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Chapter 1: Problem 10

Show that the relation "is a subformula of" is transitive.

Short Answer

Expert verified
The relation "is a subformula of" is transitive if each subformula of a subformula is also a subformula.

Step by step solution

01

Understand the Definition of Subformula

Start by considering what it means for one formula to be a subformula of another. For a formula \( \phi \) to be a subformula of another formula \( \psi \), \( \phi \) must appear in the syntactic structure of \( \psi \).
02

Define Transitivity

A relation is transitive if whenever \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) must be related to \( c \). In the context of subformulas, if \( \phi \) is a subformula of \( \psi \) and \( \psi \) is a subformula of \( \theta \), then \( \phi \) should be a subformula of \( \theta \).
03

Demonstrate Using Logical Formula Structure

To demonstrate the transitivity, consider logical formulas constructed from atomic formulas and logical connectives. If \( \phi \) is embedded within \( \psi \) (making \( \phi \) a subformula of \( \psi \)), and \( \psi \) is embedded within \( \theta \), then the structure that contains \( \psi \) in \( \theta \) must also contain \( \phi \), thus ensuring \( \phi \) is a subformula of \( \theta \).
04

Formalize the Proof

Suppose \( \phi \) is a subformula of \( \psi \) and \( \psi \) is a subformula of \( \theta \). By the definition of subformula, there exists syntactic positions in \( \psi \) where \( \phi \) is a part. Similarly, the entirety of \( \psi \) is part of \( \theta \). Hence, the positions of \( \phi \) in \( \psi \) are preserved in \( \theta \,\), making \( \phi \) a subformula of \( \theta \). Thus, proving transitivity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Formal Logic
Formal logic is a system of reasoning that lays down rules and principles to determine the validity of arguments and propositions. It uses symbols to represent logical forms and relationships, allowing for precise and clear communication. This system helps you understand how propositions interact in logical structures, such as proving the properties of logical relations like transitivity. In the context of transitive relations, mastering formal logic enables you to see how certain properties, including that of a subformula, are maintained through chains.
Formal logic uses specific syntax, akin to grammar in a language, and is fundamental for constructing valid logical statements that can be analyzed and evaluated.
  • Formal logic entails the use of logical statements, which are broken down into expressions and operators.
  • Symbols and expressions are used to codify logical arguments coherently.
  • Understanding the rules of formal logic helps to establish proofs and validate arguments systematically.
Formal logic, though abstract, forms the foundation of disciplines such as mathematics, computer science, and philosophy, where logical reasoning is indispensable.
Syntactic Structure
The syntactic structure in formal logic refers to the way symbols and formulas are organized and related. This structure determines how formulas are constructed and understood. It involves carefully arranging logical symbols, functions, and operators to form coherent and meaningful statements.
The syntactic structure is like a blueprint for understanding how complex formulas are built. For instance, if a formula is a subformula of another, it means its structure appears within a larger formula.
  • Syntactic structures are built using formalized rules and guidelines.
  • The arrangement defines the meaning and validity of logical expressions.
  • Understanding these structures allows for deeper insights into how various components relate within formulas.
Syntactic structure is crucial for determining properties like transitivity because it allows you to trace how subformulas are embedded or nested within larger formulas, maintaining logical integrity through complex systems.
Logical Formulas
Logical formulas are expressions made up of logical symbols and operators that convey logical relationships and propositions. They serve as the fundamental building blocks within formal logic to analyze and evaluate arguments. These formulas entail components such as variables, constants, logical operators (like \(\land\), \(\lor\), \(\rightarrow\)), and often follow a strict syntactic order.
Logical formulas are essential in illustrating the concepts of subformulas and transitive relations. A subformula is a formula that exists within another larger logical formula, much like a subset within a set.
  • Each logical formula can consist of multiple atomic or compound parts.
  • Atomic formulas represent the simplest statements, while compound formulas are combinations using logical operators.
  • The construction and interpretation of logical formulas rely heavily on syntax rules.
Understanding logical formulas enables you to approach complex logical scenarios by breaking them down into their constituent parts, thus simplifying analysis and proof tasks.
Subformula Relation
A subformula relation is a logical concept where one formula is contained within another. It represents a part-whole relationship, where the subformula makes up a portion of a larger formula. This concept is pivotal in formal logic for understanding and proving properties like transitivity.
The transitive property in terms of subformula relations could be likened to stacking Russian dolls, where each smaller doll fits within a larger one. If a subformula \(\phi\) is part of a formula \(\psi\), and this \(\psi\) is itself a part of a larger formula \(\theta\), then \(\phi\) is also inherently a part of \(\theta\).
  • A subformula is a foundational part that contributes to the broader logical statement.
  • Recognizing a subformula involves identifying the part as it exists within the syntactic framework of another formula.
  • This relation showcases the interconnected nature of logical statements and their dependency on structural integrity.
Subformula relations are core to proving logical properties such as transitivity, serving as a micro-scale example of how logical components interrelate and maintain systemic coherence.

