one, it is not required that we have 100 percent certainty of the truth of the premises. If the conjunction of the premises is more plausible than not, then the conclusion of a deductive argument is guaranteed to be more plausible than not. So long as the conjunction of the premises is more plausible than not, we should believe in the conclusion of a valid deductive argument. Thus a good argument for God’s existence need not make it certain that God exists. Certainty is what most people are thinking of when they say, “You can’t prove that God exists!” If we equate “proof” with 100 percent certainty, then we may agree with them and yet insist that there are still good arguments to think that God exists. For example, one version of the axiological argument may be formulated:
If God did not exist, objective moral values would not exist.
Objective moral values do exist.
Therefore, God exists.
Someone may object to premise (1) of our argument by saying, “But it’s possible that moral values exist as abstract objects without God.” We may happily agree. That is epistemically possible, that is to say, the premise is not known to be true with certainty. But possibilities come cheap. The question is not whether the contradictory of a particular premise in an argument is epistemically possible (or even plausible); the question is whether the contradictory is as plausible or more plausible than the premise. If it is not, then one should believe the premise rather than its contradictory.
What makes for a good argument?
A good argument will be formally and informally valid and have true premises that are collectively more plausible than not.
Sentential or Propositional Logic
The foundation of logic which governs the rules of inferring truth from sentential connectives.
Sentential or propositional logic is the most basic level of logic, dealing with inferences based on sentential connectives like “if . . . , then,” “or,” and “and.”
What are the nine inferential rules of propositional / sentential logic
modus ponens
modus tollens
hypothetical syllogism
disjunctive syllogism
conjunction
simplification
addition
absorption
constructive dilemma
Modus Ponens
- P → Q
- P
Therefore Q.
The rule modus ponens tells us that from the two premises P → Q and P, we may validly conclude Q. This rule of inference is one that we use unconsciously all the time, as the following examples should make clear.
Example 1:
If John studies hard, then he will get a good grade in logic.
John studies hard.
He will get a good grade in logic.
Example 2:
If John does not study hard, then he will not get a good grade in logic.
John does not study hard.
He will not get a good grade in logic.
Notice that our two examples are both valid arguments (they are both in accord with the rule modus ponens), but they reach opposite conclusions. So they cannot both be sound; at least one of them must have a false premise. If we wanted to figure out which one of these examples is a sound argument, we would need to look at the evidence for the premises.
Modus ponens focuses more on the sufficient conditionality of the “if” clause whereas modus tollens focuses more on the necessary conditionality of the “then” clause when it comes to which is the catalyst in drawing their respective conclusions.
In symbolic logic, what does an arrow stand for?
The arrow stands for the connecting words, “if . . . , then . . .”
example:
1. P → Q
2. P
3. Q.
“To read premise (1) we say, “If P, then Q.” Another way of reading P → Q is to say: “P implies Q.” To read premise (2) we just say, “P.”
In symbolic logic, what do the letters and symbols stand for and why?
Letters and symbols stand for sentences and the words that connect them. The reason letters and symbols are used is because sentences that are very different grammatically may still have the same logical form. For example, the sentences “I’ll go if you go” and “If you go, then I’ll go,” (what follows “if” if “if” is alone is always the beginning of an “if…then” construction) though different grammatically, obviously have the same logical form. By using symbols and letters instead of the sentences themselves we can make the logical form of a sentence clear without being distracted by its grammatical form. What would be helpful to note here is that in addition to positive examples that are different grammatically but mean the same thing, what you really have to watch out for is sentences that you think mean the same thing, or think they mean something you assume them to mean, on the basis of misinterpreted grammar or sentence structure. Therefore, having the tools to reflect on and double check the logical form underneath and then simplify it for accurate future reference can make all the difference.
Modus Tollens
P → Q
¬Q
¬P
It exploits the power that the necessary condition has over the sufficient condition in the opposite way that modus ponens does; like a parent telling his son, “If I’m not going, then you’re not going.” It’s just the negative inverse of modus ponens. The P and the Q stand for any two sentences, and the arrow stands for “if . . . , then . . .” The sign ¬ stands for “not.” It is the sign of negation. So premise (1) reads, “If P, then Q.” Premise (2) reads, “Not-Q.” The rule modus tollens tells us that from these two premises, we may validly conclude “Not-P.” So modus ponens is saying, “If P then Q; and P therefore Q.” Modus tollens is saying ok to all that but adds the reminder that, given “If P then Q.”, if you don’t have Q, you don’t have P either. In other words, premise 2 in modus tollens is the negation of the conclusion in modus ponens and vice versa.
