Argument

An argument is a set of statements, of which one is the conclusion and the rest are the premises.

In a deductive argument, the conclusion is necessarily true based on the premises; for example, If you spin in circles, you get dizzy. You are spinning. Therefore, you are dizzy.

In an inductive argument, the conclusion is likely true most of the time. For example, If you spin in circles, you will likely get sick. You are spinning. Therefore, you will likely get sick.

An argument is valid if and only if it is impossible for its premises to be true and the conclusion false. An argument is sound if and only if it is valid and its premises are all true.

A sound argument invariably has a true conclusion. An argument is strong if and only if it is improbable for the premises to be true and the conclusion false. An argument is reliable if and only if it is strong and its premises are all true.

Soundness is to validity what reliability is to strength.

An inductive argument is strong rather than valid. Therefore, not every invalid inductive argument is a failed argument. An inductive argument is as reliable as it is strong, and as failed as it is weak.

The major valid deductive argument forms involve truth-functional operators, universal quantifiers, existential quantifiers and identity.

The most used strong inductive argument forms include Enumeration, Statistical Syllogism, Analogical Syllogism and Inference.

Inductive arguments

Strong forms

Argument	Construct	Overview
Enumeration	A particular F is G. Another F is G. Yet another F is G. (and so on) ∴ Every F is G.	This is an example of generalizing from particular cases to obtain the conclusion, which is a universal quantification. The more particular cases that are analyzed then the stronger the argument.
Statistical syllogism	Most F are G. α is F. ∴ α is G.	This is a probabilistic version of Universal Syllogism. The more F's that are ascertained to be G, the stronger the argument.
Analogical syllogism	α and β are both X. α and β are both Y. (and so on) α is F. ∴ β is F.	Attributes in common between α and β must not just be abundant for a strong argument, but be relevant. For example, if two individuals are the same height then one person will not be good at math just because the other is; however, a relevant criteria would be meticulousness.
Inference	P. Q best explains P. ∴ Q.	For Q to be the best explanation, it must be better than any other explanation. The size of the set of rival explanations, therefore, determines the strength of the argument. Furthermore, the best explanation at one time may not be the best at another.

Prediction

Explaining a phenomenon usually gives rise to predictions about recurrence of that phenomenon. For example, finding that retinoblastoma is explained by Rb^-/- mutations allows predictions that retinoblastoma will form in Rb^-/- individuals. Thus, explanations and predictions oft go in tandem. An explanation that gives rise to poor predictions is a poor explanation. A prediction that is supported by a poor explanation is not trustworthy.

Causal Inference

In causal inference, the best explanation for a phenomenon is sought by inquiring about its cause. For example, retinoblastoma researchers combed through countless genes to find a cause for retinoblastoma. After finding a mutation present in all and only retinoblastoma cases, researchers created an Rb^-/- population. Its constituents invariably formed retinoblastoma. Rb mutations were causally inferred as the best explanation for retinoblastoma.

Deductive arguments

Reductio ad absurdum

An especially powerful valid deductive argument form is reductio ad absurdum.

If you are unsure how to directly argue P, you may use reductio ad absurdum. First, assume it is not the case that P; second, include known truths amongst your premises; lastly, draw a conclusion that is a contradiction. It is impossible for a valid argument with true premises to result in a contradiction.

Ergo, the premise P must not be true.

Suppose your friend maintains that every opinion is equally correct. You disagree but do not know how to argue against her directly. So you try reductio ad absurdum. You say to her, "Let us assume that you are right, that is, that every opinion is equally correct. Now, my opinion that you are wrong is an opinion, so it is correct. That is, it is correct to say that you are wrong. So, you are wrong. Thus, you are right and you are wrong, which is a contradiction. Therefore by reductio, the initial assumption that you are right must be rejected. This, you are wrong. McHenry & Yagisawa, p 22

Truth-functional operators

Truth-Functional Operators: What Are They?

Operator	Construct	Overview
Conditional	If P, then Q.	P is the antecedent and Q is the consequent.
Disjunction	Either P or Q.
Negation	It is not P.
Conjunction	Both P and Q.

Necessary Conditions & Sufficient Conditions

Operator	Construct
Necessary condition	If not P, then not Q.
Sufficient condition	If P, then Q.
Neccesary & Sufficient	Q if and only if P.

To get an A in math, you must ace the midterm and the final. Acing your midterm and acing your final are each necessary for an A in the class, yet neither is sufficient on its own. To get an A in literature, the only requirement is to ace the one and only paper. Acing the paper is necessary and sufficient for an A in the class. You ace the class if and only if you ace the paper.

