Argument reconstruction clarifies what the arguer actually said and to supplement the arguer, as some statements are implicit. The reconstruction of an argument can be presented as a bad argument or as a good argument, depending on the goal (e.g. political debate). A bad argument does not mean that the proposition is false. The principle of charity states that people should also choose the best reconstruction of an argument to discover reasons for accepting or rejecting particular prepositions, advancing the cause of knowledge.

TRUTH
The truth-value of a proposition is the truth of the proposition. The two truth-values are true and false.

DEDUCTIVE VALIDITY
A conclusion is valid if the conclusion would be true, given that the premises are also true. The truth of premises is, in principle, not relevant for the validity of an argument. Validity should be judged by disregarding the truth-values of the premises.

PRESCRIPTIVE CLAIMS VS DESCRIPTIVE CLAIMS
Statements that state facts are descriptive claims and statements which express desires, norms or moral rules are prescriptive claims.

CONDITIONAL PROPOSITIONS
Conditional propositions use the ‘if-then’ format. A double negative in logic is equal to a positive and it is called contraposition (e.g: if not A, then not B = if B then A). The word ‘or’ can be used in the inclusive sense or in the exclusive sense. In the inclusive sense, it means that ‘A or B’ that either A is true or B is true or they are both true. That statement is only false if both A and B are false. In the exclusive sense ‘A or B’ means that either A is true or B is true but not both. The word ‘either’ is often used here too.

The words ‘if only’ create a conditional proportion that sets a necessary condition for it to apply, but it does not implicate that it is true the other way around. For example, ‘A only if B’ implies that A only occurs if B occurs, but B can occur without A occurring. A does not necessarily have to occur if B occurs, but if B does not occur A will never occur. The ‘if and only if’ statement creates a necessary condition. It means that either both happens or neither happens. For example: ‘A if and only if B’ means that A will never occur without B and that when B occurs A also occurs. In this case, B is the sole condition for A. If and only if means the same as either both A and B or neither. Unless (‘A unless B’) implies that A if not B. It does not mean that A will occur if B does not occur, but it does mean that A will not occur if B does not occur.

THE ANTECEDENT AND CONSEQUENT OF A CONDITIONAL
Formal logic can be denoted in another way.

A →B

This implies that if A happens, B happens. The first part of the conditional proposition which sets the condition (A in this case) is called the antecedent. If the condition is sufficed, then the consequent (B in this case) will occur. The difference between conditional propositions and arguments is that conditionals are true or false and arguments are valid or invalid. Arguments can have conclusions that are conditionals.

ARGUMENT TREES
Argument trees are devices that can be used for representing arguments in the form of a diagram. Arguments that contain conditionals are not sufficient to lead to a conclusion by itself but are together. Arguments that do not contain conditionals can sometimes lead to a conclusion by itself. This often makes use of implicit premises.

Conditional propositions can be represented by using Venn-diagrams.

DEDUCTIVE SOUNDNESS
A deductively sound argument is a valid argument in which all the premises are true. An argument can be valid, but not sound, but it cannot be sound and not valid. If the conclusion of an argument is false, then the argument is deductively unsound.

THE CONNECTION TO FORMAL LOGIC
There are several basic forms of formal logic:

1. Modus ponens
   If A then B
   A
   ----------------
   B
2. Modus tollens
   If A then B
   Not B
   ----------------
   Not A
3. Disjunctive syllogism
   A or B
   Not A
   ----------------
   B
4. Argument by cases
   A or B
   If A then C
   If B then C
   ----------------
   C
5. Chain (hypothetical syllogism)
   If A then B
   If B then C
   If C then D
   ----------------
   If A then D

PROPOSITIONAL LOGIC
There are several symbols that are used in propositional logic:

An argument with the same logical form as the original argument that has true premises and a false conclusion is a form of refuting the argument and is called refutation by counterexample. It shows that an argument is invalid.