Example 1:
If Joan has been working out, then she can run the 5K race.
She cannot run the 5K race.
Joan has not been working out.
Example 2:
If it is Saturday morning, then my roommate is sleeping in.
My roommate is not sleeping in.
It is not Saturday morning.
Modus ponens is primarily about affirming a premise and modus tollens involves negating a premise. If the premise is already a negation, then we have double negation, which is logically the same as an affirmative sentence. Thus ¬¬Q is equivalent to Q. So from the premises
¬P → Q
¬Q
Therefore, ¬¬P which is logically the same as P
Example:
If it is not Saturday morning, then my roommate is sleeping in.
My roommate is not sleeping in.
It is not not (therefore it is) Saturday morning.
Another example of double negation in action would be
P → ¬Q
Q
In order to use modus tollens we first convert (2) to “¬¬Q which is the negation of ¬Q. That allows us to use modus tollens to conclude to
¬P
Example:
If it is Saturday morning, then my roommate is not sleeping in.
My roommate is sleeping in.
It is not Saturday morning.
Modus ponens and modus tollens help to bring out an important feature of conditional sentences: The antecedent “if” clause states a sufficient condition of the consequent “then” clause. The consequent “then” clause states a necessary condition of the antecedent “if” clause. For if P is true, then Q is also true. The truth of P is sufficient for the truth of Q. At the same time P is never true without Q: if Q is not true, then P is not true either. So in any sentence of the form P → Q, P is a sufficient condition of Q, and Q is a necessary condition of P. When you understand this, you can make conceptual sense of the both rules.
Modus ponens focuses more on the sufficient conditionality of the “if” clause whereas modus tollens focuses more on the necessary conditionality of the “then” clause when it comes to drawing their respective conclusions.
It sharpens the mind to contrast affirming the consequent with modus tollens.
What is the difference between necessary and sufficient conditions?
Sufficiency means one entails the other every time it is realized but the former doesn’t necessarily attain in every instance of the latter. Sufficiency means that one might be true (the then clause) without the other being true but that at least one of them (the if clause) requires the other. When a necessary condition is true, then both the necessary and sufficient conditions are true. Think of it in terms of dependency. If Q requires P to be true, then P is a necessary condition of Q. But if Q can be true or exist without P, but P, when it does exist, implies Q, then P is a sufficient but not necessary condition of Q, whereas Q is a necessary condition of P even though it is in the “then” part of the sentence. When Q can only be true when P is also true, then P is a necessary condition for Q. When Q can be true with or without P, but P always entails Q when P is true, then P is a sufficient but not necessary condition for Q. Knowing this, you can see that the “if” clause in an “if…then” conditional sentence states a sufficient condition for the subsequent “then” clause; and the consequent “then” clause states a necessary condition (admits dependency for) of the antecedent “if” clause.
In the sentence, “Extra credit will be permitted only if you have completed all the required work.”, do you have a clause stating a sufficient condition, a clause stating a necessary condition, or both? Which is which and why?
Both. One thing that trips me up about this sentence is that I keep unconsciously thinking that a necessary condition exists necessarily and always. It does not. After all, they are both CONDITIONS. It is only necessary in the sense that the sufficient condition requires it and that it does not necessarily require the sufficient condition in order to exist. Reality itself does not require it! It is possible for it and the sufficient condition to be false. Necessary in this sense only means that, if the sufficient condition is true, then the necessary condition is true. In other words, it is NECESSARY for the NECESSARY condition to be true IF the sufficient condition is true. In a world where the sufficient condition attains, the necessary condition also attains; but in a world where the necessary condition attains, the sufficient condition DOESN’T NECESSARILY attain. If P is true, then Q has to be true; but if Q is true, P is not necessarily true. This is the case, almost counter intuitively, because P depends on Q. The necessary condition sounds all independent and strong and yet, wherever you see P, you also see Q, as if Q is bound to P, tied to P. Yet it is P who cannot live without Q. Since the say P was born, you never see her without Q but you can see Q out and about without P. You wonder where P is during these times. She doesn’t exist during these times! You can never say the opposite, that there are times when P exists without Q. Q is independent but P is not. The part that trips me up sometimes however is that Q seems not to be independent when P exists because, like a whooped boyfriend, he’s always there if she is there. But the truth is, it is he who truly holds the cards because she cannot live without him. If she dies, he can continue to live but if he dies, then she dies too, just like what appears to be the case in the first part of the suicide scene from Romeo and Juliet.
