I need to formalize this sentence and decide whether the logical consequence applies

Tom drove either by bus (b) or tram (t). If he drove by tram (t) or by car (a) then he arrived late (p) and did not hear the lecture (v). He did not arrive late (non p). It follows that he rode the tram (t) or missed the lecture (v)

I create this:

$T \vee B$, $(T \vee A)\Rightarrow(P \wedge V)$, $\neg P$, ? $T \vee V$

with the help of a semantic tree, I decided that is not consequence. Is that right?

  • $\begingroup$ Are you familiar with the usual notation for logical connectives? Would it be okay to edit $T\vee B, (T\vee A)\rightarrow(P\wedge V), \neg P\vdash T\vee V$ into your question? And, as far as I am aware, semantic trees are mostly used in CS, and not to test arguments, do you mean a sequent? $\endgroup$ – GVT Feb 10 at 13:06
  • $\begingroup$ Correct; the purported conclusion is not consequence of the premise. Consider the first premise: $T \lor B$; if case $T$, we are Ok, but if case $B$, there is no way to "reach" the conclusion. $\endgroup$ – Mauro ALLEGRANZA Feb 10 at 13:08
  • $\begingroup$ The formulation "consequences were not true" is wrong. You can say that the consequences are true or false only when you are given an evaluation. I think, you meant "$T\vee V$ is not a consequence". $\endgroup$ – Hume2 Feb 10 at 13:09
  • $\begingroup$ Yes, i meant it is not consequence. $\endgroup$ – Aaron7 Feb 10 at 13:14
  • $\begingroup$ and is the wording of the statement correct? $\endgroup$ – Aaron7 Feb 10 at 13:21

Your formalization of the problem sounds correct to me, although such formalizations depend heavily on your interpretation of the text.

Now, consider the case in which $B$ is true and $A$, $T$, $V$ and $P$ are false:

  • $T\vee B$ is true, since $B$ is true;
  • $(T\vee A)\rightarrow(P\wedge V)$ is true, since $T\vee A$ is false (since $T$ and $A$ are false);
  • $\neg P$ is true, since $P$ is false;
  • $T\vee V$ is false, since both $T$ and $V$ are false.

This shows one can have all premises true while the consequence is false, meaning that the consequence does NOT follow logically from the premises.


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