Truth-Functional Operators: Truth-Fuctionally Valid Forms

Argument	Construct	Overview
Modus ponens	If P, then Q. It is P. ∴ It is Q.	Modus Ponens (Affirming Mode) is perhaps the most prevalent argument form. It contains one premise that is a conditional, which is affirmed by the other premise.
Modus tollens	If P, then Q. It is not Q. ∴ It is P.	Modus Tollens (Denying Mode) contains one premise that is a conditional statement, which is negated by the other premise.
Hypothetical syllogism	If P, then Q. If Q, then R. ∴ if P then R.	Hypothetical Syllogism contains premises and conclusions that are all conditionals. The consequent of one premise is identical with the antecedent of the other premise. The antecedent of the former and consequent of the latter are identical to the antecedent and consequent of the conclusion.
Dilemma	Either P or Q. If P, then R. If Q, then S. ∴ Either R or S.	Dilemma can be viewed as a beast with two horns -- seize the P horn, you get R; seize te Q horn, you get S. Therefore, either Q or S is begotten.
Simplified dilemma	EIther P or Q. If P, then R. If Q, then R. ∴ R.	Simplified Dilemma lists all possible conditionals; since these conditional share the same consequent, the conclusion is the consequent.
Disjunctive syllogism	Either P or Q. Not P. ∴ Q.

Truth-Functional Operators: Fallacies

Argument	Construct	Overview
Affirming the Consequent	If P, then Q. Q. ∴ P.	Affirming the Consequent is an invalid argument, which is regardless of the premises' truthfulness. The conclusion is the antecedent of the conditional, as opposed to the consequent or the negation of the antecedent.
Denying the Antecedent	If P, then Q. Not P. ∴ Not Q.	Denying the Consequent is an invalid argument, which is regardless of the premises' truthfulness. The conclusion is the negation of the consequent of the conditional premise.
Begging the Question	P ∴ P.	Begging the Question is a valid fallacy whereby the conclusion is included among the premises. Begging the Question is oft misunderstood; it actually refers to stealing (begging) the conclusion and smuggling it into the premises.

Universal and existential qualifiers

Quantificationally Valid Forms

Not all valid argument forms are truth-functional. Some use universal quantifiers (ie, every) and some use existential quantifiers (ie, some). However, it is imperative that a universal quantification is not ambiguous, as in "everything is not blue" -- is everything non-blue, or is it that some things non-blue?

Argument	Construct	Overview
Universal instantation	Every F is G. If α is F, Then α is G.	A counterexample refutes a universal instantiation, for example by providing an α that is F but not G.
Existential quantification	Some F is G. another example, Most F are G.	An existential quantification is not subject to counterexamples; it can only be disproven by examining every F and showing that none are G. This is because existential quantifications yield no entailments concerning any given item.
Universal syllogism	Every F is G. α is F. ∴ α is G.	Universal Syllogism is valid, as assured by the validity of Universal Instantiation (If &alpha is F, it is G) and Modus Ponens (&alpha is F, ∴ &alpha is G).

Quantificationally Invalid Forms

Argument	Construct	Overview
Universal Negation	Everything is not G another example, Every F is not G.	Is everything a non-G, or are some things G and some things non-G? This form is ambiguous.
"Some" and "Not All"	Some F are G. ∴ Not all F are G.	By reductio ad absurdum, this form is clearly invalid. For example, at an all-girls school where all its students are female, it is still true that some students are female.

Identity and Liebniz's Laws

Numerical identity entails absolute indiscernibility. If x and y are indiscernible, they share all the same traits and x = y. Identical twins are not one and the same and thus are not numerically identical; they are instead extremely similar, thus sharing qualitative identity.

Liebniz's Law of Indiscernibility of the Identical

For every x, for every y,
If x = y, then x and y have exactly the same properties.

Leibniz's Law of Indiscernibility of the Identical is useful in demonstrating the distinctness of two easily confused things. For example, some materialists argue that mental phenomena are simply neurophysiological phenomena. Their opponents typically attempt to show some mental phenomena have properties that do not occur neurophysiologically. If their opponents succeeded, then Liebniz's Law of Indiscernibility of the Identical woud entail, via Universal Instantiation and Modus Tollens, that mental activities are not identical with neurophysiological activities.

LIebniz's Law of Identity of the Indiscernible

For every x, for every y,
If x and y have exactly the same properties, then x = y.

This argument does not prove that identical twins are identical. To be numerically identical, identical twins would have to share the same spatial location, a feat which is possible for existing objects.