In a sentence, the clause that follows a simple “if” is the antecedent clause symbolized P, a sufficient condition. The clause that follows “only if” is the consequent clause symbolized Q, the necessary condition. So it could be reworded as, “If extra credit has been permitted, then you have completed all the required work.” or, “You may find yourself doing extra credit work if you have completed the required work.”
“Only if” should make your brain think immediately that this connective is referring back to the preceeding clause. The preceeding clause is what exists “only if” the next condition attains; and we know that a sufficient clause is that clause that exists “only if” the necessary clause is also true. You would never say “only if” reflexibly referring to a necessary clause as an introduction to a sufficient clause AS IF…!
So it could also be reworded as, “If you do extra credit work, then you have already, at the very least, completed all the required work.” Or, “If you find yourself doing extra credit, then you have had to have already completed all of the required work.” The extra credit (P) depends on having completed the required work (Q). But it’s not, “If Q, then P”, because you could do all the required work without being further permitted to do the extra credit. In other words, Q can exist without P, so we would never conclude from “If P, therefore Q”, therefore, “If Q, therefore P.” No. That is inherently contradictory to the logical structure of P → Q via the rule modus ponens.
He is saying that completing the required work is a necessary condition of doing extra credit work. Therefore, if we let P = “You may do extra credit work” and Q = “You have completed the required work,” we can symbolize his sentence as P → Q. This is tricky because when the beginner sees the words “only if,” he might think that we should symbolize the clause that comes after them as P. But that is incorrect. When he sees the words “only if,” he should think immediately “necessary condition” and realize that he should symbolize what comes afterward as Q.
You might conclude from your professor’s above statement that if you complete the required work, then you may do extra credit work. But that is not, in fact, what he said! He stated a necessary condition of your doing extra credit work, not a sufficient condition (in which case it would have been, “if you do all the required work then you may do extra credit work.” He asserted P → Q, but he did not assert Q → P (the first clause being a sufficient condition). There may be other conditions that have to be met as well before one may do extra credit work. So if you concluded on the basis of his statement that you could do extra credit work after completing the required work, you would be guilty of an invalid inference, which might prove ruinous to your grade!
What is another way of saying “P implies Q”?
If P then Q.
When the logical form used is correct, such as the correct use of modus ponens in a premise and argument, but the conclusion is still wrong, is it because the argument is invalid or is it because the argument is unsound?
It will be because the argument is unsound. When the structure obeys the rules of logic but nonetheless one premise is false (for example being the result of bad research), then the argument is valid but unsound. In order to be a sound argument, it has to both be a valid argument and have all true premises. Valid means that the structure of the argument is such that if the premises are true, then the conclusion just about has to be true and that the conclusion flows logicially from the premises in accord with the rules of logic. So always ask two questions for every argument you hear: 1. Is this a valid argument? and 2. Are the premises true?
Affirming the consequent
It is valid reasoning to either affirm the antecedent or deny the consequent and draw the appropriate conclusion. It is not vaid reasoning to affirm the consequent and conclude the antecedent because the antecedent requires the consequent but the consequent does not require the antecedent.
This common fallacy is basically swapping the logical roles of sufficient and necessary conditions to draw a conclusion that would be the case if the roles were reversed. In particular, it tries to make the necessary condition clause of the first premise play what should be the role of the sufficient condition clause in the second premise in an attempt to make the original sufficient clause of the first premise a deduced fact / conclusion rather than a premise. It’s affirming the necessary condition clause (the consequent) as a second premise to conclude the sufficient condition clause (the antecedent) rather than the other way around. For the reason that you can have Q without P but you cannot have P without Q, you can deny Q and thereby conclude not-P. However, you cannot affirm Q and conclude P.
