CST 329 - Week 1

This was our first week in CST 329 - Reasoning and Logic.

Our textbook for this class is A Concise Introduction to Logic, which is available for free.

Chapter 1: Developing a Precise Language

Propositional Logic: Meaning the logic of sentences. Sometimes also called "sentential logic" or "sentential calculus". In propositional logic the smallest independent parts of a language are sentences.

Declarative Sentence: A sentence that describes the world and can be either true or false. An example would be "It rained in Central Park on June 26, 2015". A declarative sentence cannot be ambigous, for example "Tom is kind of tall" is too vague to be declarative (how can we tell if Tom is tall?).

 

Bivalence: Also called the principle of non-contradiction. The belief that there are two truth values (true and false) and that they exlude each other, meaning that something must either be true or false, and cannot be both or neither.

 

Primitives: Something added to a logical language that is not defined in detail.

 

Atomic: A sentence is atomic if it cannot be split up into smaller parts. For example, "Malcolm Little is tall" can't be split up into smaller sentences, but "If Lincoln wins the election, Lincoln will be president can (Lincoln wins the election. Lincoln will be president.).

 

Symbols for Atomic Sentences: Upper case letters starting from P (P, Q, R, S...) are commonly used to represent atomic sentences. For example, "Malcolm Little is tall" might be represented by P.

 

Truth Table: A table that describes the conditions of a sentence being true or false.

 

Figure 1 - Basic layout of a truth table.

 

Truth Table for atomic sentence P.

 

 Metalanguage: Language describing a logical language. Greek letters (such as Φ and Ψ) are often used as variables or placeholders for atomic sentences.

 

Definition of Truth Table for atomic sentences.

 

Vagueness: A sentence is vague if the conditions under which it may be true are "fuzzy". For example, "Tom is tall" is vague because we do not know what being "tall" means.

• Often we can make a vague sentence precise by defining a specific interpretation of the meaning of an adjective, term, or other element of the sentence. For example, we could make the above sentence precise by specifying that "is tall" is true for anyone six feet or taller. 

Ambiguity: A sentence is ambiguous when it has one or more possible interpretations. For example in "When Karen had a baby, her mother gave her a pen" the word pen could either refer to a writing implement or a baby crib. An ambiguous sentence could be true for one interpretation but false for another.

• Often we can make an ambiguous sentence precise by specifying which of the possible meanings we intend to use.

Chapter 2: "If... then..." and "It is not the case that..."

Conditionals: Conditionals such as "if... then..." and "not" allow us to explain the relationship between multiple atomic sentences.

• If we have the sentence "If Lincoln wins the election, then Lincoln will be President.", we can represent "Lincoln wins the election" as P and "Lincoln will be President" as Q.
• We can then represent the their relationship as if P then Q.
• We use -> to represent "if... then..." so the relationship would be P->Q.

Parenthesis: We can show that a sentence is a single sentence using parenthesis (P->Q).

The sentence P->Q is called a conditional. The first constituent sentence (P) is the antecedent. The second constituent sentence (Q) is the consequent. Sometimes the conditional is represented with P⊃Q.

Truth Table of  atomic sentences P and Q 

Number of rows in a Truth Table: For n atomic sentences, a truth table has 2^n rows.

Truth Table of -> (If Then) relationship 

-> Truth Table: True except when the if is true and the then is false.

Not Conditional: The conditional "not" is represented by ¬. So not P would be ¬P. In programming it is often represented by ! (!P). ¬ is sometimes represented by ~ (~P).

Truth Table of ¬ (Not) relationship 

 

Chapter 3: Good Arguments

Argument: An ordered list of sentences; we call one of these sentences the "conclusion", and we call the other sentences "premises". The premises of the argument are meant to show us that the conclusion is true.

Valid argument: An argument for which, necessarily, if the premises are true, then the conclusion is true.

 

Sound argument:  a valid argument with true premises.

 

Figure 7 - Truth Table for a valid argument (P->Q) P | Q. 

In the condition where both premises are true, the conclusion is true.

Figure 8 - Truth Table for invalid argument (R->S) ¬R | ¬S

In a condition where both premises are true, the conclusion is false. 

Figure 9 - Truth Table for valid argument (T->U) ¬U | ¬T

 In the condition where both premises are true, the conclusion is true.

Scientific Method

1. We develop a hypothesis about the causes or nature of a phenomenon.
2. We predict what (hopefully unexpected) effects are a consequence of this hypothesis.
3. We check with experiments to see if these predictions come true:
• If the predictions prove false, we reject the hypothesis.
• If the predictions prove true, we conclude that the hypothesis could be true.  We continue to test the hypothesis by making other predictions (that is, we return to step 2).

If we have multiple hypotheses that may be true, we prefer one following the below criteria (in order of relative importance):

1. Predictive power: the more that a hypothesis can successfully predict, the better it is.
2. Productivity:  a hypothesis that suggests more new directions for research is to be preferred.
3. Coherence with Existing Theory: if two hypotheses predict the same amount and are equally productive, then the hypothesis that coheres with (does not contradict) other successful theories is preferable to one that does contradict them.
4. Simplicity: if two hypotheses are equally predictive, productive, and coherent with existing theories, then the simpler hypothesis is preferable.