Chapter 1: Introduction to Philalethic Logic

This chapter introduces the basic ideas behind Philalethic Logic and previews a few of its symbols before Chapter 2 begins the practical work of symbolizing ordinary claims. The goal is not to make you memorize a page of technical vocabulary. The goal is to give you a clear picture of what reasoning is for and what its basic parts are.

Main points in this chapter:

• Philalethic Logic helps us examine claims and arguments so that we can better distinguish truth, falsehood, and honest uncertainty.
• A universe is the totality of facts admitted within a specified domain. A situation is a coherent, relevant set of facts that could obtain within that universe. The factual situation is the one that does obtain.
• An idea is something thought about. A claim expresses an idea by presenting its subject matter as being a certain way.
• A valid argument must pass two tests: its premises must be able to be true together, and they must not be able to be true while its conclusion is false.
• Three basic principles support all proper reasoning: non-contradiction, excluded middle, and identity.
• Belief, knowledge, understanding, and wisdom are related, but they are not the same.

Philalethic Logic is a broad, practical system. Later chapters will examine claims about one, some, all, or none; claims joined together into larger claims; claims about what is necessary, possible, or impossible; and symbolic methods for testing arguments. We will build those tools gradually.

Origins and Destinations

There is a journey ahead of us. It has a starting point, a destination, and several stages between them.

The starting point is where you are now. You may be intelligent, sincere, and experienced in life while having little or no formal training in philosophy or logic. That is normal. Reasoning is something everyone does, but most people have never been taught to do it carefully and deliberately.

The destination is not perfect certainty about everything. No method can give us information we do not have. The destination is a better ability to tell truth from error, to notice weak or misleading arguments, and to recognize when the honest answer is, “I do not know yet.”

We will begin with simple ideas and claims. We will then learn how to restate, symbolize, combine, and test them. Eventually, we will use truth trees to determine whether arguments are properly structured. We will also examine common mistakes and consider how good reasoning can guide practical decisions.

Philosophy in General

The word philosophy is commonly translated as “love of wisdom.” Philosophy is the careful pursuit of truth about life and Reality, together with the effort to live according to what is true.

Philosophers ask questions such as these:

• What can we know?
• What makes an action right or wrong?
• What kinds of things exist?
• What makes something beautiful?
• Does God exist, and what can be known about God?
• What gives human life meaning?

We also use the phrase “a philosophy” for a group of beliefs, values, and habits of thought. In that sense, everyone lives according to some philosophy, even if it has never been written down. A person may believe, for example, that money is the main measure of success, that every person has equal value, that loyalty matters more than truth, or that God’s will should guide every part of life.

These beliefs shape decisions. They may also conflict. Someone may say, “Always tell the truth,” but then excuse lying whenever honesty becomes costly. Discovering such conflicts is part of learning to reason well.

Philosophers use special names for several broad areas of study. Epistemology concerns knowledge, belief, evidence, and truth. Ethics concerns what is good and how we ought to act. Metaphysics concerns what exists and how Reality is organized. Aesthetics concerns beauty and art. You do not need to memorize those terms now. This book is mainly concerned with truth, knowledge, and reasoning, although those questions touch every other area.

Reality, Universes, and Facts

Reality is everything that genuinely exists or obtains. Philalethic Logic treats Reality as the primary universe: the totality of all facts that genuinely obtain. In technical passages, we may use the Greek letter Ω, called omega, as a short symbol for Reality.

A fact is something that obtains within the universe under discussion. A Reality-level fact is something that genuinely obtains in Reality.

• The front door is locked.
• The kettle is boiling.
• Peter denied knowing Jesus.
• A certain package was delivered at 10:15 this morning.

Each of these is a Reality-level fact only when it really obtains. A sentence about the door or the package may be true or false. Facts themselves are not true or false; they simply obtain. Claims about them are true or false.

This distinction is simple but important:

• Reality-level facts belong to Reality.

• Claims concern their intended situations.

A claim is reality-directed when it is intended to describe Reality itself, rather than what obtains only within a story, parable, game, hypothesis, or other specified universe. Most ordinary claims are reality-directed and are true or false according to how well they match the relevant facts. Claims directed to other universes are evaluated within their intended situations.

We can also speak about facts within another specified universe. For example, Arthur is King of England within the universe presented by the legends of Camelot. Context determines the universe under discussion. A story, parable, game, or hypothesis may supply such a universe, but that universe is not thereby Reality.

