Phylosophyze: Everyday Logic
The following article provides a comprehensive overview of logic and context to the insights you may have already deduced in everyday life.
We start by exploring the relationship between truth and language, defining what “sentence” and “proposition” entail before delving into overviews of informal and formal logic. We explore consistency, inconsistency, and the validity of arguments before explaining some common fallacies such as “begging the question”, “cherry picking”, and “ad hominem”.
Informal Logic
Purpose of informal logic in argumentation
The function of informal logic is to be able to distinguish good arguments from bad ones. An argument is a set of propositions linked in a particular way, leading to a conclusion.
An argument is valid iff(if and only if) the conclusion follows from the premises.
If you accept the premises, you must accept the conclusion. You cannot have an argument with true premises and a false conclusion. Validity is truth-preserving; if you start with true premises and argue validly, you can be assured that the conclusion will be true.
Truth and Language
What is truth?
We must first show what exactly can be bearers of truth-values. We need to first distinguish between sentences and propositions.
What is a sentence?
A sentence is a meaningful sequence of words with a main verb. Governed by syntax and semantics. Syntax refers to the rules governing which sequences of uninterpreted words count as uninterpreted sentences. Semantics refers to the interpretation of words and sentences.
Central concepts of truth, meaning and reference, a language shows semantical complexity. There are several types of sentences: declarative (statements), interrogative and imperative(commands, wishes). We deal with declarative sentences which simply state that something is true or not “truth-apt.” We test this with the “it is true that …” prefix.
ABSOLUTISM AND RELATIVISM IN REGARD TO TRUTH
Absolutism — some say that there is no such thing as absolute truth. As such, there are no absolute standards. Relativism, post-modern thought talks of no absolute standards.
The notion of meta-narratives — everything is relativised to particular discourses. Everyone is entitled to their opinion, but not every opinion is known to be correct. The notion of “truth is a matter of opinion” is itself an opinion — an apparent contradiction.
Propositions are the bearers of truth-values; we use them to determine the crux and premises of arguments.
WHAT IS A PROPOSITION?
Propositions seem to be precisely declarative sentences. They are also context-relative and speaker-relative.
The same proposition can be expressed in different sentences. The same sentence can also express different propositions. Such as “Grass is nice to smoke.” It is unclear what is meant by the word “grass”; it may be lawn grass or marijuana; it is unclear of its context if it is spoken by a landscaper or marijuana vendor.
Words which point to the meaning/context of the sentence are called indexicals. Some propositions are ambiguous. There are two types of ambiguity: lexical and syntactical.
Lexical ambiguity is where a word has multiple meanings, and it is unclear which meaning is being alluded to.
Syntactical ambiguity occurs when a sentence is ambiguously structured and unclear about how it is intended to be read.
I never said we should kill him.
I never said we should kill him.
I never said we should kill him.
Punctuation plays a vital role in syntactic disambiguation, and the main reason for using formal logic language is to avoid this syntactical ambiguity.
We know what can be bearers of truth-values, but what is truth?
WHAT IS TRUTH?
The most basic tenet is from Aristotle: a proposition is true iff (if and only if) the world says it is at is.
There are four common “mainstream” views on truth:
Coherence Theory: A proposition is true iff it coheres with an ideal belief system. Locke: An idea is veridical iff it accurately represents an external material object. Berkeley: An idea is veridical iff it coheres with most of our other ideas, but then which one constitutes truth?
Correspondence Theory: A proposition is true iff it corresponds to the facts. We need to explain what is meant by facts. Do facts occupy space and time as they are part of physical reality? If not, what is the nature of facts? Does a fact need establishing? If so, there is not much correspondence.
Pragmatist Theory: Pragmatists like William James and C.S. Pierce assert that a proposition is true iff it is useful to believe it. Beliefs exist as a guide to action. If the theory works, it is true. Practicality is the main focus.
There is no difference that makes no difference. William James
Deflationist Theory:
P is true iff P (for all propositions P)
To say something is true is to assert it. It is true that no Hobbits is equivalent to Hobbits not existing. Paul Horwich says that there is nothing common in them (in all propositions P); and that ‘truth’ is a useful linguistic device. Truth is an abstract concept that is less important in reality.
Spiritual truth. Truth from supernatural sources, external to three-dimensional space-time, which informs of truths of our reality. Esoteric understanding of truth is derived from various sources, some claiming divinity and origin from God.
We know the common theories of truth, but what does it mean to be consistent in an argument or debate?
Consistency
Definition of consistency: a set of propositions is consistent iff (if and only if) there is a logically possible world where all are true at once. Pure thought alone can determine logical consistency.
So, we know what consistency means, but what do we mean when a proposition is impossible?
Types of impossibility
LOGICAL IMPOSSIBILITY: X is logically impossible iff it is ruled out by deduction.
METAPHYSICAL: X is metaphysically impossible iff there is no world where X holds.
CAUSAL: X is causally impossible iff X is ruled out by laws of nature.
TECHNOLOGICAL: X is technologically impossible, ruled out by current technology.