Example 1:
If George and Barbara are enjoying soft-boiled eggs, toast, and coffee, then they are having breakfast.
George and Barbara are having breakfast.
They are enjoying soft-boiled eggs, toast, and coffee.
This is wrong because the form of the breakfast may change for George and Barbara, but eggs, toast, and coffee always means breakfast.
Example 2:
If God is timeless, then he is intrinsically changeless.
God is intrinsically changeless.
He is timeless.
What is wrong with this reasoning is that in both examples (1) states only a sufficient, not a necessary, condition of (2). If George and Barbara are eating those things, then they are having breakfast. But it does not follow that if they are having breakfast, then they are eating those things! If God is timeless, then he is intrinsically unchanging. But that does not imply that if he is intrinsically unchanging, he is therefore timeless.
Per the rules modus ponens or modus tollens respectively, with P → Q you can either affirm the antecedent to draw a conclusion or we can deny the consequent and draw a conclusion. But based on the logical structure of the sentence, we cannot affirm the consequent and then draw a conclusion. As Craig and Moreland put it, “If P → Q, modus ponens tells us that if we affirm that the antecedent P is true, then the consequent is also true. Modus tollens tells us that if we deny that the consequent Q is true, then the antecedent P must also be denied. Thus, if P → Q, it is valid reasoning to either affirm the antecedent or deny the consequent and draw the appropriate conclusion. But we must not make the mistake of affirming the consequent. If P → Q, and Q is true, we may not validly conclude anything.”
Hypothetical Syllogism
P → Q
Q → R
P → R
“The third rule, hypothetical syllogism, states that if P implies Q, and Q implies R, then P implies R. Since we do not know in this case if P is true, we cannot conclude that R is true. But at least we can know on the basis of premises (1) and (2) that if P is true, then R is true.”
“Example 1:
If it is Valentine’s Day, Guillaume will invite Jeanette to dine at a fine restaurant.
“If Guillaume will invite Jeanette to dine at a fine restaurant, then they will dine at L’ Auberge St. Pierre.
If it is Valentine’s Day, then Guillaume and Jeanette will dine at L’ Auberge St. Pierre.
Example 2:
If Jeanette orders médallions de veau, then Guillaume will have saumon grillé.
If Guillaume has saumon grillé, he will not have room for dessert.
If Jeanette orders médallions de veau, then Guillaume will not have room for dessert.
I actually use this one a lot. I state an If…then and leave out a lot of the middle. I often employ this because I’m always trying to think of ultimate logical consequences, for better or worse, in order to get to the heart, or full implications of the issue.
Which logical rules are used in the following argument?
P → Q
Q → R
P
P → R
R
hypothetical syllogism and modus ponens
P → Q
Q → R
P
P → R (HS, 1, 2)
R (MP, 3, 4)
“The first three steps are the given premises. Steps (4) and (5) are conclusions we can draw using the logical rules we have learned. To the right we abbreviate the rule that allows us to take each step, along with the numbers of the premises we used to draw that conclusion. Notice that a conclusion validly drawn from the premises becomes itself a premise for a further conclusion.”
Which logical rules are used in the following argument?
P → Q
Q → R
¬R
P → R
¬P
hypothetical syllogism and modus tollens
P → Q
Q → R
¬R
P → R (HS, 1, 2)
¬P (MT, 3, 4)
Conjunction
P
Q
P & Q
This is the, “two things can be true at once” rule. Here we introduce the symbol &, which is the symbol for conjunction. It is read as “and.” This rule is perspicuous: If P is true, and Q is true, then the conjunction “P and Q” is also true. The symbol & symbolizes many more words than just and. It symbolizes any conjunction. Thus the logical form of sentences having the connective words but, while, although, whereas, and many other words is the same. We symbolize them all using &. For example, the sentence “They ate their spinach, even though they didn’t like it” would be symbolized P & Q. P symbolizes “They ate their spinach,” Q symbolizes “they didn’t like it,” and & symbolizes the conjunction “even though.