The Human Mind and Ideas

Reasoning occurs in the mind. We do not need a complete theory of the mind before we can observe some of what it does. The mind notices, remembers, imagines, compares, questions, and judges.

An idea is something thought about. You can have an idea of a tree, justice, tomorrow’s weather, the number seven, or Moses standing before Pharaoh.

Some ideas simply present a subject for thought. Others present that subject as being a certain way:

• The tree is dead.

• The decision was unjust.

• It will rain tomorrow.

• Seven is an odd number.

• Moses spoke to Pharaoh.

Some ideas merely present a subject for thought. Others present their subject matter as being a certain way and thereby assert something. When we express such an idea in words or symbols, we call the expression a claim.

Most simple claims have a subject and something said about that subject:

“The dog is barking.”

“The soup is too salty.”

“Jesus entered Jerusalem.”

“The meeting did not begin on time.”

Later chapters will show how claims can be quantified, negated, combined, and modified. For now, the key point is this: asserting a claim presents it as true, but asserting it does not make it true.

A person may sincerely say, “The medicine is safe,” and still be mistaken. A witness may confidently identify the wrong person. The crowd before Pilate may speak with great certainty and still judge unjustly. Confidence, sincerity, and popularity can affect what people believe, but they do not determine Reality.

Ideas, Clarity, and Truth

Before a claim can be evaluated, it must be clear enough for us to tell what it says.

Consider the statement: “He is tall.”

Who is “he”? Tall compared with whom? Is the claim about a twelve-year-old, a professional basketball player, or a horse? Until the subject and standard are clear, the claim may be too vague to evaluate.

Now consider: “Daniel is six feet tall today.”

The subject, quality, and time are much clearer. We can investigate whether the claim matches the facts.

A clear claim does not need to say everything. “The cat is on the table” does not tell us the cat’s color, weight, or mood. It says enough to identify the central idea.

A claim is truth-apt when it is definite enough to be true or false. Context, time, place, standard, relationship, and respect may be part of its meaning even when they are not repeated in every sentence.

For example:

“The soup is too spicy for Maria.”

“The soup is not too spicy for Jacob.”

These claims may both be true because they concern different people. That is different from one and the same claim being “true for Maria” and “false for Jacob.” The relevant standard is built into each claim.

For a claim about Reality, truth consists in the idea matching the relevant facts. A false claim fails to match them.

Suppose the oven is actually on.

“The oven is on” is true.

“The oven is not on” is false.

Our knowledge does not create the truth value. You may be upstairs and unable to check the oven, but the oven is still either on or not on.

A claim can also concern a story, parable, or hypothesis. In Jesus’ parable of the Good Samaritan, a traveler is attacked on the road. Saying this is true of the situation presented in the parable does not require us to claim that Jesus was reporting a particular documented crime. The target of the claim is the story as told.

Likewise, we may say, “Suppose the bridge is unsafe.” Within that hypothesis, the bridge’s being unsafe is treated as obtaining so that we can examine what would follow. Treating it as true within the hypothesis does not establish that the actual bridge is unsafe.

A future-directed sentence requires additional care. Philalethic Logic does not treat the future as a present collection of facts in Reality. Such a sentence may express a prediction, promise, schedule, intention, requirement, potential, or necessity. Its content may also be treated assertorically within a stipulated or hypothetical future situation. Chapter 15 develops these uses.

Every truth-apt claim, once its intended target and use are fixed, has exactly one truth value: true or false. This does not mean we always know which value it has.

“There is intelligent life somewhere outside our solar system” is either true or false. Our present inability to settle the question does not produce a third truth value. It shows that the truth is unknown to us.

Ideas and Claims

An idea exists in the mind. A claim expresses in words, writing, or symbols an idea presented as true.

The same idea can be expressed in several ways:

“The front door is locked.”

“The entry door has been secured.”

“The door we use at the front of the house is not unlocked.”

These sentences may express the same central claim in a given context.

Different words can express the same idea, and similar words can express different ideas. “The bank is closed” may refer to a financial institution or the side of a river. Accurate reasoning requires us to identify the intended meaning.

In this book, we will often use claim for both the idea presented as true and its accurate expression. The distinction remains real, but repeatedly separating the thought from its expression would make the prose needlessly cumbersome.

A claim does not become true merely because it is spoken, printed, assumed for discussion, or placed among the premises of an argument. Those actions tell us how the claim is being used. Truth still depends upon the claim’s intended target.