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Most popular questions from this chapter

Check which of the following sets are consistent. (a) \(\left\\{\neg p_{1} \wedge p_{2} \rightarrow p_{0}, p_{1} \rightarrow\left(\neg p_{1} \rightarrow p_{2}\right), p_{0} \leftrightarrow \neg p_{2}\right\\}\), (b) \(\left\\{p_{0} \rightarrow p_{1}, p_{1} \rightarrow p_{2}, p_{2} \rightarrow p_{3}, p_{3} \rightarrow \neg p_{0}\right\\}\), (c) \(\left\\{p_{0} \rightarrow p_{1}, p_{0} \wedge p_{2} \rightarrow p_{1} \wedge p_{3}, p_{0} \wedge p_{2} \wedge p_{4} \rightarrow p_{1} \wedge p_{3} \wedge p_{5}, \ldots\right\\}\)

Show that the connective \(\downarrow\), with valuation function \([\varphi \downarrow \psi]=1\) iff \([\varphi]=[\psi \rrbracket=0\), forms a functionally complete set (neither \(\varphi\), nor \(\psi)\).

Simplify the following propositions (i.e. find a simpler equivalent proposition). (a) \((\varphi \rightarrow \psi) \wedge \varphi\), (b) \((\varphi \rightarrow \psi) \vee \neg \varphi\), (c) \((\varphi \rightarrow \psi) \rightarrow \psi\), (d) \(\varphi \rightarrow(\varphi \wedge \psi)\), (e) \((\varphi \wedge \psi) \vee \varphi\), \((\mathrm{f})(\varphi \rightarrow \psi) \rightarrow \varphi\)

Consider the full language \(\mathcal{L}\) with the connectives \(\wedge, \rightarrow, \perp, \leftrightarrow \vee\) and the restricted language \(\mathcal{L}^{\prime}\) with connectives \(\wedge, \rightarrow, \perp\). Using the appropriate derivation rules we get the derivability notions \(\vdash\) and \(\vdash^{\prime}\). We define an obvious translation from \(\mathcal{L}\) into \(\mathcal{L}^{\prime}\) : \(\begin{aligned} \varphi^{+} &:=\varphi \text { for atomic } \varphi \\\\(\varphi \square \psi)^{+} &:=\varphi^{+} \square \psi^{+} \text {for } \square=\wedge, \rightarrow, \end{aligned}\) \((\varphi \vee \psi)^{+}:=\neg\left(\neg \varphi^{+} \wedge \neg \varphi^{+}\right)\), where \(\neg\) is an abbreviation, \((\varphi \leftrightarrow \psi)^{+}:=\left(\varphi^{+} \rightarrow \psi^{+}\right) \wedge\left(\psi^{+} \rightarrow \varphi^{+}\right)\), $$ (\neg \varphi)^{+}:=\varphi^{+} \rightarrow \perp . $$ Show (i) \(\vdash \varphi \leftrightarrow \varphi^{+}\), (ii) \(\quad \vdash \varphi \Leftrightarrow \vdash^{\prime} \varphi^{+}\), (iii) \(\varphi^{+}=\varphi\) for \(\varphi \in \mathcal{L}^{\prime}\). (iv) Show that the full logic, is conservative over the restricted logic, i.e. for \(\varphi \in \mathcal{L}^{\prime} \vdash \varphi \Leftrightarrow \vdash^{\prime} \varphi\).

Show (a) \(\varphi \models \varphi\); (b) \(\varphi \models \psi\) and \(\psi \models \sigma \Rightarrow \varphi \models \sigma\); (c) \(\models \varphi \rightarrow \psi \Rightarrow \varphi \models \psi\).
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