Example 1:
Charity is playing the piano.
Jimmy is trying to play the piano.
Charity is playing the piano, and Jimmy is trying to play the piano.
Example 2:
If Louise studies hard, she will master logic.
If Jan studies hard, she will master logic.
If Louise studies hard, she will master logic, and if Jan studies hard, she will master logic.
As example 2 illustrates, any sentences can be joined by &. When the premises in our arguments get complicated, it helps to introduce parentheses to keep things straight. For example, you would symbolize the conclusion (P → Q) & (R → S).
Simplification
P & Q
P
and also:
P & Q
Q
If conjunction is the “two things can be true at once” rule, then simplification is the, “if two things can be true at once, then they can also be true on their own (independently of each other” rule. Again, one does not need to be a rocket scientist to understand this rule! In order for a conjunction like P & Q to be true, both P and Q must be true. So simplification allows you to conclude from P & Q that P is true and that Q is true.
Example 1:
Bill is bagging groceries, and James is stocking the shelves.
James is stocking the shelves.
Example 2:
If Susan is typing, she will not answer the phone; and if Gary is reading, he will not answer the phone.
If Gary is reading, he will not answer the phone.
The main usefulness of this rule is that if you have the premise P & Q and you need either P by itself or Q by itself to draw a conclusion, simplification can give it to you.
For example:
P & Q
P → R
P (Simp, 1)
R (MP, 2, 3)
What logic rules are involved in arguing toward the following conclusions?
P & Q
P → R
P
R
Simplification and modus ponens
The main usefulness of simplification is that if you have the premise P & Q and you need either P by itself or Q by itself to draw a conclusion, simplification can give it to you.
For example:
P & Q
P → R
P (Simp, 1)
R (MP, 2, 3)
Absorption
P → Q
P → (P & Q)
This is a rule that one hardly ever uses but that nonetheless states a valid way of reasoning. The basic idea is that since P implies itself, it implies itself along with anything else it implies. It’s yet another, though indirectly, “two things can be true at once.” rule.
Example 1:
If Allison goes shopping, she will buy a new top.
If Allison goes shopping, then she will go shopping and buy a new top.
Example 2:
If you do the assignment, then you will get an A.
If you do the assignment, then you do the assignment and you will get an A.
The main use for absorption will be in cases where you need to have P & Q in order to take a further step in the argument. For example:
P → Q
(P & Q) → R
P → (P & Q) (Abs, 1)
P → R (HS, 2, 3)
What logical rules are involved in establishing the following premises?
P → Q
(P & Q) → R
P → (P & Q)
P → R
Absorption and hypothetical syllogism
The main use for absorption will be in cases where you need to have P & Q in order to take a further step in the argument. For example:
P → Q
(P & Q) → R
P → (P & Q) (Abs, 1)
P → R (HS, 2, 3)
Addition
P
P v Q
The result or conclusion of addition is always a disjunction. For this rule we introduce a new symbol: v, which is read “or.” We can use it to symbolize sentences connected by the word or. A sentence that is composed of two sentences connected by or is called a disjunction.
Addition seems at first to be a strange rule of inference: It states that if P is true, then “P or Q” is also true. What needs to be kept in mind is this: in order for a disjunction to be true only one part of the disjunction has to be true. So if one knows that P is already true, it follows that “P or Q” is also true, no matter what Q is!
Example 1:
Mallory will carefully work on decorating their new apartment.
Either Mallory will carefully work on decorating their new apartment, or she will allow it to degenerate into a pigsty.
Example 2:
Jim will make the honor roll.
Either Jim will make the honor roll or his dad will fly to the moon.
Addition is another one of those “housekeeping” rules that are useful for tidying up an argument by helping us to get some needed part of a premise. For example:
P
(P v Q) → R
P v Q (Add, 1)
R (MP, 2, 3)
What logical rules are involved in establishing the following premise?
P
(P v Q) → R
P v Q
R
Addition and modus ponens
Addition is another one of those “housekeeping” rules that are useful for tidying up an argument by helping us to get some needed part of a premise. For example:
P
(P v Q) → R
P v Q (Add, 1)
R (MP, 2, 3)
