Logic: deductive validity - summary of chapter 3 of Critical thinking: A concise guide by Bowell & Kemp (4th edition)

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Chapter 3
Logic: deductive validity

Argument reconstruction: the representation of arguments in standard form, so as to give us a clear and comprehensive view of them.

Argument assessment: the determination of whether or not arguments provide good reasons for accepting their conclusions.

The principle of charity

An argument is a system of propositions.
Propositions: a set of premises advanced in support of a conclusion.
People succeed in expressing the propositions they have in mind in varying degrees of clarity. An argument may depend upon premises that the arguer does not state at all, but which he or she is implicitly assuming.

Since the purpose of argument-reconstruction is to determine exactly what argument has been given, part of the task is to clarify what the arguer actually said, and to supplement what the arguer actually said (to make explicit what was merely implicit in the arguer’s statements).

• The sentences we use in a reconstruction of the argument need not to be the very same sentences as used by the arguer in giving their argument. We may employ sentences that more clearly or precisely express the propositions that constitute the argument.
• Our reconstructed version of the argument may contain premises that are not expressed by any of the sentences actually used by the arguer.

Argument-reconstruction is essentially a task of interpretation.

The principle of charity.

In such facts pertaining to the context in which the argument is given, together with the specific words used by the person, will constitute the total evidence you have for reconstructing the argument.

• In some cases, the context is known, and makes it obvious what the arguer was implicitly assuming.
• In other cases, we may have to learn more about the context.
• In some other cases, however, we may learn all the relevant contextual factors, yet it remains possible to represent the person’s argument in more than one way.

If, in the third case, you have to chose what representation of the argument is true, it depends on your purpose.

• If you are hoping to convince others that the person is wrong, you are most likely to succeed if you represent it as a bad one.
• If what matters to you is whether or not the conclusion of the person’s argument is true, then you should choose the best representation of the argument.

Otherwise you could conclude nothing.
The fact that someone has given a bad argument for some proposition is not, in itself, a reason to reject the proposition as false. → if you reject the argument, the proposition is unchanged. If you accept, you have a different proposition, it is true.

The principle of charity: we should always choose the best reconstruction of a given argument.
In that way, we discover reasons for accepting or rejecting particular propositions, advancing the cause of knowledge.
And because of respect for others. You want to understand others, also when they struggle with finding the right words.

The principle of charity has a certain threshold.
If our task is to reconstruct the argument actually intended by the person, then we must not go beyond what, based upon the evidence available for us, we may reasonably expect the arguer to have had in mind.
Once we go beyond what we may reasonably assume the arguer have had in mind, then we are no longer in the business of interpreting their argument. Instead, we have become the arguer.

If our concern is with how well a particular person has argued, we should not overstep this boundary.
If our concern is with the truth of the matter in question, then to overstep this boundary is perfectly all right.

It often happens that, in reconstructing an argument, we hit upon another, similar or related argument for the same conclusion which is better than the one we are reconstructing.

• If what concerns us is simply the best arguments on either side of an issue, then we will want to give a representation of this better argument.

Truth

Logic gives us some very clear answers as to what does make arguments good or bad.

The fundamental concept of logic is the concept of truth.

• The overarching concern of the critical thinker is typically with the truth (or lack of it) of the conclusions of arguments.
• Truth is the concept in terms of which the logician attempts to explain everything else.

Truth: (for this book) the way things are.
To say a proposition is true is to say it is the way things are.

If A and B are equivalent is that, necessarily, if A is true, then so is B. And if B is true, then so is A.

Discomfort with the word ‘true’ is sometimes due to a failure to distinguish truth from belief.
Beliefs depend on a person, not on the world.

Truth-value of a proposition: the truth of the proposition, if its true, or its falsity, if it is false.
There are two truth-values. True and false.

Deductive validity

Deductive validity (or simply validity)

The concept of validity pertains to the connection between the premises and conclusion of an argument, not their actual truth-values considered individually.
It pertains to inferences. Extended arguments may contain more than one inference, and each one is subject to being valid or invalid.

A single proposition can be true or false, but not valid or invalid.
An argument can be valid or invalid, but not true or false.

To say that an argument is valid is to say: if the premises are (or were) true, the conclusion would also have to be true.

If the condition specified by the definition does not hold, then the argument is invalid.

The following cases of valid argument are possible

• The premises are all (actually) true, and the conclusion is (actually) true
• The premises are all (actually) false, and the conclusion is (actually) false
• The premises are all (actually) false, and the conclusion is (actually) true
• Some of the premises are (actually) true, some are (actually) false and the conclusion is (actually) true
• Some of the premises are (actually) true, some are (actually) false and the conclusion is (actually) false.

