Analyze the natural language arguments below and match each to its precise formal symbolic… — LSAT

Analyze the natural language arguments below and match each to its precise formal symbolic structure. Use variables $X$ , $Y$ , and $Z$ to represent the distinct conditions in the order they appear.

If the chemical plant reduces its emissions, the local water quality will improve. Furthermore, the endangered fish population will recover if the water quality improves. The chemical plant clearly did not reduce its emissions, because the fish population is not recovering.Premises: $X \rightarrow Y$ , $Y \rightarrow Z$ . Fact: $\sim Z$ . Conclusion: $\sim X$ (Valid Contrapositive)
Class sizes will not decrease. After all, the university decided against increasing tuition. Yet we know that if tuition increases, more faculty can be hired, and if more faculty are hired, class sizes decrease.Premises: $X \rightarrow Y$ , $Y \rightarrow Z$ . Fact: $\sim X$ . Conclusion: $\sim Z$ (Invalid Negation)
If the new software is intuitive, employees will adopt it quickly. A rapid adoption rate guarantees that overall productivity will rise. Thus, since overall productivity has indeed risen, the new software must be intuitive.Premises: $X \rightarrow Y$ , $Y \rightarrow Z$ . Fact: $Z$ . Conclusion: $X$ (Invalid Reversal)
The governor signed the bill. Since signing the bill ensures the transportation budget will expand, and an expanded budget means the highway repair project will begin, the highway project will definitely begin.Premises: $X \rightarrow Y$ , $Y \rightarrow Z$ . Fact: $X$ . Conclusion: $Z$ (Valid Transitive)

Answer

The chemical plant argument is a Valid Contrapositive. The university tuition argument is an Invalid Negation. The software argument is an Invalid Reversal. The governor argument is a Valid Transitive chain.

Each natural language argument corresponds exactly to one symbolic representation by mapping its sufficient and necessary conditions in order. The correct match depends on whether the given premise affirms or denies either the beginning or the end of the conditional chain.

Step-by-Step Solution

Translate the 'chemical plant' argument into symbolic logic.

Plant reduces (

X

)

\rightarrow

Water improves (

Y

). Water improves (

Y

)

\rightarrow

Fish recover (

Z

). Fact: Fish do not recover (

\sim Z

). Conclusion: Plant did not reduce (

\sim X

Tracking the argument formally reveals a valid application of the contrapositive across a conditional chain.

Translate the 'university tuition' argument into symbolic logic.

Tuition increases (

X

)

\rightarrow

Faculty hired (

Y

). Faculty hired (

Y

)

\rightarrow

Class sizes decrease (

Z

). Fact: Tuition does not increase (

\sim X

). Conclusion: Class sizes will not decrease (

\sim Z

This highlights the logical flaw of illegal negation, as the absence of a sufficient condition does not guarantee the absence of a necessary condition.

Translate the 'new software' argument into symbolic logic.

Software intuitive (

X

)

\rightarrow

Adopted quickly (

Y

). Adopted quickly (

Y

)

\rightarrow

Productivity rises (

Z

). Fact: Productivity rose (

Z

). Conclusion: Software intuitive (

X

This uncovers the flaw of illegal reversal, where affirming a necessary condition is incorrectly used to deduce that the sufficient condition was met.

Translate the 'governor bill' argument into symbolic logic.

Signs bill (

X

)

\rightarrow

Budget expands (

Y

). Budget expands (

Y

)

\rightarrow

Project begins (

Z

). Fact: Signs bill (

X

). Conclusion: Project begins (

Z

This represents a straightforward, valid transitive chain application (Modus Ponens) moving from a sufficient trigger to the ultimate necessary outcome.

Key Concept

Parallel Conditional Logic

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