What does this entail when we say a person is inconsistent? In everyday language, a person who is inconsistent is fickle. This person asserts propositions that could not be true at once; they are mutually exclusive.
This is the definition of an inconsistent person.
Yogisms are single propositions that are logically inconsistent but considered okay in everyday understanding.
An example would be: “Nobody goes there anymore; it is too crowded.”
We know what it means to be consistent in an argument, but does this make the argument necessarily valid?
How do we define validity in terms of consistency?
An argument is valid iff the premises, together with the negation of the conclusion, form an inconsistent set of premises.
Validity is fundamentally consistent.
How do we define consistency in terms of logical inconsistency?
A set of propositions is consistent iff the negation of their conjunction is not logically necessary.
By conjunction, we mean putting the propositions together with and.
How do we define validity in terms of logical necessity?
An argument is valid iff the conditional where the antecedent is the conjunction of the premises and whose consequent in the conclusion is logically necessary.
If Socrates(F) is a man(G) and all men(G) are mortal(M), then Socrates (F) is mortal (M).
Ok, that was a lot of “if’s” and “onlys”, we now know about necessity, consistency and validity.
Sometimes, in an argument, we are presented with various analogies; perhaps we even present some as counter-points in a debate, but how should we validate their use? Through logical equivalence.
NOTION OF LOGICAL EQUIVALENCE
Two propositions are said to be logically equivalent iff the bi-conditional formed from them is logically necessary.
If two propositions are synonymous, then they are logically equivalent.
Some views on Logical Necessity
When we talk of logical necessity, we mean the following:
A proposition is logically necessary iff it is true in all possible worlds. Leibniz
P is logically necessary iff we cannot imagine it to be false. Hume
P is logically necessary iff language demands it. Wittgenstein
How bad is it to be inconsistent?
Some hold it as the worst intellectual crime. Others point out that the inconsistency is not as significant if it was unobvious at first. Some mistakes, like factual inconsistencies, are not as blameworthy.
But what rules govern consistency itself?
THREE LAWS OF CONSISTENCY IN PROPOSITIONS
THE LAW OF NON-CONTRADICTION
No proposition can be both true and false. Both it and its negation cannot be true at once (CANNOT BE P AND NOT P).
THE LAW OF IDENTITY
Everything is what it is and not something else.
THE LAW OF EXCLUDED MIDDLE
Every P is either true or false. Either it or its negation is true.
Most hold that the law of non-contradiction underpins the foundation of logical argumentation. Below are some views on this fundamental principle.
VIEWS ON NON-CONTRADICTION
IBN SINA, in METAPHYSICS 1 (1973)
Anyone who denies the Law of Non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned.
Walt Whitman (1855), Leaves of Grass (pg 51):
Do I contradict myself? Very well, I contradict myself (I am large, I contain multitudes).
It is acceptable to have some inconsistencies as it proves human error and isn’t overly formulaic. Dialethisms are exceptions to the law of non-contradiction:
This proposition is false
The view of non-contradiction’s critical importance comes from the fact that you can derive any conclusion from false premises, ex falso quodlibet.
THE EXPLOSION PRINCIPLE, EX FALSO QUODLIBET
The Latin for explosion principle literally means: “from falsehood, anything”.
From an inconsistent set of premises, we can derive ANY conclusion whatsoever.
An argument is valid iff the premises, taken together with the negation of the conclusion, form an inconsistent set.
General proof of ex falso quodlibet is as follows.
Prove the validity of the following statement:
“I am a ski instructor (A), and unicorns exist(B)”. Suppose we assume that both are true and its negation (A) is true; then, we can derive any conclusion and substitute whatever we want for (B).
A, Not- A, A or B, B
We assume A (I am a ski instructor)
We also assume Not-A ( I am not a ski instructor)
Therefore, we deduce that;
Either A or B (unicorns exist)
We assume Not-A (I am not a ski instructor)
therefore B (Unicorns exist), as we must keep the above statement true.
Q.E.D (Quod Erat Demonstradum), Which was to be demonstrated.
This argument form is called a dysjunctive syllogism.
CONDITIONAL SENTENCES
A conditional sentence is a complex sentence with the form “If …. then … “
The sentence which forms the “if” part is the antecedent. The sentence which forms the “then” is the consequent. Again, limitations of language do limit their comprehension; the context must be made clear and explicit.
NECESSARY AND SUFFICIENT CONDITIONS
One statement is said to be a sufficient condition for a second if the truth of the first guarantees the truth of the second.
If Socrates is a man and all men are mortal, then Socrates is mortal; Socrates being a man is a sufficient condition of his mortality.
A statement is a necessary condition for a second if the truth of the first is required for the truth of the second.
Only if Socrates is a man and all men are mortal, then Socrates is mortal
Here, “Only if” reverses the antecedent and consequent, Socrates being a man is a necessary condition for his mortality.
In formal notational logic, it can be expressed as:
Sufficient condition: R -> N
Necessary condition: R ←-> N.
We can further explore the different types of conditionals as explained below.
INDUCTIVE AND SUBJUNCTIVE CONDITIONALS
Grammatical style and mood can have a substantial impact on the meaning of a conditional sentence.