Belief, Knowledge, Understanding, and Wisdom

A belief is an idea that a person judges to be true.

Human beings are fallible. We have all believed things that later proved false, and we probably hold some false beliefs now. This need not involve dishonesty. A person can be sincere and mistaken.

When faced with a claim, we have three basic choices:

1. Judge it true.
2. Judge it false.
3. Withhold judgment.

Suppose the weather forecast gives a 40 percent chance of rain. You may believe it will rain, believe it will not rain, or withhold judgment and carry an umbrella. Withholding judgment is not weakness. When the evidence does not settle the question, it may be the most reasonable choice.

This matters in disagreements. Saying, “I do not believe the accusation,” can mean either “I judge it false” or “I have not been given enough evidence to judge it true.” Those are different positions. Careful reasoning helps us state which one we mean.

For this book, we will use the traditional definition of knowledge as justified true belief. To know a claim:

1. You must believe it.
2. You must have adequate reason or evidence for believing it.
3. The claim must actually be true.

Suppose you believe the pharmacy closes at 8:00 p.m. because you checked its current posted hours, and it really does close at 8:00. Your true belief has reasonable support and may count as knowledge. If you merely guessed correctly, your belief is true, but it is not knowledge in the same sense.

Understanding goes beyond knowing isolated facts. It connects them. You may know the pharmacy’s hours, your travel time, when the prescription will be ready, and the effect of holiday closures. Understanding brings these facts together into a useful picture.

Wisdom goes further still. Wisdom applies understanding well, within the context of love and genuine human good. If a family member needs medicine tonight, wisdom does not merely recite the pharmacy’s hours. It acts in time to help.

Biblical figures often illustrate the difference. Solomon possessed great knowledge and understanding, yet his later choices also show that possessing insight does not guarantee that a person will always act wisely. Wisdom must be practiced, not merely admired.

Arguments, Validity, and Soundness

An argument is a group of claims in which some claims, called premises, are offered as reasons for another claim, called the conclusion. For example:

The first two claims are premises. The final claim is the conclusion.

A valid argument is a properly structured argument. In Philalethic Logic, that means it passes two tests.

1. The premises must be able to be true together. They must form a coherent starting point.
2. There must be no coherent case in which all the premises are true and the conclusion is false.

The James-argument passes both tests. The premises fit together, and there is no case in which all apostles are disciples and James is an apostle while James is not a disciple. Now consider:

The premises can both be true, but the conclusion can still be false. The first premise says only that some birds fly, not that every bird flies. The argument is invalid because a counterexample is available.

Premise coherence matters too. Suppose someone begins with these premises:

You might wonder “what kind of a fool would think nonsense like this?” Sadly, it is quite common. Those premises conflict. They do not provide a coherent starting point. Philalethic Logic does not treat such an argument as valid merely because no ordinary counterexample can be built from impossible premises. Contradictory premises do not prove whatever conclusion we please.

A sound argument is a valid argument whose premises are reality-directed and actually true. Validity concerns structure. Soundness adds factual truth in Reality.

We will return to arguments, validity, and soundness in much greater detail. For now, remember the two questions:

1. Can the premises all be true together?
2. Could they all be true while the conclusion is false?

Situations

Having considered claims, arguments, validity, and soundness, we can now consider the idea of a situation. A situation is a coherent, relevant set of facts that could obtain within a specified universe. It is limited to the subjects, qualities, events, times, and relationships introduced by—or required for understanding—the claims under consideration. The universe may be Reality or another specified universe, such as the world presented by a story, parable, game, or hypothesis. A situation may be very small—perhaps consisting of a single fact—or extend across millions of related facts. The factual situation is the situation that obtains within that universe. The factual situation that a claim is intended to describe is its target situation.

As explained above, an argument presents premises as reasons for a conclusion. Taken together, its claims concern particular facts within one or more universes and assert one situation—or, in some arguments, several related situations. The asserted situation is what those claims present as being the case. The argument is not itself a situation; it is a collection of claims describing one or more situations.

The claims may determine one complete situation, or they may leave several coherent variants open. In the latter case, the asserted situation includes several asserted possible situations. Each satisfies the constraints established by the argument and concerns the same bounded factors; unrelated variations elsewhere in the universe are excluded. The argument does not assert that every variant is factual. It asserts only what they share and leaves the remaining alternatives open.

