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Conditional sentences

If p, then q p- antecedent q- consequent

The original conditional is: "If p, then q."
The converse of the original conditional is: "If q, then p."
The inverse of the original conditional is: "If it is not the case that p, then it is not the case that q."
The contrapositive of the original conditional is: "If it is not the case that q,
then it is not the case that p."

Modus Ponens
If p, then q
P
Therefore, q
Modus Ponens:
Combining of a conditional with the assertion of its antecedent and validly derive the consequent. Valid argument pattern. Affirms the antecedent of the conditional.
Conclusion affirms the consequent of the conditional.

Modus Tollens
If p, then q
Not q
Therefore, not p
Modus Tollens:

• The conditional is paired with another premise that denies the consequent of the conditional

• The conclusion denies the antecedent of the conditional

Rights in the strict sense (that is, claim-rights) always refer to someone else's action: they refer to the duty holder's action
If I have a right, all that means is that someone else has a duty: my right is a right that they do something, which means that they have the duty to do it 

Liberties always refer to the liberty-holder's own actions (or omissions)
e.g. I am not at liberty to disclose…
If I have a liberty, all that means is that I do not have a duty: my liberty is a liberty to do something, which means that I do not have a duty not to do it

Negations
It is not the case that p. Not p.

Conjunctions
P and q

Disjunctions
P or q

Hypothetical Syllogism
Hypothetical Syllogism:
Another valid form of argument involving conditional sentences is very easy to grasp:
(1) If p, then q
(2) If q, then r
Therefore (from (1) and (2)):
(3) If p, then r

• involves three conditional sentences: both the premises and the conclusion are conditionals

• Any argument of this form is a valid argument

Disjunctive Syllogism
Disjunctive syllogism:
(1) p or q
(2) Not p
Therefore (from (1) and (2)),
(3) q

• This is called the disjunctive syllogism

• It combines, as premises, (a) a disjunction, and (b) the negation of (any) one of the disjuncts, and derives, as its conclusion, the remainder of the disjunction

• Any argument of this form is a valid argument 

Fallacy

a fallacy is an argument which is (a) in some respect not a good one, but which is (b)
likely to look (at least at first sight) as a good argument
So, a fallacy is an argument that is flawed in some crucial respect, but whose flaw is not readily apparent
Formal- validity (form)
Informal- soundness (content)

 Formal Fallacy
Internally flawed
Flaw affects the validity of the argument
Flaw lies in their structure
"Non sequitur" ("it does not follow")- Conclusion does not follow from the premises
Argumentative flaw that once exposed reveals the argument not to be valid

Equivocation
Using an ambiguous term in more than one sense, thus making an argument misleading.
In logic, equivocation ('calling two different things by the same name') is a formal fallacy resulting from the use of a particular word/expression in multiple senses throughout an argument leading to a false conclusion.
I want to have myself a merry little Christmas, but I refuse to do as the song suggests and make the yuletide gay. I don't think sexual preference should have anything to do with enjoying the holiday.
The word, "gay" is meant to be in light spirits, joyful, and merry, not in the homosexual sense

o Denying the antecedent
If it barks, it is a dog.
It doesn't bark.
Therefore, it's not a dog.
Since it doesn't bark, we cannot conclude with certainty that it isn't a dog -- it could be a dog who just can't bark.
The arguer has committed a formal fallacy, and the argument is invalid because the truth of the premises does not guarantee the truth of the conclusion. If I have cable, then I have seen a naked lady.
I don't have cable.
Therefore, I have never seen a naked lady.
The fallacy is more obvious here than in the first example. Denying the antecedent (saying that I don't have cable) does not mean we must deny the consequent (that I have seen a naked lady).
The arguer has committed a formal fallacy, and the argument is invalid because the truth of the premises does not guarantee the truth of the conclusion.

o Affirming the Consequent
An error in formal logic where if the consequent is said to be true, the antecedent is said to be true, as a result.

If taxes are lowered, I will have more money to spend.
I have more money to spend.
Therefore, taxes must have been lowered.
I could have had more money to spend simply because I gave up crackcocaine, prostitute solicitation, and baby-seal-clubbing expeditions.

Summary
Modus ponens
(1) If p, then q
(2) p
Therefore (from (1) and
(2)),
(3) q
Denying the antecedent
(1) If p, then q
(2) Not p
Therefore (from (1) and
(2)),
(3) Not q

Modus tollens
(1) If p, then q
(2) Not q
Therefore (from (1) and
(2)),
(3) Not p
Affirming the consequent
(1) If p, then q
(2) q
Therefore (from (1) and
(2)),
(3) p

Disjunctions: exclusive vs inclusive
Disjunctions - sentences of the form "p or q" - can be either exclusive or inclusive
That is because the word "or" in English can be used in two different ways

• Sometimes it means "p or q but not both": this is the exclusive reading

• "Wanted: dead or alive"

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