Truth table of the conditional

Question

I am currently studying the conditional. I found myself a bit puzzled with its truth table until I cooked up a few examples in my head. I am not sure if my thinking is concrete, so I post my thinking here to invite corrections.

Let $\mathrm{P: x}$ is a triangle; $\mathrm{Q: x}$ has at least one pair of sides that are mutually perpendicular.

The conditional $\mathrm{P} \rightarrow \mathrm{Q} $ is true when $\mathrm{Q} $ follows from $\mathrm{P} $ or when $\mathrm{P} $ leads us to $\mathrm{Q} $(am I right?).

Let me take a few cases into consideration:

When $\mathrm{x}$ is a right-angled triangle, both $\mathrm{P}$ and $\mathrm{Q} $ are true and it is evident that $\mathrm{Q} $ follows from $\mathrm{P} $. We have, $\mathrm{T} \rightarrow \mathrm{T} $ is $\mathrm{T} $.

When $\mathrm{x}$ is an equilateral triangle, $\mathrm{P} $ is true but $\mathrm{Q} $ is false. The conditional here is false, for an equilateral triangle can never lead us to conclude that one of its angles is a right angle. We have, $\mathrm{T} \rightarrow \mathrm{F} $ is $\mathrm{F} $.

When $\mathrm{x} $ is a square, $\mathrm{P} $ is false but $\mathrm{Q} $ is true. The conditional is true, for a square always leads us to conclude that there is at least one right angle in it. We have, $\mathrm{F} \rightarrow \mathrm{T} $ is $\mathrm{T} $.

When $\mathrm{x} $ is a circle, both $\mathrm{P} $ and $\mathrm{Q} $ are false. A circle leads us to conclude that there is not a single right angle in it. Here, $\mathrm{P} $ leads us to $\mathrm{Q} $. We have, $\mathrm{F} \rightarrow \mathrm{F} $ is $\mathrm{T} $.

I hope that someone spots an error in my thinking.

Answer 1

The conditional $\mathrm{P} \rightarrow \mathrm{Q} $ is true when $\mathrm{Q} $ follows from $\mathrm{P} $ or when $\mathrm{P} $ leads us to $\mathrm{Q} $ (am I right?).

No: the conditional is True when either the consequent $\mathrm{Q}$ is True or the antecedent $\mathrm{P}$ is False.

Regarding your example :

Let $\mathrm{P: x}$ is a triangle; $\mathrm{Q: x}$ has at least one pair of sides that are mutually perpendicular,

maybe we want to find the truth value of the corresponding universally quantified formula :

$\forall x \ (Px \to Qx)$.

If so, your reasoning is correct : if $x$ is a right-angled triangle, both antecedent and consequent are True, while if $x$ is an equilateral triangle, the antecedent id True and the consequent is False.

Thus, the formula $(Px \to Qx)$ is not always True, i.e. $\forall x \ (Px \to Qx)$ is False.

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why the truth table of the conditional is the way it is.

The propositional connectives are a very simple mathematical model of natural language, suited for modelling Mathematical arguments.

Their definition is through truth-table that are "proxy" for the corresponding natural language mechanisms.

Someone works better (negation and conjunction), someone with some arbitrariness (disjunction, inclusive : vel instead of aut); someone with a big approximation : "if..., then...".

The mathematical model of "if $A$ then $B$" represented by truth-tables does not require any sort of "link" between the two statements : antecedent and consequent.

What may help (IMO) is the interaction between the truth-functional conditional and the inference rule of Modus ponens that allows us to infer from the premises $A$ and $A \rightarrow B$, the conclusion $B$.

We must read it as Gottlob Frege did in his Begriffsschrift (1879) :

assuming as true both the premises, the assumption that $A \rightarrow B$ is true, rule-out the row $T-F$ in the truth-table for implies, while the assumption that also $A$ is true rule out two other rows ($F-F$ and $F-T$, respectively). Then, the conclusion that $B$ is true is licensed.