Consider this argument:

The asserted situation is limited to the keys, the three relevant locations, and the time under consideration. The claims afford two asserted possible situations:

1. The keys are in the coat pocket.
2. The keys are inside the car.

Each satisfies the constraints established by the argument. If the claims accurately describe Reality, the factual situation must be one of these two. When the keys are found, the asserted possible situations can be compared with that factual situation.

The example describes only part of the larger universe of facts. It says nothing about the owner’s wallet, the weather, or countless other matters. Their omission does not assert their absence or negation. A partial description of a situation leaves unmentioned matters unsettled.

Situations may also overlap or nest. Joseph’s imprisonment can be considered as a situation in its own right, while its facts may also belong to a broader situation involving his family’s later migration to Egypt and survival during the famine. The smaller situation is included within the broader one because some of the same facts are relevant to both.

Basic Axioms of Logic

Proper reasoning rests on three basic principles. Other rules depend upon them.

Non-contradiction

A truth-apt claim cannot be both true and false in the same context, at the same time, and in the same respect. “The garage door is fully open at 9:00” and “The garage door is not fully open at 9:00” cannot both be true when they concern the same door and the same meaning of “fully open.”

The qualifications matter. The door may be open at 9:00 and closed at 9:15. A witness may see the front door open while another sees the back door closed. Those are not contradictions.

Excluded middle

Every truth-apt claim is either true or false. There is no third truth value between them. “The package was delivered by noon” is true or false, even when the tracking system is down and no one in the house has checked the porch. “Unknown” describes our knowledge, not a third state between truth and falsehood.

Excluded middle applies to individual claims. A religion, political platform, or scientific theory normally contains many claims. Some may be true and others false. We should not treat a complicated system as though it were one simple sentence with only one undivided truth value.

Identity

Each thing is what it is. Changing a name does not change the thing named. Saul of Tarsus was also called Paul. Learning the second name does not introduce a second man. When the two names refer to the same person, what is truly said of that person under one name remains true of that same person under the other, provided the time and respect remain the same.

Identity does not mean that a thing never changes. Paul could be in Jerusalem one year and elsewhere another year. It means that we must keep track of whether we are speaking about the same subject.

First Symbolic Distinctions

Philalethic Logic uses symbols to remove distracting wording and make the structure of claims easier to see. Symbols are tools, not mysterious objects.

We may write Ω for Reality, the totality of all facts. You do not need to use that symbol in ordinary conversation.

Let the capital letter A stand for any complete claim:

A

Written by itself, A presents the claim as obtaining in its intended situation.

A tilde placed before the symbol negates the complete claim:

~A

This means “A is not the case.”

Suppose A stands for “The porch light is on.” A presents “The porch light is on.” ~A means “The porch light is not on.”

The tilde denies the whole claim. Later, Chapter 5 will distinguish complete negation from the more limited operation of negating or complementing one part of a claim.

Other Systems of Logic: Aristotle

Before turning to the chapter summary, it is worth placing Philalethic Logic briefly within the longer history of logic.

Aristotle, who lived in ancient Greece, developed the earliest comprehensive system of logic for which substantial records survive. His writings were later collected under the title Organon, meaning “tool” or “instrument.”

Aristotle studied quantified claims—claims about all, none, or some members of a group. Later teachers used the letters A, E, I, and O for four common forms:

A: All dogs are animals.

E: No squares are circles.

I: Some kings are wise.

O: Some kings are not wise.

He also examined modal claims about what is necessary or possible. He treated non-contradiction as essential to reasoning and called properly structured arguments syllogisms.

Consider a familiar form:

Aristotle’s work shaped education for many centuries. Philalethic Logic shares his broad aim: to analyze ordinary reasoning while accounting for quantity, modality, and the structure by which conclusions follow from premises. It also develops a different symbolic language and validation method for modern practical use.

In Summary

Reasoning well helps us distinguish truth from error, recognize the limits of our knowledge, and make decisions that fit Reality rather than illusion.

A universe is the totality of facts admitted within a specified domain. Reality is the universe of all facts that genuinely obtain. A situation is a coherent, relevant set of facts that could obtain within a specified universe. The factual situation is the one that obtains, and the factual situation a claim is intended to describe is its target situation. The asserted situation is what a claim or set of claims presents, including any coherent variants it leaves open.

The mind forms ideas. Claims express ideas presented as true by describing their subjects as being a certain way. A truth-apt claim is true or false, even when its truth value is unknown to us.