Inductive conditional: concerns only what is the case.
If Oswald did not kill JFK, then someone else did
Subjunctive conditional: concerns what might have been the case.
If Oswald hadn’t killed JFK, someone else may have
CONDITIONALS AND VALIDITY
Validity is determined by the logical form of the argument. Two common valid forms of argument, modus ponens and modus tollens, as well as two common fallacious forms of argument.
Assume that A: I am a ski instructor, B: I have a job.
Modus ponens (putting forward):
A → B, A, B
Modus tollens (taking away):
A → B, Not-B, Not-A
Two common fallacious forms of argument:
The fallacy of affirming consequent:
A → B, B → A
The fallacy of denying the antecedent:
A → B, Not- A → Not-B
Substitute A for Ski instructor, B have a job
FORMAL PROOF OF MODUS TOLLENS
If we assume A, then B, modus ponens, but we know B is not true; therefore, A is not true.
If I am a ski instructor (A), I have a job (B). I don’t have a job (Not B), then I am not a ski instructor (Not A). (But this is a contradiction, I have a job)
METHODS OF DERIVING CONCLUSIONS IN AN ARGUMENT
We arrive at conclusions within an argument, either through deduction or induction. But what are the differences?
INDUCTION
Inductive logic is non-monotonic, meaning the addition of premises may weaken the conclusion.
The inductive strength of an argument comes in degrees, as probability comes in degrees.
An argument is said to be inductively strong iff it is improbable given the premises; the conclusion is false.
Inductive arguments are ampliative; they expand/generalise on what is contained in the premises.
The problem of induction is showing such generalisation is permissible. Inductive logic is difficult to formalise with classical notational logic.
DEDUCTION
Deductive logic is monotonic; the addition of premises does not weaken the conclusion.
An argument is either valid or invalid.
An argument is said to be deductively valid iff the conclusion follows from the premises and logically impossible for the premises to be true and the conclusion false.
Deductive arguments merely unpack what is implicitly held in premises.
So now you know the material behind the everyday logic you encounter, the above should give a brief overview of informal logic and argumentation and provide you with important context for everyday life.
Informal Fallacies
PETITIO PRINCIPII, BEGGING THE QUESTION
If you tacitly assume what you set out to prove, you are said to be “begging the question”.
For example, athletes are healthy, and I am healthy, so I am an athlete.
IGNORATIO CLENCHIS — IGNORANCE OF REFUTATION
Refers to a situation where a conclusion is established, soundly perhaps, but it is NOT what the arguer set out to prove.
Error is committed if the argument is misread as an argument for something else.
TU QUOQOE “YOU TOO”
An attack on the person making the argument in the form of “if X says NOT-A, But then X does A, it does not mean A is concluded to be right”. This is an invalid argument.
If you say, do not drink and drive,
then you drink and drive,
then I say: drinking and driving is okay since you do it.
This is an invalid form of argument.
SIMILAR RELATION TO AD HOMINEM
Refers to argument type directed against the person themselves rather than the argument.
Criticising a personal trainer for advocating the benefits of exercise.
FALLACY OF EQUIVOCATION
A fallacy of equivocation takes place when an argument tracks on ambiguity.
This may be syntactical (grammar structure) or lexical ambiguity (concerning the definition and application of words in sentences).
Difficult to identify as words in certain contexts have varying definitions and meanings (no precise definitions [perhaps multiple definitions])
This is why Ludwig Wittgenstein held the view that “Philosophy is a battle against the bewitchment of our intelligence by means of language.”
BAD ARGUMENTS FALLACY
Usually expressed in the form: “There are bad arguments for X to be true, therefore Not-X.”
Just because X has not yet had good arguments for it, does not mean we can assume Not-X.
RELATED TO: ARGUMENTUM AD IGNORANTIUM
X has not been proven true; therefore, X is false.
But I always say, the absence of evidence is not the evidence of absence.
The Boondocks (2010), Series 3, Episode 15, “It’s Going Down”
Just because something has not yet been proven does not mean we can assume it is false.
WEAK ANALOGY
Fallacy assumes that as X is similar to Y in some aspects, it is similar in all aspects.
FALSE DILEMMA
A widespread (ubiquitous) fallacy includes limiting positions to fewer than available and limiting the other side to picking from these outlined positions.
PERFECTIONIST FALLACY
Placing excessive demands on a proposal and rejecting it purely as it does not completely solve the problem.
For example, some may reject quantum mechanics and the uncertainty with the general theory of relativity as it does not have a perfect synthesis.
CHERRY PICKING
Selecting carefully only the evidence that supports your premises whilst knowingly omitting opposing evidence.
Inductive reasoning is non-monotonic, so additional premises can weaken arguments.
This is as opposed to monotonic logic, whereby additional premises do not weaken the conclusion.
Arguments are usually presented in the form of conditional sentences. But what exactly are conditionals?
SUMMARY
We have learned the following:
Purpose of informal logic in argumentation
Truth and its relationship with language
Consistency and Inconsistency
Informal fallacies and their features
Conditionals and their use in argumentation
Deduction and induction as methods of argumentation