The only case in which an argument cannot be valid is the case when the premises are all (actually) true, but the conclusion is (actually) false.

How to judge validity

The way to determine whether or not an argument is valid is to ignore the actual truth-values of the premises and the actual truth-value of the conclusion.
Whether or not they are actually true, suppose or pretend that the premises were all true, then in that situation, could the conclusion conceivably be false? If it could not be false, then the argument is valid. If it could be false, then the argument is valid.

Prescriptive claims vs descriptive claims

Descriptive claims: fact-stating claims.

Prescriptive claims: claims which state or express desires, norms or moral values.

To claim a proposition is true, is the same thing as agreeing with it.
The distinction between descriptive and prescriptive claims is not be made out in terms of truth.

Conditional propositions

• If not… then not

Contraposition: saying ‘if not-Q then not-P’ is equivalent to saying ‘if P then Q’ in logic

• Either-or

Usually, when wo statements are joined by ‘either-or’, or just by ‘or’ to form a compound statement, the compound is equivalent to a statemetn using if-then, and vice versa. But to pass from the version using ‘or’ to the version using ‘if-then’, or vice versa, we have to insert the word ‘not’.
The word ‘or’ is used in either of two ways.

•  
  • Exclusive sense
    Either P is true, or Q is true, but not both.
  • Inclusive sense.
    Either P is true, or Q is true, or they are both true.

• Only if

P only if Q.
‘P only if Q’ means the same as ‘If P then Q.’
‘If and only if P then Q’ means the same as ‘If P then Q’, and if ‘if Q then P’.

• Unless

The trick for dealing with ‘unless’ is to think of it as meaning ‘if not’.

The antecedent and consequent of a conditional

In general, a conditional is a compound proposition consisting of two parts, each of which is itself a proposition, where these two parts are joined by some connecting words.
What a conditional says is that the truth of one proposition ensures that of another.
In formal logic it is →

Antecedent: the one form which the arrow points.
Consequent: the one to which the arrow points.

Conditionals vs arguments

A conditional is said to be true or false, rather than valid or invalid.
A conditional is not itself an argument.

A conditional: one proposition that comprises two propositions as parts, joined by ‘if-then’ or a similar device.
An argument cannot consists of just one proposition.

A conditional does not assert either its antecedent or its consequent.
An argument asserts its premises and its conclusion.
But, many arguments have conditional conclusions.

Argument trees

Argument trees: devices that can be used for representing arguments in the form of a diagram.
They are helpful when we are reconstructing arguments, particularly complex ones, because they provide a means of showing the ways in which the different parts of an argument are related to each other.
The process of constructing an argument tree is especially useful before you have supplied missing premises and before you have settled upon a reconstruction of the argument in standard form.

Deductive soundness

Knowing that the argument is valid, is not enough to show you that the conclusion is true.
In order to determine that, you must determine the truth-values of the premises.

To say that an argument is deductively sound is to say: the argument is valid, and all its premises are (actually) true.

The conclusion of a deductively sound argument must be true.
An argument that is not deductively sound is deductively unsound.
Deductive soundness pertains to whole arguments.

If we know that the conclusion of an argument is false, then we know that the argument is deductively unsound.
If the argument is deductively unsound, it follows that either the argument has (at least) one false premise, or the argument is invalid.

If the argument is valid, but unsound, (at least) one of the premises must be false.
If you found that the argument is invalid, you can conclude nothing about the premises.

The connection to formal logic

Modus ponens:

P1) If P then Q
P2) P
C) Q

Modus tollens:

P1) If P then Q
P2) Not-Q
C) Not-P

Disjunctive syllogism:

P1) P or Q
P2) Not-P
C) Q

Argument by cases:

P1) P or Q
P2) If P then R
P3) If Q then R
C) R

Chain (or hypothetical syllogism):

P1) If P then Q
P2) If Q then R
P3) If R then S
C) If P then S

Formal logic:

→ If then

V: inclusive

‘or’, ‘&’, ‘Ʌ’ : and

¬ (or ~): not

Letters (like P and Q) are placeholders for whole sentences or whole statements.

Quantificational logic.
Representing arguments in rigorous ways has certain advantages

• there are mechanical or algorithmic means of identifying the validity of an argument, no matter how complex or how long. The validity of argument expressed in symbolic notation is determined by precise rules, and the forms of the argument are always exactly determined.

Formal logic claims that logical relationships are made by abstracting from the particular subject matter we talk about and concentrating on the logical forms of arguments.
But logical forms are not completely without content or meaning.

Refutation by counterexample: an argument with the same logical form as the original content, thereby proving that the logical form of the original argument is invalid, and therefore that he original argument itself is invalid.