A belief is an idea that has been judged or accepted as true. Knowledge adds adequate justification and truth. Understanding connects known facts. Wisdom applies understanding well within the context of love.

A valid argument must begin with premises that can be true together, and those premises must not be able to be true while the conclusion is false. A sound argument is valid and has true premises.

The principles of non-contradiction, excluded middle, and identity support every later part of the system.

Most people can improve their reasoning through deliberate instruction and practice. That is the purpose of everything that follows.

Chapter 2 begins that practical work by showing how ordinary claims can be reduced to subjects and predicates and represented with clear, stable labels.

Discussion Questions

1. Why is asserting a claim different from making it true?
2. What is the difference between a universe, a factual situation, a target situation, an asserted situation, and an asserted possible situation?
3. How can two claims about the same subject differ without contradicting each other?
4. Why is “I do not know” not a third truth value?
5. What is the difference between judging a claim false and withholding judgment?
6. What must be added to a true belief before it counts as knowledge?
7. Why must a valid argument begin with coherent premises?
8. Can a valid argument have a false conclusion? Under what circumstances?
9. How does wisdom differ from knowledge and understanding?
10. Give one practical example in which poor reasoning could cause real harm.

Informal Fallacies

Reasoning can fail even when its defect is not captured by the bare symbolic form of the claims. An informal fallacy is a recurring pattern in which the reasons offered do not support the conclusion as they are presented. The defect may depend upon ambiguity, relevance, omitted information, unfair representation, weak evidence, or a shift in context rather than upon the visible arrangement of operators alone.

Fallacy names are useful diagnostic tools, but they are not magic words. Calling an argument “ad hominem” or “circular” does not prove that its conclusion is false. A fallacious argument may accidentally reach a true conclusion, and a person may identify a fallacy incorrectly. The task is not merely to name a pattern, but to explain exactly how the offered reasons fail to support the conclusion.

Formal and informal evaluation therefore answer related but different questions. Formal validation examines whether the represented premises can be true together and whether a counterexample to the proposed conclusion remains available. Informal analysis also asks whether the premises were stated fairly, whether the words kept stable meanings, whether relevant evidence was omitted, and whether the premises provide the kind of support that the speaker claims for them. An argument may pass a structural test while still begging the question, relying on a distorted restatement, or hiding an important qualification.

A careful fallacy diagnosis normally has four parts. First, state the conclusion and the reasons offered for it. Second, interpret the argument as charitably and clearly as the language allows. Third, identify the precise break in support. Fourth, explain what additional evidence, qualification, or reformulation would repair the reasoning. This keeps fallacy analysis from becoming a substitute for reasoning or a rhetorical weapon against people we dislike.

Each later chapter will close with one common informal fallacy connected, as closely as practical, with that chapter’s subject. These closing questions are invitations to reflection, not additional answer-keyed exercises. The purpose is not to build a catalogue of insults. It is to practice recognizing how otherwise intelligent reasoning goes wrong in ordinary life.

Recurring question. What exactly is supposed to make this conclusion follow, and does that support survive careful clarification?

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Exercise: Belief, Doubt, or Denial?

For each statement, decide whether the speaker is judging a claim true, judging it false, or withholding judgment. Note any ambiguity or confusion.

1. “I have not seen enough evidence to decide whether the supplement works.”
2. “I do not believe the defendant is guilty,” says a juror who means that the prosecution has not proved its case.
3. “I am almost sure the flight will be late, but I am leaving early in case it is on time.”
4. “No one has proved that the bridge is safe, so it must be unsafe.”
5. Thomas says that he will not believe Jesus has risen unless he receives direct evidence.
6. “The package probably arrived, but the tracking page is down and no one has checked the porch.”

Exercise: Valid or Invalid?

For each argument, decide whether its premises can be true together and whether they could all be true while the conclusion is false.

1. All apostles were disciples of Jesus.
   Peter was an apostle.
   Therefore, Peter was a disciple of Jesus.
2. Some vehicles are electric.
   This truck is a vehicle.
   Therefore, this truck is electric.
3. Every item in this box is a book.
   This item came from this box.
   Therefore, this item is a book.
4. The same lamp is on at 8:00 p.m.
   The same lamp is not on at 8:00 p.m. in the same respect.
   Therefore, the moon is made of cheese.
5. Every warrior in Achilles’ company carried a shield.
   Patroclus was a warrior in Achilles’ company.
   Therefore, Patroclus carried a shield.